FUNCTIONAL ANALYSIS, HOLOMORPHY AND APPROXIMATION THEORY I1
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NORTH-HOLLAND MATHEMATICS STUDIES
86
Notas de Matematica (92) Editor: Leopoldo Nachbin Centro Brasileiro de Pesquisas Fisicas and Universitv of Rochester
Functional Analysis, Holomorphy and Approximation Theory I1 Proceedings of the Seminario de Analise Funcional, Holomorfia e Teoria da Aproxima@o, Universidade Federal do Rio de Janeiro, August 3-7,1981 Edited by
Guido 1. ZAPATA lnstituto de Maternatica Universidade Federal do Rio de Janeiro
1984
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Library of Congress Cataloging in Publication Data
Semindrio de An6lise Funcional, Holomorfia e Teoria da AproqimacSo (1981 : Universidade Federal do Rio de Janeiro) Functional analysis, holomorphy, and approximation theory 11. (North-Holland mathematics studies ; 66) (Notas de matemctica ; 92) 1. Functional analysis--Congresses. 2. Holomorphic functions--Congresses. 3. Approximation theory-Congresses. I. Zapata, Guido I. (Guido Ivan), 194011. Series. 111. Series: Notas de matemgtica (Amsterdam, Hollandd ; 92. QA32O.Sb56 1981 515 7 83-25454 ISBN 0-444-66645-3
.
-
PRINTED IN T H E NETHERLANDS
In Memory of
SILVIO MACHADO
Born on September 2, 1932 in Porto Alegre, RS, Brazil Died on July 28, 1981 in Rio de Janeiro
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vii
FOREWORD
This volume is the Proceedings of the Seminsrio de
Anslise
Funcional, IIolomorfia e Teoria da Aproximaszo, held at the Instituto de M a t e d t i c a , Universidade Federal do Rio de Janeiro (UFRJ) in August 3 - 7 ,
1981.
The participant mathematicians and
contributors
are from Argentina, B r a z i l , Canads, Chile, France, Hungary, Italy
,
Mexico, Spain, Rumania, United States and West Germany. "Functional Analysis, Holomorphy and Approximation
Theory"
includes papers either of a research, or of an advanced expository, nature and is addressed to mathematicians and advanced graduate students in mathelnatics.
Some of the papers could not actually
be
presented at the seminar, and are included here by invitation. The members of the Organizillg Committee
-
J.A. Barroso, M.C.
Matos, L.A. Ploraes, J. Mujica, L. Nachbin, D. Pisarlelli, J.B. Prolla and G.I. Zapata (Coordinator)
-
would like to thank the Conselho de
Ensino p a r a Graduados e Pesquisa (CEPG) of UFRJ, Conselho Nacional de Desenvolvimento Cientifico e Tecrlol6gico (CNPq), and Brasil for direct financial contribution.
We would also
I.B.M.
do
like
to
acknowledge the indirect financial contributions from Universidade Federal do Rio de Janeiro, C o o r d e n a ~ g ode Aperfeisoamento de Pessodl de Nivel Superior (CAPES), Financiadora de Estudos e ProJetos (FTNEP), as well as other universities and agencies. We are happy to thank Professor Sergio Neves Monteiro, president of CEPG of UFRJ for his personal support and understanding; Professor Paulo Emidio de Freitas Barbosa, Dean of the Centro de Ci#
Sncias Matemsticas e da Natureza (CCMN) of UFRJ, in whose facilities the seminar w a s comfortably held; and Professor Leopoldo Nachbin for
viii
FOREWORD
his constant friendly support and understanding.
We also thank Wil-
son Luiz de Gdes for a competent typing job. Finally, let us tell with emotion that one mathematician was sorely missed at the seminar, Silvio Machado, a member of
its
Organizing Committee, who passed away after a heart attack in July
28 of 1 9 8 1 , just a few days before the opening of the meeting.
The
loss caused by his death will surely be long felt by our community, in particular by his friends in which many participants of the seminar are included.
A s a posthumous homage, all of us wish to de-
dicate these Proceedings to the memory of Silvio Machado.
Guido I. Zapata
ix
TABLE OF CONTENTS
Rodrigo Arocena
Richard M. Aron and Carlos Herves
On generalized Toeplitz kernels and their relation with a paper of Adamjan, Arov and Krein
1
Weakly sequentially continuous analytic functions on a Banach space
23
Andreas Defant and Klaus Floret
The precompactness-lemma for sets of operators
39
Zeev Ditzian
On Lipschitz classes and derivative inequalities in various Banach spaces
57
A stratification associated to the copy phenomenon in the space of gauge fields
69
On the angle of dissipativity o f ordinary and partial differential operators
85
Francisco A. Doria
Hector 0. Fattorini
Two equivalent definitions of the density numbers for a plurisubharmonic function in a topological vector space
113
Chebyshev centers of compact sets with respect to Stone-Weierstrass subspaces
133
On the Fourier-Bore1 Transformation and spaces of entire functions in a normed space
139
John McGowan and Horacio Porta
On representations of distance functions in the plane
171
Reinhard Mennicken
Spectral theory for certain operators polynomials
203
Pierre Lelong
Jaros lav Mach
MBrio C. Matos
X
Miklos Mikolgs Petronije S . Milojevi?
Luiza A. Moraes
Vincenzo B. Moscatelli Jorge Mujica
TABLE OF CONTENTS Integro-differential operators and theory of summation
245
Approximation-solvability of some noncoercive nonlinear equations and semilinear problems at resonance with applications
259
Holomorphic functions on holomorphic inductive limits and on the strong duals of strict inductive limits
297
Nuclear Kothe quotients of Fr6chet spaces
31 1
A completeness criterion for inductive limits of Banach spaces
319
Peter Pflug
About the Caratheodory completeness of all Reinhardt domains 331
J o z o B. Prolla
Best simultaneous approximation
Reinaldo Salvitti
Abstract Frobenius theorem - Global formulation. Applications to Lie groups 359
Ivan Singex
Optimization by level set methods. 11: Further duality formulae in the case of essential constraints
383
Gerald0 S. de Souza
Spaces formed by special atoms I 1
413
Harald Upmeier
A holomorphic characterization * of C - algebras
427
A property of Fr6chet spaces
469
Manuel Valdivia
339
Functional Analysis, Holomorphy and Approximation Theory II, G.I. Zaputa (ed.) 0 Elsevier Science Publishers B . V. (North-Holland), 1984
1
ON GENERALIZED TOEPLITZ KERNELS AND THEIR RELATION WITH A PAPER OF ADAMJAN, AROV AND KREIN
Rodrigo Arocena
SUMMARY We consider the relation of the so called generalized Toeplitz kernels with some theorems of Adamjan, Arov and Krein, concerning the unicity of the best uniform approximation of a bounded function, canonical approximating functions and the parametrization of the approximations.
I, INTRODUCTION K
Let
be a kernel in
= K(j+l,n+l), sequence
K
ZxZ.
function in
n E Z)
(c(n):
definite, p.d.,
is said a Toeplitz kernel if
(j,n) E ZxZ,
Y
C
if
the set of integers, that is, a
2,
K(j,n) =
o r , equivalently, if there exists a
such that
K(j,n) = c(j-n ) .
-
K(j,n)s(j)s(n)
P
0
whenever
K
is positive j E Z)
{s(j):
j,n
is a sequence of finite support. int = e e,(t) measure in
T
Let
T
be the unit circle,
where
and integrals are over
T
m
is a complex Radon
unless otherwise specified.
The classical Herglotz theorem says that a Toeplitz kernel p.d. iff
K(j,n) = G(n-j),
Radon measure in
V
(j,n) E ZxZ, where
T, and, moreover,
m
m
K
is
is a positive
is unique.
In this paper we deal with the following extension of the notion of Toeplitz kernel. (1.1) DEFINITION.
K
K(j,n) = K(j+l,n+l),
is a generalized Toeplitz kernel, for every
j,n E Z-(-13.
GTK, if
R. AROCENA
2
This definition is equivalent to saying that there exist four sequences Krs, r , s = 1 , 2 , such that K(j,n) = Krs(j-n), def E Zrs = Zr X Z s , where Z1 = (n E Z: n 2 01, Z2 = 2 - 2
(j,n)
V
1’
For these kernels there is a natural extension of the Herglotz theorem.
A C
As usual,
T,
r,s = 1,2,
T; we write
measures in E Zrs.
M = (mrs),
Let
M
(mrs(A))
be a matrix of complex Radon Krs(j,k) = GrS(k-j),
K - M A if
is said to be positive,
M
2
(j,k)
Y
0, if for every
is a positive definite numerical matrix.
Then
the following theorem holds, a proof of which will be given in next section, (1.2) THEOREM.
Let
K
be a
GTK.
for some positive matrix measure
M = (mr s ) belongs to the class h(K). K
If
and
Then
T; Lp
are as in (1.2) we say that
= (f E Lp: ?(n) = 0, Y n nomials,
P+ =
P
n
H
1
,
dicate a matrix of four that
m12 = rii21.
MA
-K
holds for every
2
03;
P- = P
n H:
and
K
M
2
-
K A , then
we write
M
-
K M‘.
.
= 1,2)
0
is p.d.
iff
is Lebesgue normal-
?
m;
Y
is the Fourier
n <
01,
HP =
M = (m
rs
)
will always in-
Radon measures in
T
such
is p.d.;
if
M(fl,f2)
2
0, V(fl,f2)
E PXP;
r 0
it is easy to see that M(fl,f2)
(fl,f2) E P+ x P - ;
is weakly positive and we write M
5
M
is the set of trigonometric poly-
P
(r,s
-
Set
A s is well known
now, if
1 s p
LP(T,dt),
dt
HP = (f E LP: F(n) = 0 ,
f E L’.
transform of
=
K
(mrs).
Let us fix the following notations. ized measure in
is p.d. iff
K
in this case we say that
M > 0.
Conversely, if
M’ = (mks)
is such that
Then (1.2) implies the following.
M
5
M’
0
-
M and KA,
*
3
ON GENERALIZED TOEPLITZ m R N E L S
(1.4) THEOREM. M’
such that
If M
If
M
t 0,
there exists a positive matrix measure
- M.
M’
and
(1.4) we say that M‘
M’ are as in
is a posi-
M.
tive lifting of
In [4]
Theorems (1.2) and (1.4) were established in [9].
and [ 5 ] they were applied to uniform approximation by analytic functions and related moment problems, of the type considered by Adamjan, Arov and Krein [l]. the unicity of
M
2
0
These questions led to the problem of
such that
ditions that ensure that
m(K)
M*
-
K,
that is, of finding con-
contains only one element, which is
not always true, contrary to what happens in the classical case.
In [ 6 ] , a different and simpler proof of the generalized Herglotz theorem (1.2) was given, which can be extended to the vectorial case
[7] and leads to a better unicity condition.
That approach happens
to be much closer to the ideas of Adamjan, Arov and Krein.
In par-
ticular, it enables the consideration of the second paper [ 2 ] that those authors dedicated to Hankel operators, generalizing their principal theorems and extending their concepts and methods to obtain new results concerning generalized Toeplitz kernels.
That
is the subject of this paper.
In sections I1 and I11 we review the proofs given in [ 6 ] of the existence of elements in
m(K)
and of the unicity condition,
adding some details that will be needed in the sequel.
In sections
I V to XI some new results are presented, concerning the following subjects and their applications. a)
Conditions that ensure that the class
m(K)
contains only
one element, including an extension of a theorem of [ Z ] concerning the unicity of the best uniform approximation of a bounded function by analytic functions.
4
R. AROCENA
b)
Definition, characterization and properties of the canonical
elements of
h(K),
and, in particular, generalization of some theo-
rems on canonical functions given in [ 2 ] . c) given
f E L"
r
and
(M'2 0: M'
the class
11.
{h E H": IIf-hllm .s r ) ,
A remarkable parametrization of
with
> 0, is established in [ 21 , By means of it
- M),
M > 0, is parametrized.
with given
CONSTRUCTION O F ALL THE POSITIVE MATRIXES ASSOCIATED TO A POSITIVE DEFINITE GENERALIZED TOEPLITZ KERNEL
From now on,
K
GTK.
will always be a p.d.
Let
HK
be
the Hilbert space defined by the linear space P and the metric def = K(n,m). Set H . = the closed linear hull given by (en,em)K J
in
HK
of
{en: nfj] ,
Ven = e
isometry given by class
b(K)
space
HU
3
if
HK
j = 0,-1, and
n+l'
U
V: H-l -I Ho U
We say that
is a unitary extension of
the linear
belongs to the
V
to a Hilbert
.
U E L(K) generates a matrix def M(U) E h(K). F o r every v E HU, J(n,m) = (Unv,Umv) is a p . d . HU ordinary Toeplitz kernel; choosing successively v = eo,e,l,eo+e,l, We shall now see that every
eo+ie-l,
we get four such kernels,
Kll,
K22, F, G ,
which, by the
classical Herglotz theorem, are given respectively by the Fourier transforms of four positive measures,
mil,
m22, u, V.
Set
It is easy dm12 = e-ldm, m21 = G12. def M(U) = (mrS), r,s = 1,2 verifies:
m = 1 (u+iv-(l+i)(mll+m22)), to see that the matrix
If
fl = C a .e ., J J
f2 = C bjej
belong to
P,
the previous equalities
ON GENERALIZED TOEPLITZ KERNELS
5
z
imply that
r, s=l,2
M(U)
Analogously, a straight forward verification shows that
0.
2
K( j,k) = I?I ing
.
rs
(k-j),
(j,k) E Zrs.
Y
(11.2) PROPOSITION.
U E L(K)
Each
so
h(K)
V
gives rise, by means of for-
m(~).
M(U) E
to a matrix mulae (II.~), Obviously,
So we have proved the follow-
always admits a unitary extension to
HK@l 2 ,
is not empty, neither is, by the above proposition, m(K).
So we have proved the generalized Herglotz theorem (1.2). Now we shall consider the reciprocal of (11.2), each
M = (mrs) E h(K)
way that
= M.
M[U(M)]
U(M) E L(K)
we shall associate Since
M
2
the linear space
0,
that is, to in such a
PxP
and
the metric
(11.3) define a Hilbert space
HM
en + ( e , , O )
and
if
isometry from
n 2 0
HK
HM,
to
.
Since
HM
Then
-K
en + (O,en)
the correspondence
if
n < 0
defines an
s o we can identify the former with a
closed subspace of the latter. in
M”
Let
U(M)
be the unitary operator
given by:
U(M)
V,
extends
so
U(M) E I n ( K ) .
Moreover,
= ((en,~),(eo,~))M = ill(-n); from analogous veHU(M) rifications in the other cases, the following result is apparent.
(U(M)neo,e
)
(11.5) PROPOSITIOI~. If M E h(K), fine a unitary operator
formulae (11.3) and
U(M) E L ( K )
such that
M[U(M)I
(11.4) de= M.
Summing up, the construction presented in this section gives A
all the matrices
M
2
0
such that
M
- K.
R. AROCENA
6
111.
THE UNICITY CONDITION
and the theorem of F. and M.
h(K)
The very definition of Riesz imply the following. (111.1) LEMMA.
,
= urr
(mrs) E h(K).
Let
r = 1,2,
u12 = m12
+
Then
h dt,
( u r s ) E h(K)
iff
m rr =
+ fi dt, where
u 21 =
h E HI. Consequently the problem of unicity when
= 1
#[h(K)]
G12(n)
for every
-
-
that is, of knowing
is the problem of knowing when
n
2
0.
K
detrmines
Now, a straight forward calculation
shows that : (111.2)
If eel E H-l if
,
then
eo E Ho, then
Vn+ 1 e,l)K
(V-n-leo,e-l)K
'
= '12(n),
( eo,
=
G12(n),
\d
O;
n
2
0.
Obviously, any of the two conditions in (111.2) implies that determines
m12
and, consequently, there is only one
K
M E h(K).
We shall see, reciprocally, that if neither of those two conditions holds,
h(K)
contains more than one element.
(111.3) NOTATION.
j = -1,0
Let
and
ej
Hj;
let
v
j
be the
the condition
HK, perpendicular to Hj and well determined by def cj = (ej,vj)K > 0; let u be the orthogonal
projection of
e
unit vector in
j
j
on
H
j'
Define the linear operator Vt: HK
-t
HK,
(111.4) It is clear that
Vt E L(K)
and that every unitary extension of t
to
HK
has this type,
Set
(mrs)
= M(Vt);
and the above notation it follows that
(111.5)
from (11.2),
V
HVt = HK
ON GENERALIZED TOEPLITZ KERNELS
#
Since
if
C ~ C - 0 ~,
(111.6) THEOREM. equivalent:
a)
e it f eit'
For every h(K)
,
7
M(v~) # M(v~,),
K, GTK,
so:
the following conditions are
contains more than one element;
b)
Hj,
ej
j = -l,O.
Let us recall (see (1.1)) that every sequences
r , s = 1,2;
Krs,
KI1 = KZ2
if
plications) it is easy to see that V
metry in
,
HK
-
dist e - p H - 1 ) = dist(eo,Ho). (I11 7 ) COROLLARY.
Let
K
(as is usual in the ap-
,
= Hml
B(Ho)
IV.
so
Consequently: be a p.d.
GTK
e - l d Hml; c)
b)
,
defines an antilinear iso-
Kll = K22.
such that
Then the following conditions are equivalent: more than one element;
is given by four
(ej,ek) = (e-j-l,e-k-l)K
B(ae.) = ae j-1 J such that B(e ) = e-l
(j,k)E ZXZ; then
GTK
a)
eo
f$
h(K)
contains
Ho.
EXTENSION O F ADAMJAN, AROV AND KREIN UNICITY THEOREM We shall now
see
how the method employed in section 2 of [2],
for proving a theorem concerning the unicity of the best approximation in
L"
to some
GTK,
by function of
by considering our previous results.
T o each
Ti
,
K , GTK,
we associate two Toeplitz forms,
and one Hankel form,
(IV.l)
j
2
H',
= Kll(j-n)
T;(ej,en) if
can be used to extend that theorem
H",
0,
n < 0;
Ti
and
in the following way: if
j,n 2 0 ;
T;(ej,en)
H'(e.,e-,) J
= K22(J-n)
The method under consideration applies to
if GTK
= K12(jfn) j,n < 0 .
such that
the above forms are bounded, that is, given by bounded operators 2 2 2 2 2 2 T1: H + H , H: H -t H- , T2: H + HNow, we have the follow-
.
ing well known (see for example [lh])
property of Toeplitz operators.
8
R. AROCENA
(IV.2) PROPOSITION. P+ x P+
form in
Let
n E Z) C C
{b,:
T(e. e ) = b . J’ n J -n H2 x H2 iff 3 f E L”
given b y
to a bounded form in
v n 6 Z , and in that case
.
T
be the bilinear can be extended
such that
?(-n) = bn ’
.
IIfll,
IIT/l=
T
and
The corresponding result for Hankel operators is the following. (IV.3) NEHARI’S THEOREM 1131.
Let
(b,]”
c C
be given.
Then
n=1
there exists a bounded operator = b J.+ k
,
V
;(-n) = bn,
j
H: H2
-t
Hf
> 0, iff there exists
2
0, k
Y
n > 0. In that case
(He.,e-k) =
such that f 6 Lm
J
such that
IIHIl = dist(f,H”).
This theorem is a simple consequence of the generalized Herglotz theorem applied to the = /lH/jS(n), V n E Z,
and
GTK
given by
v n > 0.
= bn,
K12(n)
= K22(n) =
Kll(n)
(See f 5 ] ) .
From (IV.2) and (IV.3) we have the following. (IV.4) PROPOSITION.
Let
be a p.d. GTK.
K
The bilinear forms
defined by (IV.l) are given by bounded operators iff M = (w
rs
dt) > 0
When
K
and
K
- MA,
with
wrs E L“.
is as in (IV.4) the unicity condition (111.6) can
be given in terms of the operators
T19 T2, H.
As in [ 21
the proof
rests on the following. (IV.5) KREIN’S LEMMA. in a Hilbert space
Let
E;
let
A
be a bounded non-negative operator
EA
be the Hilbert space obtained by
completing the linear manifold E with respect to the metric def the (g,g’)A = (Ag,g‘). Then in order that, for any h (E E), linear functional
Fh(g)
= (g,h),
is necessary and sufficient that
h E E,
lim ((A
be continuous in
+ sI)-’h,h)
<
m,
h E A1/2E.
(1v.6) LEMMA,
K
Let
be a p.d. GTK.
Set
F . ( f ) = (f,ej) J
it
or,
E 40
equivalently, that
EA
I
?(j),
9
ON GENERALIZED TOEPLITZ KERNELS
V f E
e. @ H. J J
Then
P.
PROOF.
F
such t h a t
.
iff
F.(f) = (f,v.) J J K ' = 1 ; consequently
that
then
= 0,
?(j)
,
= C(f,vj)K
Fj(f)
(en,vj)K = 0,
#
since
n
f
-1,O.
v. E H. J J and
j,
vj €
3 c,
and t h e r e f o r e
v f E P;
V
=
j
HK 0 H . so e . @ H.. ReJ ' J J v . E HK 0 H . such t h a t (ej,v.) = J J J K ( f , ~ = ~0 ,) ~ f o r e v e r y f E P such 0
-
,
HK
implies t h a t there e x i s t s
so
(ej,v.) J K ciprocally, if there exists
= 1 = (ej,ej),
i s continuous i n
j
HK
continuous i n
J
F
F.(f) = J
c o n s t a n t , such t h a t
1 = (e.,e.) = c, J
J
lpj(f)l 5
IlvjllKllfllK,
W f E P .
(IV.7)
THEOREM.
Let
be a p . d .
K
T o e p l i t z ar.d Hankel forms ( I V . l ) T1,
T2, H.
Then
h(K)
GTK
such t h a t i t s a s s o c i a t e d
a r e g i v e n by bounded o p e r a t o r s
c o n t a i n s more t h a n one e l e m e n t i f f
the f o l -
lowing h o l d :
lim
(2)
([(T2+cI)
E-+O+
K
If
verifies also
PROOF.
L~
2 H ~ , H-
onto
then
p + , p-
Let
A
Kll
= K22,
,
#[h(~)] (ii)
Set
i s a bounded o p e r a t o r i n
> 1 lim
. ( 2 ) does.
that
and i t i s e a s y t o s e e t h a t
LL
#[h(K)]
7
to
F
A = ( T ~ + H ) ~ ++ ( T ~ + H * ) ~ -;
1
iff
e. J
H. J'
j
= -l,O,
being continuous i n
j
So from ( i ) and K r e i n l s lemma ( I V . 5 )
-1,O.
m
t h e n (1) h o l d s i f f
respectively.
and t h i s i s e q u i v a l e n t ( I V . 6 )
=
<
be t h e o r t h o g o n a l p r o j e c t i o n o p e r a t o r s o f
N o w , we know (111.6)
j
-1 e-l,e-l)
H(T1+~I)-lH*]
HK
'
we g e t t h a t
iff: ((A+eI)-'ej,ej)
<
m ,
j
= -1,O.
e+0+
Let
2 2 ( f l , f 2 ) E H xH-
-
[ ( T ~ + ~ I )H(T~+EI)
be such t h a t
-1 * H If,
= e -1
'
( A + c I ) ( f l + f 2 ) = e-l;
then
and t h i s l a s t e q u a l i t y i m p l i e s
R. AROCENA
10
j = -1.
that condition (2) is equivalent to (ii), with
The result
follows. EXAMPLE.
Let
f E L"
and
= dist(f,H");
s
hr E H"
such that
implies
('1,f
s+r
MA;
every matrix in
Let
K
-
f
) >
0,
a.e. s o
s+r,
5
m(K)
Consequently,
H".
,
so
= 1
#[h(K)]
For this k e r n e l
iff
K
f
3
s+r
f-hr) 2 0, which s+r def M = (7 s ) > 0.
(FZr
r > 0; therefore
V
has the form
h E H1, which is equivalent to
with
in 2 H-
[f-hrl
r > 0
for every
h E H"
z ( L G f-h) s
0,
/)f-hllm= s
and
([b]).
has a unique best approximation
we have
T1 = s I
in
T2 = sI
H2,
in
from (IV.7) see get the following.
(IV.8) COROLLARY.
Let
Hankel operator given by There is only one
s = dist(f,H")
f E L",
h E H"
(Hej,e-k) = z(-j-k), such that
dist(f,H")
and
+
H
be the
j 2 0,
= IIf-hllm
k > 0.
iff the
following holds: lim ([r21-H*H] ri s
-1 eo,eo) =
m.
This last result constitutes theorem (2.1) of [ 2 ] , proved there by the method the extension of which to
GTK
which is has been
presented in this section.
V.
CHARACTERIZATION OF CANONICAL MATRICES
For
K,
GTK
#[h(K)]
such that
> 1, we shall
use notation
(111.3) and also the following.
(V.2)
G12(0)
Clearly,
= g
+ P(V~,UV-~ for ) ~every ~ U E
I(V~,UV-~)~ s ~ 1, I
so
U E L(K)
L ( K ) and (mrs)=M(U).
can be chosen in such a
ON GENERALIZED TOEPLITZ KERNELS
11
takes any complex value of modulus not bigger HU That implies the following.
(vo,Uv- )
way that than one.
(V.4) DEFINITIONS.
U E L(K)
Let
K
be a p.d.
is a canonical element of
GTK L(K)
such that if
HU = HK
.
M E h(K)
U E L(K)
is a canonical matrix if there exists a canonical
> 1.
#[h(K)]
such
M = M(U).
that
From (111.4) it follows that the set of canonical elements of
L(K)
CVt: t E C 0 , 2 l l ) 3 .
is the same as the set
We have the following characterization of canonical matrices.
(v.5) THEOREM. M = (mrs)
Let
M
and
M = M(U),
be a p.d.
GTK
such that
> 1
#[m(K)]
and
The following condtions are equivalent:
E h(K).
(a)
K
is a canonical matrix;
(b)
lG12(0)-gl
U(H K ) = HK ;
then
(d)
= p ; (c) if U E L(K)
H K = HM
PROOF.
(a) implies (b):
Since
(mrs) = M(Vt)
for s o m e
t,
(b) follows
from (111.5).
In this case (V.2) says that
(b) implies (c): so
U V - ~ is parallel to
= HK
vo;
U(HK) = V(H-l)
then
I = 1, HU (CUV-~: cEC) =
I(vo,Uv- )
+
.
(c) implies (d): and
Since
(ej,O) = U(M)j(eo,O)
every
j
< 0, so
(d) implies (a):
PXP
=
vt/,
=
t = t'.
Then:
(v.6)
= HK
belong to HK
.
HK
(O,en) = U(M)n+l(O,e-l) for every
n
;r
0
and
.
Follows immediately from the definitions.
Note that, since u[M(vt)]
U(M)(HK)
SO
U[M(Vt)]
is canonical, it must be
M(v~) = M ( v ~ / ) and, by (111.5) and (111-4),
U[M(Vt)]
= Vt
,
V
t E [0,2T).
AROCENA
R.
12
VI.
SOME A U X I L I A R Y MATRICES
M = (m ) rs
Let
Radon m e a s u r e s i n
r,s = 1 , 2 ,
T;
set
dm
rs
b e a n h e r m i t e a n m a t r i x o f complex
= w rsd t
+
L = L(M)
W = (w
w i l l denote t h e matrices
vr s
where
rs
)
is
W = W(M)
From now on
s i n g u l a r w i t h r e s p e c t t o Lebesgue measure. and
,
dvrs
L = (v
and
rs
).
The f o l l o w i n g e q u a l i t y i s e v i d e n t ,
v fl,f2 So A
W 2 0
c T
and
L 2 0
M t 0.
imply
Conversely,
B C T
t h e n f o r any
M z 0,
if
v
be a Lebesgue n u l l s e t such t h a t s u p p o r t
E P.
rs
= (mrs(BnA))
( fI w rs d t ) =
(mrs[Bn(T-A)]) a r e p o s i t i v e , so L 2 0 'B ( V I . l ) a l s o shows t h a t W > 0 and L 2 0 i m p l y M > 0 .
and
M'-
l y , if the l a t t e r holds l e t def
=
W'
-
W(M')
L'
and
W
def
=
M
L(M')
= L,
so
W z 0
z 0.
and W
Converse-
M'z 0 ;
be such t h a t
r,s=1,2;
c A,
(vrs(B))
t h e numerical matrices
let
then
L 2 0.
and
Then : ( V I . 2 ) PROPOSITION. W
> 0
iff
and
W(M)
L 2 0;
-
W(b1')
The f o l l o w i n g r e l a t i o n s h o l d : b)
M 2 0
and
L(M)
W 2 0
iff
= L(M');
2
2 Il(f,g)llM = II(f,g)llW + l l ( f , g ) l l ~ 9
"
d)
and if
GTK,
ciated t o
M E h(K) W^.
W = W(M);
and
L 2 0;
M 2 0
denote by
too.
K'
The a b o v e c o n s i d e r a t i o n s show t h a t
d o e s n o t depend of t h e m a t r i x
M
c)
-
iff
M'
then
( f , g ) E pxp*
The l a s t a s s e r t i o n f o l l o w s f r o m ( V I , l ) , p.d.
M > 0
a)
M E h(K)
Let
K
t h e GTK
K'
be a asso-
i s p.d.,
used i n i t s d e f i n i t i o n ,
and, moreover:
(VI.3)
k(K) =
k(K') +
L = L(M).
L,
with
K -
MA,
K'-
WA,
W = W(M)
and
13
ON GENERALIZED TOEPLITZ KERNELS
distH (ej,Pj) = 0, L is the linear space generated by the ek with k f j;
Also, since the
P
where
j
vr s
are singular measures,
dist (ej,P.) = dist (ej,Pj). HK J HK/
then, by (VI.2d),
S o the problems of unicity and of describing all the elements
of
n(K)
can be restricted to the GTK generated by function ma-
trices
(wrS).
M = (m
)
rs
w
h
set
12); for any matrix
N(h)
In fact, if
w = (wll.w22) 1/2
set
0,
5:
111
Wo =
and
mll,m22
.
wll = w22
We can even suppose that
is hermitean and
N = (u
)
;
H 1(M) = (h€H1:M(h)zO).
rs
12
=
U
set
and any function
22
Then : (VI.4) PROPOSITION. Wo > 0
c)
and
9 h E H1(Wo)
Wo(h)
2
Wo
M > 0
0
0
2
iff
r2
0
iff
W
iff
Wo
0
so
L
2
2
3 h E H1(M);
L z 0. Now
and and
M
M
iff
t 0
L z 0;
and
0
2
a)
h E H1(Wo).
iff
Clearly
Obviously
0; b)
t
h E H1(M)
PROOF.
e
L
The following relations hold:
(b) follows from (VI.2). Wo > 0
also,
h E H1(M)
0 u h E H1(Wo)
W(h)
e
and
L
2
2 0;
L z 0
and
L
iff
0
and
so
(a) and (c)
2
0
have been proved. The consideration of canonical matrices can also be restricted to function matrices.
In fact:
(VI.5) PROPOSITION.
K
> 1. Then
#[h(K)] in
Let M
and
K’
be as in (VI.3)
is canonical in
h(K)
iff
W
and is canonical
h(K‘).
PROOF.
Since, for every 2
= dist ((fl,f2), P+xP-), HW the result follows.
(fl,f2) E PXP,
HK
c
HM
iff
2
((fl,f2>, p+Xp-) = HM F r o m (V.5) HK/ = HW dist
R. AROCENA
14
VII.
WPRESENTATION O F THE HILBERT SPACE ASSOCIATED TO A
A
POSITIVE FUNCTION MATRIX
We shall now see how a construction employed in [ 2 ] extended to give a representation of
W = (wrs) =
and
W(y)
p
= wll
2
E = L
Set
B
isometry
from
2
@ L (p dt).
Hw
to
E
E E
63 B ( H W )
2
Set
0.
~ ~ ~ =( 0t . )
I
is easy to see that a linear
is defined by
B;
it is not hard to see that 1 = 0 and w221? = 0,
3
is equivalent to F = 0, w22
dt-a.e., that is, to
M
w121 / w ~ ~ assuming , from now on
We shall now determine the range of (F,G)
, when
2
-
~ ~ ~ ( t )2/ w ~ ~= (0t )whenever
that
HM
can be
dt-a.e. and
G
= 0, p dt-a.e.;
consequently I
( VII .2 )
B ( H ~ )=
x r w22>o~L2
L2(p dt).
Moreover: (VII.3) THEOREM.
P =
-
Wl1
a)
Iw121
Let
M
2
0, W = (wrs) = W(M),
2
X1w22,0~/W22
Xr ~ ~ ~ ’L20€91 L2(p
HW w
.
L = L(M),
Then the following hold.
dt),
where the isometric isomorphism
is given by (VII.l). X(~22>0] L2 €9 L2(p dt) @ HL ,
‘M
b,
If, also,
c) 2
K
-
M A is such that
#[h(K)]
> 1, then
2
HW = L €9 L (P dt)
(a) has been proved already; (b) follows from (a) and (VI.2); as to (c), it stems from:
(VII.4) LEMMA.
Let
M > 0, (wrs) = W(M),
w = (wll.w22)
3.
If M
15
ON GENERALIZED TOEPLITZ KERNELS
has more than one positive lifting, then log w E L 1
or, equivalently,
log wll
log w22 E L1
and
.
This lemma is an obvious consequence of (VI.5) and the following. (VII.5) PROPOSITION.
Let
w
2
0
be an integrable and not trivial
function.
Then the following conditions are equivalent: (i) There w f exists a function f such that ) is weakly positive and has (7 w more than one positive lifting; (ii) l o g w E L1
.
PROOF.
If
(i) implies (ii): such that 2w
w
If+hll
2
and
hl
and
w z
h2
H1
are distinct elements of
If+h21, log w E L1
because
Ihl-h21.
2
3 h E H1(T) such that Ihl = w w h () 2 0; take f 0. h w
(ii) implies (i): not trivial and
VIII.
h
tion equivalent to
(VIII.~)THEOREM.
-
Let
2
we can obtain a condi-
HM
having only one positive lifting.
M 2 0
M 2 0,
I
(wrS) = w ( M ) ,
w = ( w ~ ~ . w ~and ~ )
The following properties are equivalent.
Iw121 /w.
a)
There exists a positive matrix
b)
log w E L1
M'
f M
such that
M'
-
M.
and at least one of the following conditions
holds : i)
log p E L1;
ii)
1
belonging to
H
there exist
2 1 Tl L (w dt)
hl E H1
and a not trivial
IWhl
such that w(lwl
belongs to p(t)
is
ANOTHER FORMULATION OF THE UNICITY CONDITION From the above representation of
p = w
a.e., s o
= 0.
L1
and that
w(t)hl(t)
-
-
2
Wl2h2I
-Iw121
w12(t)h2(t)
2
= 0
2
1
XEP>Ol whenever
h2
16
R.
AROCENA
w
Because o f ( V I . 4 )
PROOF.
Suppose ( a ) h o l d s .
HM
L
N
2
2 @ L (p dt).
trivial
log w E
Then
S i n c e u n i c i t y d o e s n o t h o l d , t h e r e e x i s t s a not
(F,G) E [L2 @ L2(p d t ) ] FW12/w
If
F = 0
-
l o g [ IGI 'p]
a.e.,
+
.
1
and
>
0,
so
3=
Fw
h2,
l o g w 2 log p = loglhll
5
w 12;
G E
since
L2(p d t ) ,
t h e l a s t equal-
( i i ) . So ( a ) i m p l i e s ( b ) .
Now assume t h a t ( b ) h o l d s .
If
H1
function i n
then
(*), b u t n o t t o
not hold.
If
( i i )h o l d s ,
f
h;
= enh2,
set
hl,
h2
r e p a l c e d by
to
E E) B ( H o ) ;
so
e
&'
(VIII.2)
Ho.
r2
so
hl
= en h' l
F = h;/wl/',
so
hi, hi,
,
let
hl
a.e.,
(F,G)
G = lil/p;
B(Ho)
orthogonal t o
E,
eo
be an o u t-
and s e t
q
with
Ho
and u n i c i t y does
h i E H1
such t h a t
G
and
g;(O)f
(*) holds with
i s n o t t r i v i a l and belongs
{ (F,G),B(eo,O))E =
Let
r = ) ) f - h lm )
.
r = dist(f,fl)
E Lm,
f
Then s u c h
h
the following conditions holds:
(f-h)G21 lf-hI2
1
Gi(0)
f
0 and
The r e s u l t f o l l o w s .
hl E
(ii) there e x i s t
Ihl
let
L
lhll
B(eo,O),
on t h e o t h e r h a n d
COROLLARY.
b e such t h a t one of
p =
such t h a t
because of
0;
log p E
i s a n o t t r i v i a l e l e m e n t of
(0,G)
-
2
(*) i t a l s o f o l -
i t y e n s u r e s t h e v a l i d i t y of
er
.
1
is n o t t r i v i a l , ( * ) shows t h a t
F E L2
-
Gpw = h w 1
hl,h2 E H
with
and i s n o t t r i v i a l ; m o r e o v e r , f r o m
(G d t )
lows t h a t
El
If
which i s e q u i v a l e n t t o
0 B(Ho),
Gp =
/Ihllll
E L
2 1
h2 E L
$
12). w21 and, consequently,
(VII.4)
L1
w
M = (
we may assume t h a t
H1
7 0
and
h E H"
i s n o t unique i f f a t l e a s t
1
E L ;
( i ) log[r-lf-hl]
and a n o t t r i v i a l
h2 E H2
such t h a t
2
-
'[r2-If-hl
>0)
E L1
and t h a t
- - -h
h1 =2
whenever
r = If-hi. (VIII.3)
COROLLARY.
Let
There e x i s t s a not t r i v i a l f
= ug/g,
with
u
f
be unimodular and d i s t ( f , H m )
= 1.
IIf-hll m = 1
iff
h E H"
i n t e r i o r and
such t h a t g E H2
and o u t e r .
ON GENERALIZED TOEPLITZ KERNELS
REMARK.
Clearly,
If
EXAMPLE. h E H1
in (VIII.l)
= w,
wrs
such t h a t
may b e assumed t o b e o u t e r .
r , s = 1,2,
w 2
Iw-hl
2 1 k E L (F d t ) ,
such t h a t
h2
17
there exists a not t r i v i a l
iff t h e r e e x i s t s a c o n s t a n t kf0
a.e.
1
;SE
that is, iff
a p p e a r s i n a remarkable theorem of P.
L1.
This condition
Koosis [ 1 2 ] ,
concerning
w e i g h t e d q u a d r a t i c means o f H i l b e r t t r a n s f o r m s ,
t h a t can be obtain-
ed by d e v e l o p i n g t h e a b o v e s k e t c h e d r e a s o n e m e n t
[6].
SOME PROPERTIES OF C A N O N I C A L MATRICES
IX.
Let W = W(M).
So,
M
Then
considering
(VII.l),
s o t h e same h a p p e n s w i t h
be a canonical matrix,
B(Ho)
HW
=
Ho
L
2
2
3 L (p d t )
t
HW
a s a subspace of
p = wll
with
E,
and w i t h
E.
h a s c o d i m e n s i o n one i n
Now,
B
2
= Iw121
/wz.
as in
(F,G) E E 0 B(Ho)
iff
-1 2 Fw12/w22
( I X . 1)
+
A C T,
such t h a t
real in
1 , w = ~h 2 ~
7
a l 2 w 11
2
p 2 a2
Then
i
IG1I2P
Moreover,
F1 E L2
2i,
d t L Ilhll,
IGI2p d t
[3
(F1,G1)
s o t h e dimension of
must b e
p = 0 THEOREM.
Let
.
1
Then t h e r e e x i s t s al,
A.
Proceeding as i n
h E Ha
be such t h a t
(T-A)
+
8
and
:l
k
i s not p a r a l l e l t o
a.e.,
E H
G1
h
a2,
'1-
1
= (LK1-FhGl2/w 2 2) -P
and
trivial,
M = (mrs)
in
hl,h2
and p o s i t i v e c o n s t a n t s
holds i n
F1 = hF
Set
with
does n o t h o ld a . e .
of p o s i t i v e measure,
(T-A).
A.
(IX.2)
F
3 o f [ 2 ] , l e t a non c o n s t a n t
section
in
cl,
p = 0
Suppose t h a t a set
Gp =
E
(3
B(Ho)
I F I 2 d t l , s o (F1,G1)EE
(F,G)
if
0 B(Ho).
the l a s t i s not
would n o t b e o n e ; t h e n i t
that is: K
be a p.d.
a c a n o n i c a l element o f
GTK m(K)
such t h a t and
#[h(K)]
dmrs = w r s d t
> 1,
+
dvrs,
18
R.
with
vrs
wllw22
-
s i n g u l a r w i t h r e s p e c t t o Lebesgue measure.
w12w2, =
F E L~ Q B ( H ~ ) i f f
h
-
-
i n t h e above h y p o t h e s i s
F
;
~
hl = uh,
Set
Then
a.e.
0
Consequently
h 2 E H1
AROCENA
3
~
~
=/
with
G1 W
u
F u G ~ ~ = / w 6,~ ~F u 2Fw =~ u~h 2 ;
~
and ~
F
HW x L
&w = ~h 2~'
i n t e r i o r and
h
2
and
with
hl
outer,
and
so
t h e n t h e same argument c o n c e r n i n g
u
t h e codimension employed i n t h e l a s t p r o o f shows t h a t
1
I
and
leads t o the following.
(IX.3)
THEOREM.
outer
I n t h e same c o n d i t i o n s o f (IX.2) t h e r e e x i s t two
H
functions i n
W
1
hl
and
h2'
such t h a t
W
1 2 - hl -W
-- -w 11 '
22
fi2
lhll
=
21
Note t h a t i f , moreover,
s o there e x i s t s an
w 1 2/ W
outer
= w22 = w ,
wll
h E H1
function
n
then
APPLICATIONS
[a].
=
RELATED TO THE H E L S O N - S Z E G ~ THEOREM
W = (w
Let
wI1
= w
t
rs
)
be an h e r m i t e a n f u n c t i o n m a t r i x w i t h
the kernel
0,
+
s = inf(;(O)
(x.1)
C
-
A
W
+
'
cjEkG12(k-j)
+
2 Re
2 Re
X
It i s not hard t o see t h a t
such c a s e
K
cjG(-j)
t
cjG12(-j))'
(c,}
i s p.d. iff
of f i n i t e j€
s
2
0;
in
s = d i s t ( eoyHo). Consequently:
(1) The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t : with
o,j c
g i v e n by
j
where t h e infimum i s c o n s i d e r e d on t h e sequences support.
s
and t h e number
c.c' ( k - j )
k . j >O
C
K
j .k
a.e.
I
= h/c. The l a s t theorems e x t e n d t h o s e of s e c t i o n 3 i n
x.
h2
such t h a t
L2($ d t )
h E H1;
K
i s p.d.;
( 2 ) T h e r e e x i s t s o n l y one
W > 0; w 2
Iw12-hl,
s 2 0.
h E H1
such t h a t
w 2
Iw12-hl
a.e.
ON GENERALIZED TOEPLITZ KERNELS
o
s =
iff
(see
(111.7)).
s > 0,
(3) I f
3 h
E H1
form i n
L
,
g a t i v e and i n t e g r a b l e f u n c t i o n and k-1 w = -w , k +1
with
H
Let
> 1
k
w12
= w.
let
If(
k (T
2
w dt,
k- 1 k+l w
that
V
Iw-hI
2
E P;
f
(5) k-1
k+lw
(i)
a.e.
iff
Iw-hl
a.e.
When
Rk;
let
that
w
w E R
2 h E H1
[Rk
: k
w = e
iff
i t was shown t h a t ( 4 . i i i )
w E
IIvllm <
artg
mk
,
-
3’
v,
that i s ,
3
-
S o the deduction t h a t
(4.iii). (X.2)
c
a.e.
,
IIvIIoo< n / 2
(and
? = Hv + i c ( 0 ) ) .
a positive constant,
k+ 1 ( u ( 4 arch(v)
E Rk
(W-h(
7
I n [8]
implies t h e following refinement:
2k
2k Now, i f
k-1 k+l w =
u , v E L:
with
w = c e u+v ,
iff
k-l
such t h a t
such t h a t
> 1). The Helson-Szeg8 theorem [ll] s a y s U+G
t h e harmonic c o n j u g a t e o f
(7)
h E H1
( h i ) , we s a y t h a t i t b e l o n g s t o t h e s e t
verifies
u
R =
W t 0
i t stems from ( 2 ) and ( 3 ) ,
t h e r e e x i s t s o n l y one
sk > 0 ,
If
( i )with
that :
sk = 0 2
(6)
,
Set sk = s ,
IHf12w d t L T ( i i i ) 3 h E H1 such
( i i ) sk 2 0;
(1). With t h e same n o t a t i o n ,
r e spec t i v e l y
b e a non ne-
Then:
T h i s r e s u l t f o l l o w s ( 1 9 1 ) from t h e e q u i v a l e n c e o f and o f
w
a constant.
( 4 ) The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t : 5
(IX.2).
a.e.
be t h e H i l b e r t t r a n s -
Hen = ( - i ) ( s g n ) e n ;
g i v e n by
g i v e n by ( X . l )
w = I w12-hl,
such t h a t
L e t u s c o n s i d e r a n example.
2
19
and
3
k’ > k ,
leads t o
u,v
L,:
a.e.
i t follows t h a t
sk, > 0 .
( 7 ) can s t a r t o f ( 6 ) i n s t e a d of
Thus:
THEORGM.
(fI2
w dt
0 5 w 5 L1
i s such t h a t
f o r a fixed constant
IHfI2W
k > 1
dt 5
and e v e r y
f
E
P iff
R.
20
w = ce
U+i;
,
where
i s a positive constant,
c
-, ( u I 2 k' 3
< a r t g k' -1
AROCENA
= arch(- k' + 1 v ) 2k'
a.e.,
3
of extremal measures g i v e n i n [ S ]
XI.
EXTENSION OF ADAMJAN,
such t h a t
4
I n section
E
(XI.1)
Let
rifies
IIf-ul10 s 1.
f
ix
L"
g E H"
I$
E H1,
K
and
w = ( w ll.w22)
and
outer, 0
lw12+hol,
2
T'hon
h
E
so
H1(M)
11
u E Hm v e -
F E HI,
-
+. )
1B
and w i t h
:
- gcP
assume
w = /I$[
E L1
a.e.
and f i x ;
I/ (w,,+h)/$ll,
ip
#[m(K)]
log w
Then
01
the following holds:
(l-f)(l-q)
1 (wl2+ho)/$l
1 z iff
< 1,
121
MA;
5
outer
i s g i v e n by
= {F
such t h a t h dt
= ( h E H1 : M + ( F i d t O
w
a.e.
Then t h e r e e x i s t s
((f-u(/m S
M > 0
1W-h)
= 0
i f f sk
b e s u c h t h a t more t h a n one f u n c t i o n
- z' ( F ( e i x ) ( d x ,
E Hm :
= W(M)
Bk
t h e followi n g remarkable r e s u l t i s s t a t e d .
+
eix
Let
a constant.
AROV AND K R E I N PARAMETRIZATION
121
of
such t h a t , i f
i,
IIvllm <
which i m p l i e s :
k-1 k+l w =
3 h E H1
2n
,
L:
z k,
k'
b e l o n g s t o a n e x t r e m a l r a y of t h e cone
UI
and
=&
with
E
the equality i n (6) i s related t o the characterization
Also,
(8)
u,v
> 1.
Set
E
+
13, =
(wrs)
so there exists 1 H (M) H1(M);
= then
= (w12+ho)/$,
f
= IIf
I\'p\\, 5
Set ho
set
E Ha,
f€Lm,
so
h-ho
7 1 1 0 . Let
h-ho
; s i n c e f E Lm, ( h - h o ) E H1 and $ is I$ follows t h a t u E H". C o n s e q u e n t l y h E H1(M) iff
u=-
with
u
E
H"
and
(XI.2) THEOREM. matrix,
jlf-ullm Let
S
M = (m
such t h a t , i f K
-
1.
rs
Then ( X I . l )
)
h(K)
h = ho
-
$u,
implies the following.
r,s = 1,2,
MA, then
it
outer,
b e a weakly p o s i t i v e
c o n t a i n s more t h a n
ON GENERALIZED TOEPLITZ KERNELS
one element.
IIF/(l = 1,
ho,#,F E H
There exist
such that, if
g E €1-
f 2=
then
such that
llqil,
5:
1
,
$
and
F
outer functions,
is given by
.
..
is a positive lifting of
M'
1
21
M
cp 5 H"
iff there exists
and
\
m
22
REFERENCES 1.
V.M. ADAMJAN, D.Z. AROV and M.G. KREIN,
Infinite Hankel ma-
trices and generalized Caratheodory-Fejer and Riesz problems, Funct. Anal. Appl. 2, n o . 1 (1968), 1-19. 2.
v.M. ADAMJAN, D . Z . AROV and M.G.
KREIN,
Infinite Hankel ma-
trices and generalized Caratheodory-Fejer and I. Schur problems,
3.
Funct. Anal. Appl. 2, no. 4 (1968), 1-17,
R. AROCENA, M. COTLAR and C. SADOSKY,
Weighted inequalities in
L2 and lifting properties, Adv. Math. , Supplementary studies, v. 74 (1981), 95-128.
4. R. AROCENA and M. COTLAR, On a lifting theorem and its relation to some approximation problems,
North Holland Mathe-
matical Studies 71 (1982), J. Barroso Ed.
5.
R. AROCENA and M. COTLAR,
Generalized Toeplitz kernels and
Adamjan-Arov-Krein moment problems,
Operator Theory:
Advances and Applications, Vol. 4 (1982), 37-55.
6.
R. AROCENA and M. COTLAR,
Generalized Herglotz-Bochner theorem
and L2-weighted problems with finite measures ,
Proc. Con-
ference in honour of Prof. A. Zygmund, Chicago, (1981), 258-269.
R. AROCENA
22
7. R. AROCENA and M. COTLAR, Dilation of generalized Toeplitz kernels and some vectorial moment and weighted problems, Springer L. N. in Math., 908 (1982), 169-188.
8.
R. AROCENA,
A refinement of the Helson-Szegb theorem and de-
termination of the extremal measures,
Studia Math. 71, 2
(1981), 203-221.
9. M. COTLAR and
C.
SADOSKY,
On the Helson-Szeg6 theorem and a
related class of modified Toeplitz kernels,
Proc. Symp.
Pure Math. AMS 35: I (1979), 383-407. Bounded analytic functions, Academic Press, 1981.
10.
J. GARNETT,
11.
H. HELSON and G . SZEG6, A problem in prediction theory, Mat. Pura Appl. 51 (1960)~107-138.
12.
P. KOOSIS, Moyennes quadratiques pondere'es, C.R. Acad. Sc. Paris 291 (1980), 255-256.
13*
2.
NEHARI, On bounded bilinear forms,
Ann.
Ann. of Math. 65 (1957),
153-162.
14. V.V. PELLER and S.V. HRUSCEV,
Hankel operators, Best approxi-
mations and stationary Gaussian processes,
LOMI preprint
E-4-81 Leningrad 1981.
ADDED IN PROOF.
A self contained proof of the parametrization theorem (XI.2) can be given. (R. Arocena, "On the parametrization of Adamjan, Arov and Krein", Publications Mathe'matiques dc0rsay).
Departamento de Matemgticas Universidad Central de Venezuela Mailing address: Apartado Postal 47380 Caracas 1041-A Venezuela
in
Functionul Analysis, Holomorphy and Approximation Theory Il, G I . Zapata (ed.) 0 Ekevier Science Publishers B . V. (North-Holland), 1984
23
WEAKLY SEQUENTIALLY CONTINUOUS ANALYTIC FUNCTIONS
ON A BANACH SPACE
Richard M. Aron
and
Carlos Herves
ABSTRACT Let
Hwsc(E) be the space of complex valued analytic func-
tions on the complex Banach space
H ~ ~ ~ ( contains, E ) in general
sequences to convergent sequences. properly, the space
Hw(E)
which map weakly convergent
E
of analytic functions on
weakly continuous on bounded subsets of properly contained in the space
of analytic functions which
are bounded on weakly compact subsets of
E.
closely the relation between these spaces. that for many Banach spaces
E,
if and only if its differential
a function df
which are
and is, in general,
E
Hd(E)
E
Here we study more
We show, for example, f
belongs to Hwsc(E)
belongs to the &-product
Hd(E) E E'.
INTRODUCTION F o r a complex Banach space following subspaces of the space Hd(E) = {f E H(E):
of
f
E,
H(E)
we will be interested in the of entire functions on
E.
is bounded on each weakly compact subset
E)
Hwsc(E) = {f E H(E):
f
takes weakly convergent sequences in
to convergent sequences)
Hw(E) = {f E H(E):
f
is weakly continuous when restricted to
any bounded subset of
El
E
24
RICHARD M.
Hwu(E)
= ( f E H(E):
ARON and C A R L O S HERVES
i s u n i f o r m l y w e a k l y c o n t i n u o u s when
f
r e s t r i c t e d t o any bounded s u b s e t o f
= [ f E H(E):
Hb(E)
The i n t e r s e c t i o n
pwu(%)
=
Pw(%)
i s bounded
f
n
P(%)
i s bounded
f i Hw(E)
i s a member of
on bounded s u b s e t s of
reasonable t o ask whether, E
i s denoted
Hwu(E) C Hb(E)
t h a t for r e f l e x i v e Banach s p a c e s
implies t h a t
liwsc(E)
pwsc(%;
and t h a t
[&I).
Hwu(E)
=
I t i s known ( c f . Lemma 3 . 1 o f [ 4 ] )
c H u ( E ) c Hwsc(E) c H d ( E ) .
f
on bounded s u b s e t s o f E ) .
i s d e f i n e d s i m i l a r l y ( c f . Theorem 2 . 9 of
It i s easy t o see t h a t
that a function
E)
E,
E.
i f and o n l y i f
Hwu(E)
Moreover,
it is trivial
H w ( E ) = Hw,(E).
It i s
c o n v e r s e l y , t h e e q u a l i t y of t h e s e s p a c e s
I n f a c t , i t m i g h t seem r e a s o n a b l e
i s reflexive.
t o hope t h a t t h e s e c o n d i t i o n s a r e e q u i v a l e n t ,
i n l i g h t of the f a c t
t h a t t h e c o r r e s p o n d i n g r e s u l t h o l d s f o r c o n t i n u o u s f u n c t i o n s by a t h e o r e m o f V a l d i v i a [ 2 3 ] and f o r d i f f e r e n t i a b l e f u n c t i o n s by a Such a hope w a s b r u t a l l y d a s h e d r e c e n t l y by
r e s u l t of G i l h 2 ] .
D i n e e n [ 7 ] , who showed t h a t e v e r y f u n c t i o n i n on b a l l s o f
co
Hw(co) = Hwu(co).
and t h u s
somewhat s p e c i a l , h o w e v e r ,
Hd(E)
we do n o t know i f
and
are a l l different.
i s always e q u a l t o
H e r e we c o n c e n t r a t e p r i m a r i l y on w i t h e i t h e r t h e weakly-compact
n e r a l headings.
is barreled. ‘rod
E = c
is
i n t h a t a l l of t h e above d e f i n e d s p a c e s
Hw(E)
Hw(E)
weak-ported topology
The c a s e
i s bounded
( I t i s e a s y t o s e e t h a t , for g e n e r a l
coincide i n t h i s case.
Hwsc(E),
Hw(co)
T iLld.
O n t h e o t h e r hand,
Hwu(E)
Hwsc(E),
open topology
1.
w h i c h we endow
‘rod
or t h e N a c h b i n
Our d i s c u s s i o n f a l l s u n d e r t h r e e ge-
F i r s t , we examine t h e q u e s t i o n o f w h e t h e r
I n f a c t , we show t h a t
t o p o l o g y i f and o n l y i f
s p e c i a l s i t u a t i o n s i n which
E,
E
Hwsc(E)
i s reflexive.
Hwsc(E)
i s barreled with the W e a l s o s t u d y some
( H w s c ( E ) , T W d ) i s , a n d i s n o t , barreled.
25
WEAKLY SEQUENTIALLY CONTINUOUS ANALYTIC FUNCTIONS
Next, we consider the question of completeness of and
Hw(E),
Hwsc(E)
9
Hd(E)
and approximation by finite-type holomorphic functions.
Here we make use of the recently defined concept of bounded-weak approximation property due to Gil and Llavona. known that a function belongs to
Finally, it is
if and only if its res-
Hwsc(E)
E
triction to each weakly compact subset of
is weakly continuous.
This observation permits us to obtain a useful characterization of Hwsc(E)
(for many spaces
E) in terms of
Hd(E),
ogous to the one given by the first author r2] of of
Hb(E).
which is analHwu(E)
in terms
We conclude the paper with some specific observations
and questions concerning two special Banach spaces, the James quasireflexive space
J
and the Cartesian product
coxT
T
where
is
Tsirelson's original space. Throughout much of o u r discussion, we will be assuming that our space E
= L1
E
does not contain a copy of
will be treated separately.
L,,
and the special case
It would be very interesting
to know the relationship of the assumption
E $ C1
to the results
we obtain here. Our notation and terminology will follow the standard works on the subject by Nachbin [ 171 and Dineen [ 61. The second author acknowledges with thanks the support he received as a postdoctoral fellow at University College Dublin, while on a grant from the Irish Department of Education.
ON THE SPACES .Hd(E) AND
We will endow the topology of
E
rod
Hd(E)
Hwsc(E)
and its subspace
Hwsc(E)
with either
of uniform convergence on weakly compact subsets
or the Nachbin weak-ported topology
t Ulda This locally
convex topology is generated by all seminorms
p
which there is some associated weakly compact set
on
Hd(E)
K C E
for
satisfying
26
RICIIARD M. ARON and CARLOS NERVES
the following condition: For every
>
there is a constant
0
f E Hd(E),
all The
E
p(f)
I;
> 0
C(E)
such that f o r
c(c)-sup{IIf(x)// : dist(x,K) < e } .
topology has been studied by Paredes [18], who showed
Tud
that it is generated by all seminorms of the f o r m m
f = CPn E Hd(E),
for all
K c E
where
is weakly compact and
is a sequence of non-negative real numbers converging to
(an)
0.
One of the main reasons for o u r interest in
Hwsc(E) lies in the following result, which suggests that some properties of Hwu(E)
Hwsc(E).
may have analogues which hold for
PROPOSITION 1
(Proposition 3.3, [ b ] ) . Hwsc(E)
analytic functions on compact subset of
E
is the space of
which are weakly continuous on each weakly
E.
This result has been recently extended by Ferrera, Gil and Llavona [ll] to the case of continuous functions. Another reason for o u r interest in these spaces is that f o r special choices of portance.
HwU(E)
E,
Hwsc(E)
and
Thus, for example, if
where
7
=
7
Od
Or
'u d
E and
is reflexive then (HwSc(E),T)=
Hwu(E)
uniform convergence on bounded subsets of flexive
E and
E,
= Hb(E)
(Hd(E),T)
is a Schur space, then (Hwsc(E),TWd)
where
are of particular im-
Hd(E)
T
has the topology of
E.
= 7 Od
Further, for reOr
'
Wd.
Also, if
(Hwsc(E),~Od)= (H,(E),T~~) = (H(E),T~)
= (Hd(E),7Wd)
= (H(E),T~), where
' 0
and
'
UJ
are, respectively, the compact-open and the Nachbin ported topologies. Note that if Hwsc(E)
and
barreled [ 8 ] .
Hd(E)
E
is reflexive, then it is trivial that both
are barreled, while
On the other hand, neither
(Hd(C1),TWd) (Hd(Cm),TWd)
i s also nor
27
WEAKLY SEQUENTIALLY CONTINUOUS ANALYTIC FUNCTIONS
(Hwsc(Cm),Tud)
i s barreled.
co,
c l o s e d u n i t b a l l of ( f E Hd(Lm)
Thus,
recall first that the
To s e e t h i s ,
B(co),
: s u p { l l f ( x ) ) ) ,x E
B(co)}
( H d ( L m ) , ~ W d ) . If t h i s s p a c e w e r e b a r r e l e d , c o n v e x w e a k l y compact s e t E
>
there is
0,
C(E)
I ! r l l B ( c o ) = sup(llf'(X)l/:
In particular, for a l l
>
nth
C(C
,
T
n
r o o t s and l e t t i n g
~
~
and f o r a l l
n,
) . s ~ p r I i c p n ( x ) i i: d i s t ( x , ~ )
t a i n e d i n t h e w e a k l y compact s e t
(Hd(Lm
E Hd(Lm)
f
c ( ~ ) ' s u p { I / f ( x ) l l :d i s t ( x , K ) < E } .
cp E C 1 C C L
c
t h e n f o r some a b s o l u t e l y
we would h a v e t h a t for a l l
such t h a t i f
0
xEB(co)}
ll~nllB(co) Taking
Lm,
in
K
is a barrel in
1)
S
i s) n ' t b a r r e l e d .
1161.
H(Lm)
i s b o u n d i n g for
-t
we s e e t h a t
m,
K.
d.
B(co)
i s con-
T h i s c o n t r a d i c t i o n shows t h a t
Hwsc(Lm)
The p r o o f f o r
T h u s f a r our d i s c u s s i o n on b a r r e l e d n e s s o f
i s identical.
Hd(E)
h a s c l o s e l y p a r a l l e l e d t h e analogous s i t u a t i o n f o r
(H(E)
a n d Hwsc(E) An
,TW).
i n t e r e s t i n g d i f f e r e n c e f o l l o w s from t h e f o l l o w i n g simple p r o p o s i t i o n which a l s o h o l d s for Ferrera
reflexive.
(1).
(2).
reflexive or
A = A
(Compare t h i s r e s u l t w i t h
r 91 ) .
PROPOSITION 2 .
PROOF.
Hwsc(E).
(Hd(E)
If
,TOd)
(Hd(E),~wd) i s b a r r e l e d ,
i s convex, b a l a n c e d and a b s o r b i n g .
{ZX
: z E
C,
Therefore
A
i s barreled.
(Hd(E),TOd)
: IIdf(O)ll = s u p ( I d f ( O ) ( x )
( f E Hd(E)
'rod,
then e i t h e r
is
E
is
H ~ ( E )f H ~ ( E ) .
( 1 ) . Suppose t h a t
for
E
i s b a r r e l e d i f and o n l y i f
then f o r a l l unit vectors
IzI
=
13.
Thus,
is a barrel i n
I,
IIxIl
Also, x,
df(O)(x)
f
a
S
if
-+
f
Let
13
5
1.
fa
E
A
Clearly and
f
a -'f
u n i f o r m l y on
=
(Hd(E),~Od)
and t h u s t h e r e i s a n
28
ARON a n d CARLOS HERVES
RICHARD M.
K c E
a b s o l u t e l y convex w e a k l y compact s e t such t h a t
A 3 (f
E
Hd(E)
: IIfllK
i s contained i n c - I K ,
of E , B ( E ) ,
and a c o n s t a n t
E > 0
I t f o l l o w s t h a t t h e u n i t ball
< c}.
and so E is r e f l e x i v e . The c o n v e r s e i s
trivial.
(2).
Assume t h a t
= Hd(E).
IIb(E)
Since
is
(Hd(E),Twd)
b a r r e l e d , a n a p p l i c a t i o n o f t h e open mapping t h e o r e m ( s e e , f o r e x a m p l e , p a g e 299 of [ H b ( E ) -+
(Hd(E),TWd)
141)
shows t h a t t h e i d e n t i t y mapping
i s a t o p o l o g i c a l isomorphism.
a b o v e , we c o n c l u d e t h a t t h e u n i t b a l l o f a weakly compact s u b s e t o f
must b e c o n t a i n e d i n
E
and t h e r e f o r e
E
Arguing a s
must b e r e f l e x i v e .
E
..
Q .E D S i n c e D i n e e n h a s shown t h a t following
Hd(co) = H b ( c o ) ,
we h a v e t h e
. 3.
COROLLARY
(Hd(co),TWd) i s not b a r r e l e d ,
We t u r n now t o a r e v i e w o f
H w ( E ) , Hwsc(E),
t h e spaces
and
t h e q u e s t i o n o f c o m p l e t e n e s s of The b a s i c known r e s u l t s
Hd(E).
a r e summarized i n t h e f o l l o w i n g p r o p o s i t i o n .
L1
The
case i s
discussed l a t e r . PROPOSITION 4 .
i s reflexive or
E
provided a copy of
(1). H w ( E )
E
T~~
and
7
Wd'
i s s e p a r a b l e and d o e s n o t c o n t a i n
Ll.
(2).
Hwsc(E)
(3).
Hd(E)
PROOF.
i s complete f o r b o t h
(1).
i n t h i s case
i s complete f o r b o t h i s complete f o r b o t h
If
E
i s reflexive,
Hw(E) = Hwu(E)
TOd TOd
and and
T 7
wd'
wd'
then the r e s u l t i s t r i v i a l since
and t h e t o p o l o g i e s
TOd
and
T
wd
a r e b o t h t h e t o p o l o g y o f u n i f o r m c o n v e r g e n c e o n bounded s u b s e t s o f
E.
If
i s s e p a r a b l e and d o e s n o t c o n t a i n a copy o f
E
Proposition proof
of
3.3 of [ b ] ,
Hwsc(E)
= Hw(E).
(I), i t s u f f i c e s t o p r o v e ( 2 ) .
4,
t h e n by
Thus t o c o m p l e t e t h e
WEAKLY SEQUI?NTIALLY C O N T I N U O U S A N A L Y T I C FUNCTIONS
Let
(2).
E E H(E)
a function
such t h a t
f
a
argument u s i n g t h e c o m p l e t e n e s s of E lIwsc(E).
To see t h i s ,
Since t h e s e t
K = (x} U
f
for
The p r o o f
for the
T
let
u n
and hence f o r
TWd
Hwsc(E).
be a TWd-Cauchy n e t i n
(fa)
T
f
-t
by a r o u t i n e
TWd,
We c l a i m t h a t
(€I(E) , T ~ ) .
c o n v e r g e weakly t o
(x,)
{x,}
for
Then t h e r e i s
(? .I?.
which i s due t o P a r e d e s r 1 8 1 .
The a s s u m p t i o n i n P r o p o s i t i o n
f
-t
a
f ( x n ) -+
c a s e i s o m i t t e d , as i s t h e p r o o f
Od
E.
in
i s weakly compact and s i n c e f
i t i s easy t o see t h a t
Od,
x
f(x).
(3)
of
u. 4 . 1 seems t o be n e c e s s a r y , a s
t h e f o l l o w i n g example shows. EXAMPLE
5.
Let
n xn E
f ( x ) = Cn
H(.Ll)
x = (x,)
where
a',
an e n t i r e f u n c t i o n converge t o t h e f u n c t i o n f o r
IIowever,
f
$?
€IW(L1).
and s u p p o s e t h a t t h e r e i s
Lm
T
W
.
Thus
s i n c e t h e f i n i t e Taylor s e r i e s terms of
IIw,(L1).
such t h a t i f
( f ( x ) ( L 1.
Indeed,
> 0
E
x E D4
i = l , . .. , k .
i
\vj
-
Taking
i
j,
lf(x)
1
I
0
as
j
and
j'
j , j'
,
1
then
we g e t t h a t
I
< c
x
,
E R4
(1sil:k)
can be a r b i t r a r i l y l a r g e .
There a r e two i n t e r e s t i n g
observations
l a t i o n s h i p o f t h e c o m p l e t e n e s s of
Hw(E)
concerning the r e -
t o whether
contains
E
F i r s t , we do n o t know w h e t h e r t h e above argument c a n ed t o t h e c a s e o f a r b i t r a r y Banach s p a c e s show t h a t i n t h i s s i t u a t i o n e i t h e r topology complete when
k)
in
(j,j' E N' )
m
Icpi(x)
1<4},
of t h e na-
N'
-+
11
,...,rpk}
(1 < i s
Irpi(x)l < e
x = 2(ej-ej,)
and for s u f f i c i e n t l y w e l l - c h o s e n and y e t
(rp
and a f i n i t e s e t
satisfies
are in
f
= ( x E L1:
Bh
let
It i s easy t o f i n d an i n f i n i t e subset
t u r a l numbers s u c h t h a t f o r each
It
t h a t t h e f i n i t e sums o f a T a y l o r s e r i e s e x p a n s i o n o f
i s well-known
f E
E Ll.
TOd
E
or
Hw(E)
7Wd.
E
containing
el,
to
i s n o t complete with r e s p e c t t o
Second,
i n proving t h a t
i s s e p a r a b l e and d o e s n o t c o n t a i n
the f a c t t h a t i n t h i s case
lie e x t e n d -
HT.,sc(E) = H w ( E ) .
C1,
Hw(R)
is
we u s e
T h i s makes u s e o f a
30
ARON a n d CARLOS HERVES
RICHARD M .
r e s u l t of F e r r e r a [lo], w h i c h w e now d e s c r i b e . with the r e l a t i v e
topology,
c o n s i s t s of t h o s e f u n c t i o n s
C(X)
in
u(E,E')
C(Xn)
f o r each
s u l t is that
if
i s a k-space.
E
n;
thus
and l e t on
f
Hw(E)
E
Xn = B n ( E )
Let
X = l i m Xn. Then n whose r e s t r i c t i o n s a r e
= H(E) n C(X).
Ferrerals re-
i s s e p a r a b l e and d o e s n o t c o n t a i n
The p r o o f
of t h i s r e s u l t makes u s e ,
L1,
then X
i n turn,
E,
s u l t of R o s e n t h a l [ 2 0 ] , t h a t f o r a s e p a r a b l e Banach s p a c e d o e s n o t c o n t a i n a copy o f
i f and only i f
E
e v e r y bounded s u b -
i s weakly s e q u e n t i a l l y d e n s e i n i t s weak c l o s u r e .
E
s e t of
L1
of a r e -
We t u r n now t o t h e q u e s t i o n o f d e s c r i b i n g some d e n s e s u b s p a c e s of t h e above s p a c e s .
Recall t h a t
Pf(E) =
C
bf(%),
where
n
b(%)
i s t h e subspace of
Pf(%)
(cp"
g e n e r a t e d by
: cp
E
E'}
.
The f o l l o w i n g c o n c e p t , d u e t o G i l a n d L l a v o n a r13], w i l l b e u s e f u l i n our d i s c u s s i o n .
E
We s a y t h a t a Banach s p a c e
h a s t h e bounded
K C E,
weak a p p r o x i m a t i o n p r o p e r t y i f f o r a n y weakly compact s e t
E
(u,):
there i s a net
-P
E
of f i n i t e r a n k o p e r a t o r s w i t h t h e p r o -
perties:
(u,)
(a)
c o n v e r g e s t o t h e i d e n t i t y u n i f o r m l y on
K
with
r e s p e c t t o t h e weak t o p o l o g y , a n d
u ua(K)
(b)
i s a bounded s u b s e t o f
E.
a
G i l and L l a v o n a h a v e shown t h a t i f
t i o n property then
E
E'
h a s t h e bounded a p p r o x i m a -
h a s t h e bounded weak a p p r o x i m a t i o n p r o p e r t y ,
and t h a t a r e f l e x i v e Banach s p a c e h a s t h e a p p r o x i m a t i o n p r o p e r t y i f a n d o n l y i f i t h a s t h e bounded weak a p p r o x i m a t i o n p r o p e r t y .
PROPOSITION
E
and t h a t is
6 . (1) Assume t h a t does not c o n t a i n
Hwsc(E) (2)
Pf(E)
If
E'
4,.
with respect t o e i t h e r
E
has t h e approximation property Then t h e c o m p l e t i o n o f T~~
or
T
wd'
h a s t h e bounded weak a p p r o x i m a t i o n p r o p e r t y ,
i s TOd-dense i n
Hw(E).
If
E'
Pf(E)
then
has t h e approximation pro-
WEAKLY SEQUENTIALLY CONTINUOUS ANALYTIC FUNCTIONS
Pf(E)
perty, then PROOF. (1)
Since
E
f E Hwsc(E),
Let
By [18, p.641
is Twd-dense in
Hw(E).
with Taylor series expansion f = CP,.
this series converges to
does not contain
dl
for the topology
f
Pn E Pwsc(%)
and since each
it follows by Proposition 2.12 of 14 ] that each closure with respect to the norm topology of this case is completed by noting that on relative
topologies coincide.
Tw d
7
wd' [2]
,
m,
Pn E
The proof in
P(%),
the norm and the
P(%)
T h e proof for the
case
TOd
follows from the above argument and from Proposition 4 above. (2)
(cf [ls]).
Assume that
f E Hw(E).
tion property and that compact subset of
E,
ua u a ( K ) .
Since
Let
y E Br(E)
assumption, there is Hence see '
If(x)
that
-
x-y E V,
and
a
a (x)I <
f o u
V
fou E pf(E)Tod,
be a ball containing
of
then
such that
x
for all
E
be the associated net
a simple compactness argument shows
that there is a weak neighborhood x E K,
be an arbitrary weakly
(u,)
Br(E)
f E Hw(E),
K
Let
E > 0, and let
of finite rank operators. K U
has the bounded weak approxima-
E
0
in
-7
(2) Pf(dl)
wd
entire functions
dl
By o u r
- u,(x)
E V
x E K.
Since it is easy to
for all
x E K.
the proof of the first part is complete. Q.E.D.
case is somewhat different,
= (Hc(&l)l~w), where f = CPn
< E.
If(x)-f(y)l
The second part is proved as in ( 1 ) above. A s before, the
such that if
E
Hc(dl)
such that for each
is the space of n,
Pn E fff(%),
dl
is a Schur space,
closure in the norm topology. PROOF.
The basic fact w e make use of is that
so that weakly compact sets are compact.
Hwsc(L1) = H(L1)
and that both
To
=
+iOd
From this it follows that and
7w
=
7 wd'
Thus (l)
32
RICHARD M.
ARON and CARLOS HERVES
f o l l o w s f r o m t h e well-known approximation p r o p e r t y ,
i s easy.
(2)
p r o o f of
f a c t t h a t i f a Banach s p a c e
Pf(E)
then
has t h e
E
H(E).
i s TO-dense i n
The
Q.E.D.
From t h e above p r o p o s i t i o n and e x a m p l e , we s e e t h a t f o r many
-'0d
E,
spaces
= (Hwsc(E),TOd).
Pf(E)
We do n o t know w h e t h e r t h i s
E
h o l d s i n g e n e r a l ( i , e . f o r a r b i t r a r y Banach s p a c e s
Ll),
e v e n i f one a s s u m e s t h a t
and
E
E'
containing
h a v e some t y p e of a p p r o -
ximation property. has played an important r o l e i n t h e preceeding d i s -
Hwsc(E)
A s we now show, t h i s s p a c e h a s s e v e r a l a d d i t i o n a l i n t e -
cussion.
resting properties. PROPOSITION 8 .
tl
and t h a t
Hwsc(E)
Assume t h a t
Then f o r a l l
$ i E!.
= Hwu(E)
f o r a Banach s p a c e E
Banach s p a c e s
Hwsc(E;F) =
F,
= Hwu(E;F). Note t h a t t h e a s s u m p t i o n s a r e s a t i s f i e d i f E = c
or i f PROOF.
f =
Let
Pn
CP, E H w s c ( E , F ) . n,
for e a c h
E Pwu(%;F).
Also,
E
F'
such t h a t
function
cpof
cpof
By [ 2 ] , i t f o l l o w s t h a t
a n d s o by P r o p o s i t i o n 2 . 1 2 f
I n d e e d i f t h i s were n o t t h e c a s e ,
cp
i s reflexive
0.
Pn E P w s c ( % ; F ) each
E
t h e n t h e r e would b e a n e l e m e n t
Hwsc(E)\Hwu(E),
The r e s u l t f o l l o w s b y Lemma
r4] ,
i s bounded on bounded s u b s e t s of E.
i s unbounded on some b a l l o f
is then i n
of
3 . 1 of r b ] .
E.
Since the
we h a v e a c o n t r a d i c t i o n . Q.E.D.
The f o l l o w i n g r e s u l t g i v e s a n i n t e r e s t i n g c h a r a c t e r i z a t i o n of
Hwsc(E) f o r many s p a c e s
characterization of THEOREM 9. Hwsc(E)
=
d f ( K ) C E'
Hwu(E)
Assume t h a t ( f E H(E)
E
E,
which i s analogous t o t h e
given i n [ 2 ] .
L1.
d o e s not c o n t a i n a copy o f
: f o r e a c h w e a k l y compact s e t
i s r e l a t i v e l y compact].
K
in
E,
Then
WEAKLY SEQUENTIALLY CONTINUOUS ANALYTIC FUNCTIONS
33
The proof makes use of the following lemma, which is similar
.
to one given in [ 31 LEMMA 10.
P
Let
E
If
P(%),
K
for every weakly compact set versely, if
E $ L1
pact subset
K
E,
PROOF OF THE LEMMA. K
E
of
P
dP(K) c E ’ dP
then f o r every weakly com-
takes weakly compact subsets E’.
Using Proposition
it is equivalent to show that
P E Pwsc(%)
is weakly continuous on each such K. Arguing as in [3,5], we first {A(xl,,..,x n-l,*) : x . E K ,
observe that the set
a relatively compact subset o f
E‘, where
n-linear mapping associated to
P.
rep,
finite set there is
cpj
IIKII
(where Icpj(x-y)I
+...+
,...,cpk)
c E’
= sup(llx11
< e/2n X)
lA(y, ...,y,x-y)l.
5
-
< e
weakly compact i n
E.
Then
K,
IIKII
be such that =
IP(x)-P(y)l
cpj(x-Y)
I +
lA(x,
A,
we see that
...,x,y,...,y,x-y)l
Icpj(x-y)l
+
S
Hence
< c/n.
P E PwSc(%).
and so
Conversely, let
is
~ A ( x - Y , x , - . - , x )+~ IA(y,x-Y,x, ...,x)l
each of the above terms is of the form
IP(x)-P(y)l
,...,xn- 1 E
x1
Using the symmetry of
IA(x,...,x,y,...,Y,x-y)
n-l]
> 0, there is a
E
x,y E K
Now, let
= l,...,k.
j
,Y)I
, . a .
Thus given
such that for all
E K]).
: x
S
is the symmetric
-
for all A(Y
A
1L i
I l ~ ( x ~ , . . . , x ~ - ~ , * )yjll < c / 4 n
satisfying
..., -
= ~A(x,
.5;
Con-
is relatively compact.
Assume that
show
Pwsc(%).
P E
then
to relatively compact subsets in
p+],to
2.13 of
E,
in
P € Pwsc(%),
and
in
is relatively compact in E’
dP(K)
P
E pwsc(”E)
E $
where
el
and let
K
be
F o r simplicity of notation, we will not use
additional subscripts to denote subsequences in the following argument.
T o show that
dP(K)
is relatively compact, it suffices
to show that each infinite sequence vergent subsequence. a point
x E K
Using the weak compactness of
to which a subsequence of
Thus it suffices to show that
(xj)
IIdP(xj)-dP(x)((
has a con-
E K]
: xj
(dP(xj)
K,
there is
converges weakly.
+ 0
as
j
-t
m.
That
34
RICHARD M .
i s , we must show t h a t -t
as
0
j
-t
ARON and CARLOS HERVES
SUP( l A ( x j ,
I f t h i s f a i l s , then there i s
m.
e a c h p o i n t o f a subsequence of vector
yj
...,x j t Y ) - A ( x , . . . , x , Y ) l
above terms t e n d s t o Hence
weakly compact.
...,x j - x , y j ) .
,
that
dPn(K)
each
f = CPn E H w s c ( E )
Let Since
f
K,
E’,
and t h e p r o o f
i s r e l a t i v e l y compact i n
df(K)
K C E.
f = C Pn
K
E
C
Thus
u n i f o r m l y on
i n t h e a b s o l u t e l y convex h u l l of
u n i f o r m l y on
E’
for a l l
K
K.
i s compact i n
d f = CdP, K.
Since
f(K)
E’
for e a c h
u n i f o r m l y on e a c h s u c h dPn(K)
s contained
and i s t h e r e f o r e r e l a t i v e l y Pn
is
F i n a l l y t h e u n i f o r m convergence of t h e
gives the r e s u l t .
Q.E.D.
We remark t h a t we have no example of a Banach s p a c e which t h e lemma f a i l s .
K.
Theref ore
n.
a n o t h e r a p p l i c a t i o n o f t h e lemma shows t h a t e a c h
weakly c o n t i n u o u s on
be
and t h e p r e c e e d i n g lemma t e l l s u s
Pn E 6 ’ w s c ( % ) ,
weakly compact s e t
and l e t
df = CdPn
and t h u s
C o n v e r s e l y , suppose t h a t
s e r i e s on
e a c h of t h e
i s bounded on e a c h weakly compact s e t ,
E‘,
compact,
,
o f [4]
which c o n t r a d i c t s o u r assump-
-t m ,
i s r e l a t i v e l y compact i n
and s o
I n e a c h of t h e s e
while the ramaining coor-
0
i s r e l a t i v e l y compact i n
u n i f o r m l y on
By [ 21
K
J
A(x,
j
+
= A(x.-x,xj,*..,xjsYj)
Q.E.D.
PROOF O F T H E O F E M .
f = CPn
We
F i n a l l y , we may w r i t e
Thus by Lemma 2 . 4
as
0
dP(K)
i s complete.
df(K)
(yj).
one c o o r d i n a t e t e n d s weakly t o
d i n a t e s a r e weakly Cauchy.
C .
t o c o n c l u d e t h a t t h e r e i s a weakly
Ll# E
A ( x , x j - ~ , ~ j , . . . , ~ j , y j+ )a , , +
tion.
>
(A(xj
...,x j , Y j ) - A(x, ...,x , y j )
terms,
such t h a t f o r
t h e r e corresponds a u n i t
(xj)
Cauchy subsequence of t h e sequence
+
> 0
,...,x J. , y J. ) - A(x, ...,x , y j ) \
f o r which
now u s e t h e f a c t t h a t
A(xj,
E
: llY/l=l)
E
for
T h u s , we do n o t know of any s i t u a t i o n i n
which t h e c o n c l u s i o n o f t h e theorem f a i l s .
Note t h a t i f
t h e n t h e lemma and h e n c e t h e theorem a r e t r i v i a l l y t r u e .
E = 41 I n any
WEAKLY SEQUENTIALLY CONTINUOUS ANALYTIC FUNCTIONS
35
case, the above proof shows the following. COROLLARY 11.
If K
df(K) in
E,
f E H(E)
Let
is relatively compact in
E
Note that if
E‘
f o r each weakly compact set
for all
is such that
for every weakly compact subset is compact in
[4]), and it is unknown if
In other words, using [a],
E.
f E H(E)
known if whenever
L 1, then
is separable and does not contain
(Proposition 3.3,
Hwsc(E) = Hw(E)
df(B)
is an arbitrary Banach space.
f E Hwsc(E).
then
Hw(E) = HwU(E)
E
where
K c E
df(K)
it is un-
is compact in
E’
it necessarily follows that
E’ for every bounded subset
B C
E.
Theorem 10 and its corollary are interesting, in light of the following c-product representation theorem (cf [Zl]),
whose
proof is omitted. THEOREM 12.
For arbitrary Banach spaces E and F ,
= {f E H(E;F) : f ( K )
is relatively compact in
weakly compact subsets
K
in
El,
E = L1),
f E Hwsc(E)
for all
with the topology of uniform
convergence on weakly compact subsets of
(or if
F
(Hd(E) ,T Od) E F
Therefore, if E $ L1
E.
if and only if
df E Hd(E) e
E’.
Finally, we mention some related open problems in connection with two special Banach spaces. ive James space
J
First, consider the quasi-reflex-
1151, which is a non-reflexive Banach space
with basis which is isomorphic to its second dual.
Hwsc(J) = Hw(J)
Cauchy sequences in element of
Hwu(J)
not know whether
J
= P ~ ( ~ J )for all pwsc ( n ~ ) and every entire function which takes weak
is separable and does not contain n,
Thus, since
J
L1’
to convergent sequences of scalars is an
(cf. Proposition 3 . 3 of [ 41). Hw(J)
= Hwu(J),
this, take the polynomial
P(x) =
# x 2il 2
although
Hw(J)
m
-
C i=1
(x2i-1
However, we do Hd(J).
in
T o see ~(‘5).
36
RICHARD M .
E = coxT
Second, l e t space [ 2 2 ] ,
ARON and CARLOS HERVES
where
Lp,
15 p
h
.
OD
s p a c e of c o n t i n u o u s n - l i n e a r r e f l e x i v e and, i n f a c t ,
,
I n [l] f o r m s on
that
= HwU(coXT) = Hd(coXT),
P("coXT)
= P f ( n coxT)
co-A(xk)
Y E T.
$
n.
for all
E P(n-kT),
Indeed,
Cf(
P(nT)
COY
53 ( n ' k T ) ) .
co, pf(n-kT)),
are
We d o n o t know i f
let
P
A ( x k ) ( y ) = A(,x
where
E
p ( n c o ~ T ) and
P.
,...,x r y ,..,,,y,)
for k n-k c a n b e e x p r e s s e d a s a f i n i t e sum
P
P(kco, P(n'kT)).
i t f o l l o w s by [ 1 9 ] t h a t
t h a t e a c h e l e m e n t of k
and
c a n b e e x p r e s s e d a s a f i n i t e sum o f t h e f o r m
P
That i s , t h e a c t i o n of
P(n-kT),
= Pf(
S(nT),
a l t h o u g h we do know t h a t
o f e l e m e n t s e a c h of w h i c h i s i n some co
T,
b e t h e s y m m e t r i c n - l i n e a r mapping c o r r e s p o n d i n g t o
Then t h e a c t i o n o f x
i t i s shown t h a t b o t h t h e
P(nT) = P f ( n T ) .
Hb(coXT)
A
is Tsirelsonfs (original)
a r e f l e x i v e Banach s p a c e w i t h u n c o n d i t i o n a l b a s i s w h i c h
c o n t a i n s no
let
T
Moreover, s i n c e
P ( k c o , P(n'kT))
Since
=
= Pf(n-kT),
6'(n-kT)
we s e e
i s a l i m i t of e l e m e n t s of
P ( k c o , P(n'kT))
which completes t h e p r o o f .
REFERENCES
1.
R . ALENCAR, R.'PLRON,
and S . DINEEN,
A r e f l e x i v e space of holo-
m o r p h i c f u n c t i o n s i n i n f i n i t e l y many v a r i a b l e s , Proc. A.M.S. 2.
R.
ARON,
Weakly u n i f o r m l y c o n t i n u o u s and w e a k l y s e q u e n t i a l l y
continuous e n t i r e functions, e d . ) , N o r t h - H o l l a n d Math.
3.
R.
ARON,
i n Adv.
Studies
North-Holland R.
Math.
A R O N , C . HERVES,
Studies
and M.
R.
(J.A.
Barroso,
ARON a n d J . B .
PROLLA,
(J.B.
Prolla,
Shilw,
ed.),
( 1 9 7 9 ) , 1-12.
VALDIVIA,
p i n g s on a Banach s p a c e ,
5.
i n Hol.
34 ( 1 9 7 9 ) , 47-66.
Polynomial approximation and a q u e s t i o n o f G.E.
i n Approx. T h e o r y and F u n c t . A n a l .
4.
t o appear i n
Weakly c o n t i n u o u s map-
t o appear i n J. Funct.
Anal.
Polynomial approximation o f d i f f e r -
e n t i a b l e f u n c t i o n s on Banach s p a c e s , J . r e i n e angew. Math. (Crelle),
313, (1980), 195-216.
37
WEAKLY SEQUENTIALLY CONTINUOUS ANALYTIC FUNC’TIONS
6.
S. DINEEN,
Complex a n a l y s i s i n l o c a l l y c o n v e x s p a c e s ,
H o l l a n d Math.
7.
DINEEN,
S.
Studies
North-
57 ( 1 9 8 1 ) .
E n t i r e f i m c t i o n s on
c0
t o appear i n J . Funct.
’
Anal.
8.
S . DINEEN,
9.
J . FERRERA,
(co,Xb)
modules,
Math.
S p a c e s of w e a k l y c o n t i n u o u s f u n c t i o n s ,
Pac.
J.
1 0 2 ( 2 ) , ( 1 9 8 2 ) , 285-291.
Math., 10.
H o l o m o r p h i c f u n c t i o n s on
1 9 6 ( 1 9 7 2 ) , 106-116.
Ann.,
J . FERRERA,
E s p a c i o s de f u n c i o n e s debilmente c o n t i n u a s s o b r e
e s p a c i o s de Banach, t h e s i s ,
Univ. Complutense de Madrid
.
(w80) 11.
J.
FERRERA, J . GOMEZ G I L a n d J . G .
LLAVONA,
12.
3 , n o . 54 (1983), 260-264.
J . GOMES G I L ,
E s p a c i o s de f u n c i o n e s debilmente d i f e r e n c i a b l e s , C o m p l u t e n s e de M a d r i d ( 1 9 8 1 ) .
t h e s i s , Univ.
13.
J . GOMEZ GIL a n d J.L.G.
LLAVONA,
P o l y n o m i a l a p p r o x i m a t i o n of
w e a k l y d i f f e r e n t i a b l e f u n c t i o n s on Banach s p a c e s , Royal I r i s h Acad.,
14.
J . HORVATH,
vol.
15.
R.C.
16.
B.
82 A ,
Proc.
2 ( l 9 8 2 ) , 141-150.
T o p o l o g i c a l v e c t o r spaces and d i s t r i b u t i o n s ,
1, A d d i s o n - W e s l e y , M a s s .
(1966).
B a s e s and r e f l e x i v i t y o f Banach s p a c e s ,
JAMES, Math.
15,
Bull. L.M.S.
s p a c e s of w e a k l y c o n t i n u o u s f u n c t i o n s , part
O n completion of
Ann.
5 2 , n o . 3 ( 1 9 5 0 ) , 518-527. Bounding s u b s e t s of &
JOSEFSON, Uppsala
m
(A),
t h e s i s , Univ.
of
( 1 9 7 5 ) , a n d J . Math. P u r e s e t A p p l . 57 ( 1 9 7 8 ) ,
397-421.
17.
L.
NACHBIN,
Erg.
18.
T o p o l o g y on s p a c e s o f h o l o m o r p h i c m a p p i n g s ,
d e r Math.
J . PAREDES ALVAREZ,
holomorfas,
19.
47, Springer-Verlag
A.
PELCZYNSKI, operators,
(1969).
E s t u d i o de algunos e s p a c i o s de f u n c i o n e s
t h e s i s , Univ.
d e S a n t i a g o , S p a i n (1981).
A theorem of D u n f o r d - P e t t i s B u l l . Acad. P o l .
Sc. X I ,
t y p e f o r polynomial
6 ( 1 9 6 3 ) , 379-386.
RICHARD M. ARON and CARLOS HERVES
38
20.
H. ROSENTHAL,
Some recent discoveries on the isomorphic
theory of Banach spaces, Bull. A.M.S.
84, no. 5 (1978),
803-831. 21.
L. SCHWARTZ,
Theorie des distributions a valeurs vectorielles
I, Ann. Inst. Fourier 7 (1957), 1-142. 22.
TSIREL'SON,
B.S.
N o t every Banach space contains an imbedding
of Cp or c o y Funct. Anal. and Appl. 8 (1974), 138-141.
23.
M. VALDIVIA,
Some new results on weak compactness,
Anal. 24 (1977), 1-10.
R.M.
ARON
Trinity College, Dublin, Ireland Current address:
Dept. of Mathematics Kent State University Kent, Ohio
44242
U.S.A.
C
. HERVES
Colegio Universitario de Vigo Apartado
14
Vigo, Spain
J. Funct.
Functional Analysis, Holornorplry and Approximation Theory 11, G I . Zapata l e d . ) 0 Eheoier Science Plcblislrers B. K (North-HuNand),1984
39
THE PRECOMPACTNESS-LEMMA FOR SETS OF OPERATORS
Andreas Defant and Klaus Floret
SUMMARY Grothendieckls lemma on precompactness in dual systems of vector-spaces is generalized to sets of operators.
We want to focus
the readerls attention to the method of proving compactness results
In par-
by using "individual" duality statements on precompactness.
ticular, certain localization results in spaces of semi-precompact operators are deduced in this manner; for the special case Schwartz' e-product
G c F
these localization properties show when
K
1.
C
G
and
L
C
H
G c F
C
F
such that
C
of a (real or complex)
(K0@Lo)O.
C
INTRODUCTION For an absolutely convex subset E
vector space
the Minkowski-functional mc(x)
subset
A c E
1
:= inf(h > o
is a semi-norm on the linear hull
span C
such that set of
A.
x E XC}
E
CO,~]
1
mc(x) <
m].
A
is called mC-precompact or C-precompact, if it is
and for every A C AC
If
mC
span C = ( x E E
precompact in the semi-normed space A C
H
For every compact
a kind of lifting holds true, namely: there are compact sets
of
+ cC;
(E1,E2)
system of vector spaces,
c
7 0
the set
[C]
:= (span C,mC),
there is a finite set AC
i.e. Ae c s p a n C
can always be chosen as a sub-
is a (not necessarily separating) A C El
any subset and
dual
40
DEFANT
A.
AO
1
:= (y E E~
the absolute polar of
A,
m,o(y)
and
K. FLORET
s 1
I(a,y)l
for all
a E A)
then the relation
=
SUP
I(a,y\l
a€A holds for all
y E E2,
PRECOMPACTNESS-LEMMA.
B c E2. pact
Then
A
The following lemma is due to A. Grothendieck
Let
(E1,E2)
be a dual system,
is Bo-precompact if and only if
B
A
c El
and
is Ao-precom-
.
The core of Grothendieckls original proof is the immediate fact that
(continuous scalar-valued functions, every A
b E B
is operating on
by the duality bracket) is always uniformly equicontinuous;
(A,mBO)
if
is a precompact space the Arzelh-Ascoli theorem applies.
This idea will be used later on.
There is also an easy direct proof
by polarity calculations (see e.g. [ 4 ] ,
where the result is called
ffth60rbmede pr6compacit6 r8ciproqueff). Unfortunately, the precompactness-lemma has not yet found the attention it deserves by its structural elegance and its usefulness.
It
is basic for the duality theory of locally convex spaces; it can (and should) be an early part of lectures on this topic.
We observ-
ed that having this lemma in mind many questions are easier to investigate,
The purpose of this note is to attack the problem of
compactness of sets of operators within the spirit of the precompactness-lemma.
THE PRECOMPACTNESS -LEMMA
2.
41
SETS OF OPERATORS
Let
(E1,E2)
and
(F1,F2) be two dual systems.
(F1,F2))
is the set of linear operators
operator
T' : F2 + E2,
u o u s operators.
i.e. the set of
T: El -+ F1
L((E1,E2\,
having a dual
a(E1,E2)-a(F1,F2)-contin-
A n immediate well-known consequence of the precom-
pactness-lemma is the following COROLLARY. set
El
,(F1,F2)),
and
F1
are normed spaces,
unit balls,
E2
= Ei,
A C El
For
and
A C El
is Bo-precompact if and only if
T(A)
If
T E L((E1,E2)
For
T/(B)
C
F, -
the
is Ao-precompact.
B c F 2 := F;
and
A c El
B
the
this is just Schauder's theorem.
and
N(A,V
H(A)
:= {T(a)
I
T E H, a E A}
:= [ T ' E L((F~,F~),(E~,E~))
H/
1
T E
HI
.
The basic result reads now as follows: THEOREM (Individual precompactness-theorem for sets of operators).
(E1,E2) and
Let
subsets. ments
(F1,F2) be two dual systems,
F o r every
H c L((El,E2) ,(F1,F2))
H(A)
is Bo-precompact
(b)
H'(b)
is A'-precompact
(2) (a)
H'(B)
is A'-precompact
(b)
( 3 ) (a) (b)
H(a)
H
for all
is Bo-precompact for all
b E B
a E A
is N(A,BO)-precompact
T(A)
is Bo-precompact for all
c F2
the following state-
(1)-(4) are equivalent:
(1) (a)
A C El and B
TE H
42
A. DEFANT
(4) (a) H' (b)
PROOF. and
is N(B,AO)-precompact
T'(B)
(4b)
-t
T E H.
is Ao-precompact for all
The implications (la)
(2a)
K. FLORET
and
(2b),
-t
are obvious;
(2a)
(lb) ,
-t
(3b) f-t (4b)
(la)
3
(3b)
holds by the corollary,
Since
the statements (3a) and (&a) are equivalent, whence ( 3 )
By the precompactness-lemma
(1) -I (2a):
c > 0
therefore for every
there are
B
bl,
-
(4).
is H(A)O-precompact;
...,bn E
B
with
Consequently n
H'(B)
since all
H' (bi)
is immediate that Therefore
(3)
-t
(la):
u H'(bi) i=1
C
are A'-precompact
H' (B)
(1) -t ( 2 )
+
cH'(H(A)O);
by (lb) and
c
H
C
C ' A
it
is A'-precompact.
and, by symmetry,
F o r every
H'(H(A)O)
(2)
-t
(1).
...,Tn E
> 0 there are
T1,
H
with
..
{T1,. ,Tn] + cN(A,Bo)
whence n
H(A) Since all
Ti(A)
C
(4) -+ (2a)
+
eBo.
are Bo-precompact (la) follows.
By symmetry the implication
(3)-
u Ti(A) i=l
-t
(lb).
(4) + (2a)
is true and therefore
T o finish the proof it is enough to show
that (2)
-+
(3a)i
By (2b) every
TE H
defines a function
fT; A -+ [BOI] a .rrt T(a)
43
THE PRECOMPACTNE SS-LEMMA
and it is easy to see that
[fT
nuous set of functions from
1
is a uniformly equiconti-
T E H]
(A,%,
(B)~) into
pactness-lemma applied to (2a) implies, that
[BOD.
(A,%'(B)~)
compact semi-metric space; since by (2b) for each
I
{fT(a)
E H) = H(a)
T
The precom-
a E A
is a prethe set
is precompact in the semi-normed space [Bolt
the Arzelh-Ascoli-theorem f o r the vector-valued setting ascertains that
I
{f,
T E H)
is precompact in the space C ( ( A , % ,
with respect to the topology of uniform convergence.
0O ) , [ B o ] ) I t was al-
ready mentioned that the equation sup m
= m
= sup sup I(Ta,b)l
(fT(a))
~ E AB O
aEA M B
(TI
N(A,BO)
holds which implies that precompact with respect to the uniform convergence j u s t means F o r Banach spaces
balls of
and
E
N(A,B')-precornpact. and
E
the sets
F,
A
and
B
being the unit
respectively, this is a well-known result on
F'
sets of collectively compact operators, [ 9 ] ; from this special case it is also clear that the extra-conditions (lb) and (2b) are not superfluous.
3.
SETS OF z-PRECOMPACT OPERATORS
F
Let
E
and
spaces and
C
a cover of
be not necessarily separated locally convex by bounded sets.
E
is precompact in
F
The space
for all
A E
C]
of all C-precompact operators is equipped with the topology of uniform convergence on all
on
PC(E,F)
all of
C
A E
C;
a basis of continuous semi-norms
where A is running through m"A,W) ' through a basis U F ( 0 ) of absolutely convex
is given by all
and
W
closed zero-neighbourhoods of
F.
The symbol
LC(E,F)
stands for
44
A.
t h e space
L(E,F)
DEFANT
K . FLORET
and
of a l l c o n t i n u o u s l i n e a r o p e r a t o r s
E
en-
F
-+
dowed a s w e l l w i t h t h e t o p o l o g y of u n i f o r m c o n v e r g e n c e o n a l l A € C. If
i s the scalar f i e l d
F
w i l l be used.
M
= LC(E,K)
E i := PC(E,IK)
the notation
The f o l l o w i n g p a r t i c u l a r c a s e s f o r
C
a r e important:
b := s e t o f a l l bounded s e t s pc := s e t of a l l p r e c o m p a c t s e t s c o := s e t o f a l l a b s o l u t e l y c o n v e x , r e l a t i v e l y compact s e t s e := s e t of a l l e q u i c o n t i n u o u s s e t s ( i f A € C
J u s t adding q u a n t i f i e r s over a l l
E
i s a dual space).
B = Wo,
and a l l
E
W
UF(0),
t h e ” i n d i v i d u a l ” theorem has t h e f o l l o w i n g ” g l o b a l precompactness theorem” a s a
For
COROLLARY.
(1)-(4)
(In
i s precompact i n
H(A)
(b)
H’(cp)
(a)
H’(w’)
(b)
H(x)
F
F
H
i s precompact i n
P~(E,F)
(4)
H‘
i s precompact i n
P e ( F b c ,E;)
(4)
the topology o f
a
T‘E H’
F’
A
for a l l
for a l l
E C cp E F’
for a l l
E.;:
i s precompact i n
i s precompact i n
for a l l
Ei
i s precompact i n
(3)
topology
all
the following statements
are equivalent:
(1) ( a )
(2)
H C L((E,E’) , ( F , F ’ ) )
w E
~
(la!)
(F’
,a) -+ E i . )
and
For special and W.
I t i s w o r t h w h i l e t o n o t e t h a t b y a simple
( l a ) i s equivalent t o
i s equicontinuous i n
H‘
d e n o t i n g by
6
)
c a n b e r e p l a c e d by any l o c a l l y c o n v e x
a r e weakly c o n t i n u o u s
.
0
*
s u c h t h a t a l l e q u i c o n t i n u o u s s e t s a r e a-bounded
R u e s s [lo] ; s e e a l s o [l]
(
x E E
c a s e s t h i s c o r o l l a r y i s r e l a t e d t o r e s u l t s of A . Geue [5]
argument
~
t h e t o p o l o g y on
E
the statement (2a) i s equivalent t o
L(“bc ,E;:)
i
which i s i n d u c e d by
(Ef)Lc,
45
THE PRECOMPACTNE S S-LEMMA
H
(2al)
L((E,G),F).
is equicontinuous in
If every precompact set in is continuous and
H
Eb
is equicontinuous then
In particular,
is equicontinuous.
L(E,F)
C
this holds true in the following cases C
E c.(E,6)
( F arbitrary):
(i)
E
is barrelled,
(ii)
E
is quasibarrelled,
(iii)
E
has a countable basis of bounded sets and every zeroEL
sequence in
arbitrary
C
3
pc
is equicontinuous (e.g. E
topological = gDF),
C =
a-locally
b.
For Schwartzt e-product of two quasicomplete locally convex separated spaces
G
F
and
G the corollary (with
0
F := Le(GLo,F)
E := G k o ,
= Pe(GLo,F) EL = G
whence
and
(E,6) =
“Lo)
is a refinement of L. Schwartzf characterization of relatively compact sets in
G E F
(see [ 8 ] ,
$44)
which was also obtained by
W. Ruess [lo].
4.
LOCALIZATION
B y definition, a subset
if there is a precompact set pact in
F;
H c L ( ( E , E / ) ,(F,F’)) C-localizes
K C Eh
if the precompact set
equicontinuous
H
such that K C Ei
C-localizes fully.
was initiated by some results of hi.
H(Ko)
is precom-
can be chosen to be
The study of this property
Ruess, e.g. in [lo].
The difference between ”C-localization” and “full C-localization” is of a more technical nature; in most of the applications every precompact set in
Ei
is already equicontinuous (compare the re-
marks after property (2al) in Section 3 . ) .
46
A. DEFANT
and
K. FLORET
PROPOSITION 1.
H c L((E,E‘) , ( F , F ’ ) )
(1) If pact in
H
is precom-
P~(E,F). If
(2)
C-localizes, then
and
G
are quasicomplete, then
F
H
C
Le(GkoyF)= G E F
e-localizes
( = e-localizes fully) if and only if there are compact
sets
and
K C G
Lc F
such that H c (KOOLO)~.
(Here
G‘OF’
PROOF.
C
(G
E
F)’
is the natural embedding).
F o r (1) check conditions (1) (a) and (b) of the global pre-
compactness result, (2) is as easy by taking
L := H ( K o ) .
Our concern is now to find conditions under which all precompact sets in
PC(E,F)
C-localize, i.e. when (1)-(4) of the global pre-
compactness-theorem is equivalent to
(5)
H
C-localizes.
Certainly this includes the problem under which circumstances each b-precompact operator is precompact
( : = there is a zero-neighbour-
hood whose image is precompact) which in general is not true.
In
the setting of e-products the question involves finding out when all compact sets
H
C
in Proposition 1 ( 2 ) .
G
E
F
can be llliftedll in the sense expressed
Note, that this is equivalent to the conti-
nuity of the natural embedding*)
The analysis will be split up in essentially two parts: calizes every precompact set (i.e. there is a
*)
U
E UE(0)
H
C
PC(E,F)
such that
When C-lo-
which is equibounded
H(U)
is bounded)?
When are
There is a close relationship between the notion of e-localization and the duality of E - and lT-topologies on tensorproducts. W e shall deal with this question in another paper.
THE PRECOMPACTNESS-LEMMA
precompact sets in
47
equibounded?
PC(E,F)
For the first question some further notation is helpful: R
of bounded sets of a locally convex space
A family
satisfies the
E
Mackey-condition (resp. the strict Mackey-condition) if f o r every A E R
B E 0
there is an absolutely convex
every (in
precompact subset of
E)
is B-precompact).
such that
A C B
and
is B-precompact (resp.
A
By the precompactness-lemma
strict Mackey-condition if and only if
F'
PC
pc
A
satisfies the
is a Schwartz-space.
One answer to the first question is given by the LEMMA 1
(1)
b = pc
or if H
C
If the family
c
H(U)
(2)
then every equibounded precompact subset
If
is Schwartz or c C pc in E then every precomPC H C P C ( E , F ) which is equiprecompact C-localizes fully.
(1)
If
b = pc
in
U E UE(0)
an equicontinuous set in
F
nothing has to be shown.
such that E'
H(U)
is bounded.
which is chosen to
the Mackey-condition it is enough to show that pact for every C
such
F'
other case take
H
U E UE(0)
there is a
is precompact.
pact subset PROOF.
F,
satisfies the Mackey-condition
Ei
is equiprecompact, i.e.
P (E,F)
that
in
in
e
Pz(E,F))
H(x),
x E E,
But (by the global result)
Uo,
is
according to is W-precom-
are W-precompact it suffices by
the individual theorem to check that
is absorbed by
H(Bo)
If B
Since (by the precompactness of
W E UF(0). all
Uo
In the
H'(Wo)
H'(Wo)
is B-precompact.
is precompact in
the Mackey-condition gives that
Ei
and H'(Wo)
H' (Wo)
is
B-precompact. (2)
Take
U E U,(O)
such that
every equicontinuous set in ness-lemma, whence
H
Eh
H(U)
is precompact.
If C
C
pc
is precompact by the precompact-
C-localizes fully.
48
A.
T h e other case runs as follows: condition of
H(U)
D
C
pc
and
Obviously
in
According to the strict Mackey-
is D-precompact.
i s contained in
H(Ko)
Take
is D-precompact.
compactness of pact.
H
such that
:= H'(Do) c Uo.
K
U
So it
for every
is A'-precompact
to apply the individual theorem note that H(A)
D
whence precompact.
DOo
K = H'(Do)
remains to show that
FLORET
there is a precompact set
F
H(U)
K.
and
DEFANT
absorbs
A
A
E
C:
whence
Again the individual theorem (and the pre-
in
shows that
Pz(E,F))
H'(Do)
is Ao-precom-
U
With rather the same arguments the following lemma can b e shown: LEMMA 2
(1) If in
E
b
in
F
satisifes the Mackey-condition or if
following property ( * ) :
K
in (2)
H
then every equibounded precompact set
If
b = pc
in
has the
There is an equicontinuous precompact set
H(Ko)
such that
E i
C Pz(E,F)
C c pc
F
is bounded.
or if the family Ei
continuous precompact sets in
e
n
pc
of all equi-
satisfies the strict Mackey-
condition then every precompact subset
€I C P C ( E , F )
with ( * )
Z-localizes fully. Collecting the results of the two lemmata gives the PROPOSITION 2.
In each of the following cases (a)-(.)
bounded precompact subset of (a)
C
(b)
Fi
(c)
e
c pc
in
E
and
Pz(E,F)
b = pc
in
every eyui-
C-localizes fully: F.
is a Schwartz-space. in
Ei
satisfies the Mackey-condition and
F'
PC
is
Schwartz, (d)
E
(e)
e fl pc b
is a Schwartz-space.
in
in
F
Ei
satisfies the strict Mackey-condition and
satisfies the Mackey-condition.
49
THE PRECOMPACTNESS-LEMMA
(a) and (c) follows from Lemma 1
PROOF.
ing the following fact: b = pc
Schwartz and
I?;
we11
(b) by ohserv-
as
is Schwartz if and only i f
F’
is PC The statements (e) and (d) come from
F.
in
as
Lemma 2 noticing for the latter that the following holds by the
E
precompactness-lemma: Ei
e fl pc
is Schwartz if and only if C C pc
satisfies the strict Mackey-condition and
in
E.
in
Later on it turns out that the assumption of one of the spaces being Schwartz is not at all artificial,
H C PC(E,F)
H’c Pe(F’ , E i ) PC e-localizes the following result is a corollary of Proposition 2 (d)
Since
and (a)
C-localizes if and only if
(and, of course, the global theorem):
PROPOSITION 3.
Let
F’ PC
every precompact subset
be Schwartz or I1 C PC(E,F)
b = pc
in
Ei.
Then
H‘C L(F‘ ,E&) PC
such that
is
equibounded, C-localizes. Coming back to the original question “When C-localizes every precompact set
H
C
PC(E,F) 7“
(la!) of the global theorem by ( 2 a f ) the set
H
note first that according to condition
H’c L(Fbc,Ek)
is equicontinuous and
itself is equicontinuous in
L((E,6),F)
in most cases implies that it is equicontinuous in
L(E,F).
which
In
view of Propositions 2 and 3 it is therefore reasonable to investigate under which circumstances a given pair
(M,N)
of locally
convex spaces satisfies the following localization principle: Every equicontinuous subset of PROPOSITION
4.
(M,N)
L(M,N)
i s equibounded.
satisfies the localization principle in each
of the following five cases:
N
(a)
M
or
(b)
M
has the countable-neighbourhoods-property (i.e. for
is normed.
every sequence
(Un)
in
which is absorbed by each
UM(0) Un)
there is a and
N
U E UM(0)
is metrizable.
A.
50
(c)
DEFANT
is Baire-like [ll] and
M
K. FLORET
and
N
has a countable basis of
bounded sets. ly
(d)
M
is metrizable and
NL
(or even the completion
NL)
is Baire-like. (e)
M
is metrizable and
N
has a countable basis of bounded
sets.
PROOF.
(b)
(a)
If
is obvious.
(Wn)
is a neighbourhood basis of
N
and
A
C
L(M,N)
n T-l(Wn) and a neighbourhood TE H according to the definition; then H ( U ) is bounded. equicontinuous take
(c)
If
(B,)
Un :=
U
is a basis of closed, absolutely convex, bounded
sets, consider
n
D~ :=
T-'-(B~).
TE H
(d)
H
If
C L(M,N)
continuous; since
Mk
T'
is equicontinuous in
L(N~,M;).
there is in both cases a bounded set
H'(Ao)
is bounded = equicontinuous in
(e)
H
such that
B y dualizing follows
(an alternative p r o o f can be found
4.2.).
There are even pairs
(M,N)
localization principle! type and 209).
Mk.
A c N
is equibounded.
is a special case of (d)
i n [31,
of all
rcI
By ( c )
that
is equi-
5
is a complete (DF)-space the set
rc/
extensions
H'C L(Nk,ML)
is equicontinuous then
N
e.g.
of Frbchet-spaces which satisfy the M
a power sequence space of finite
one of infinite type (see V.P.
Zahariuta n3], p 2 0 8 ,
The localization principle for pairs of Frechet-spaces was
recently charactericed by D. Vogt [12]. I t is not too difficult to see, that for an arbitrary locally convex space
E
and a quasibarrelled space
F
the pair
satisfies the localization principle if and only if on
E
(E,Fk) Q
F
the
THE PRECOMPACTNESS-LEMMA
51
projective and the (b,b)-hypocontinuous topologies coincide.
5.
APPLICATIONS I t was shown
(A)
If
satisfies the localization principle then in any of
(E,F)
the cases of Proposition 2 every equicontinuous precompact subset of
P (E,F) C-localizes fully. C
(For the equicontinuity recall the remarks at the end of 3 . )
(B)
(Fkc,Ei) satisfies the localization principle then in both
If
cases of Proposition 3 every precompact subset of
PC(E,F)
C-localizes.
For the following examples note first, that for metrizable
F
standard manipulations with precompact sets show that
is
F'
PC
Schwartz and has the countable-neighbourhoods-property.
E
for every quasinormable space
the family
e fl pc
Moreover,
in
EL
sa-
tisfies both the Mackey-condition and the strict one. The assumptions of (A) hold true in the following cases: (a)
E
is quasinormable,
F
normed,
I:
= b
(Prop. &(a),
Prop. 2(e)). (b)
E
is Schwartz,
F
normed,
C
arbitrary (Prop. &(a),
Prop. 2(d)). (c)
E
has the countable-neighbourhoods-property (e.g.
0-locally topological),
(d)
is metrizable,
F
b = pc
C
arbitrary (Prop. 4(b),
E
has the countable-neighbourhoods-property and is
Schwartz,
F
metrizable,
E in
is F,
Prop. 2(b)).
C
arbitrary (Prop. 4(b),
Prop. 2(d)). Since 0-locally topological spaces are quasinormable (see [ 71 ,p.260)
52
DEFANT
A.
E
(d) includes the case that
in (e)
E,
E
F
K. FLORET
and
is 0-locally topological,
metrizable and
C
arbitrary.
is metrizable and Schwartz, C
o f bounded sets,
b = pc
F
has the countable basis
4 (b), Prop. 2(d)).
is arbitrary (Prop.
The assumptions of (B) hold true for
(f)
E
has a countable basis o f bounded sets,
C = b (g)
E
(Prop.
Fi
Schwartz,
F
of bounded sets (e.g. (Prop. 4(e),
F
has a countable basis
a n (LS)-space),
3 since
Prop.
is metrizable,
Prop. 3).
4(b),
is metrizable,
F
Fi
C
is arbitrary. F'
Schwartz implies
PC
Schwartz. ) F o r e-products
G E F = Pe(Gbo,F)
PROPOSITION 5.
Let
G
convex spaces such that ciple and: subset
F
be quasicomplete separated locally
(Fbo,G) satisfies the localization prin-
is semi-Monte1 or
H C G E F
(1) H (2)
G
and
the principle (B) gives
Fko
is Schwartz then for every
the following two statements are equivalent:
is relatively compact.
There are compact sets
K c G
and
L
C
F
such that
H c ( K ~ B L ~ ) ~ .
The assumptions of this result are satisfied e.g.
in the following
G
is Banach and
G
and
G
has a countable basis o f bounded sets and
F
Fko
Schwartz.
are FrBchet-spaces. F
is a n
(LS)-space.
G
and
F
both have a countable basis of compact sets
(this implies that p.
266).
G
and
F
are semi-Montel, see [ 7 ] ,
53
THE PRE COMPACTNE SS -LEMMA
$44).
((b) was already treated in [ 8 ] ,
G
i.e. the assumptions o n
and
F
F,
and a n (LS)-space
G
is relatively compact in
F
and
F = F
E
G,
Note that
are semi-Monte1
then for every compact set
H C C ( X ) E F = @ ( X , F ) there is a compact set
(,( fdw
E
For a n illustration of (a) take a
0-locally topological spaces.
X
G
can be interchanged.
(d) comprises the case that both spaces
compact set
Note that
I
f
K C @(X)
such that
5 H , CI E KO}
F.
6. NECESSITY RESULTS By the very nature of C-localization it is clear that once it holds in L((E,6),F)
PC(E,F)
L(Fkc,Eb).
and
assumption of
certain sets have to be equibounded in
E
or
F' PC
However it is surprising that the being Schwartz which appears frequently
is sometimes even necessary. The key for the following results is the external characterization of Schwartz-spaces ( [ 2 ] , 12.4.):
A locally convex space
Schwartz if and only if for every Banach-space
E + G
linear mappings PROPOSITION G
6.
(1)
If
G
E
is
all continuous
are compact.
C
every one-point set in
C pc
in
PC(E,G)
and for every Banach-space
E
E
C-localizes fully, then
is
a Schwartz-space. (2)
G
F
Let
be a semi-Montel-space such that for all Banach-spaces
every one-point set in
Pb(G,F)
b-localizes, then
FL
is
Schwartz. PROOF.
(1)
directly by the external characterization, (2)
simple additional duality argument.
with a
W
Since there is a widely known Frkchet-Montel-space which is not
54
A.
and
DEFANT
K.
FLORET
Schwartz this result shows that there is a Banach-space Montel-(LB)-space
G
and a
such that not every precompact set in P b ( G , F )
F
b-localizes,
In the setting of the c-product (1) implies the following COROLLARY.
a quasicomplete separated locally convex space
If
has the property that for every Banach-space
TE Fc G T E
there are compact sets then
(K0@Lo)O
Fbo
K C F
and
G
F
and for every L C G
with
is a Schwartz-space.
Together with the above-mentioned example the last result shows that the lifting-property of
F a Montel (LB)-space and tion 5(a) and (c)).
G c F
G
may be false in the case
a Banach-space" (compare Proposi-
Since it is immediately clear that in the ex-
ternal characterization of Schwartz-spaces only spaces of the form G = C(X)
(with an arbitrary compact set
this also means, that the property for of 5. does in general not hold if
X
X)
C(X,F)
have to be checked stated at the end
is compact and
F
is only a
Montel (LB)-space. However:
if
H
C F e G
= Pe(Fbo,G)
is precompact it follows ( e . g .
by (2al) of the global precompactness theorem) that tinuous in
L(Fbo ,G)
.
If
(Fbo,G)
H
is equicon-
satisfies the localization prin-
ciple (which is obviously true in the case just mentioned) equibounded which readily means there is a compact bounded
B
C
G
K c F
H
and a
such that
Hc
(KO~BO)O.
BIBLIOGILAPHY 1.
A. DEFANT,
Zur Analysis des Raumes der stetigen linearen
Abbildungen zwischen zwei lokalkonvexen Rtiumen; Dissertation Kiel 1980
is
55
THE PRECOMPACTNESS-LEMMA
2.
K. FLORET, Lokalkonvexe Sequenzen mit kompakten Abbildungen; J. reine angew. Math. 247 (1971) 155-195
3. K. FLORET, Folgenretraktive Sequenzen lokalkonvexer RLume; J. reine angew. Math. 259 (1973) 65-85
4. H.G. GARNIR, M. de WILDE, J. SCHMETS, Analyse fonctionnelle, Tome I; Birkhkuser 1968
5.
A.S. GEUE, Precompact and Collectively Semi-Precompact Sets o f Semi-Precompact Continuous Linear Operators; Pacific J. Math. 52 (1974), 377-401
6. A. GROTHENDIECK,
Sur les applications line'aires faiblement compactes dfespaces du type C(K); Canadian J. Math. 5
(1953) 129-173
7. H. JARCHOW, 8.
G.
KBTHE,
Locally Convex Spaces;
Teubner 1981
Topological Vector Spaces I and 11; Springer 1969
and 1979
9. T.W. PALMER, Totally Bounded Sets of Precompact Linear Operators; Proc. Am. Math. SOC. 20 (1969) 101-106 10.
W. RUESS, Compactness and Collective Compactness in Spaces of Compact Operators; J. Math. Anal. Appl. 84 (1981) 400-417
11.
S.A. SAXON,
Nuclear and Product Spaces, Baire-like Spaces and
the Strongest Locally Convex Topology; Math. Ann. 197 (1972)
87-106 12.
D. VOGT,
Frgchetrlume, zwischen denen jede stetige lineare
Abbilding beschrlnkt ist;
13
V.P. ZAHARIUTA,
preprint
1981
On the Isomorphism of Cartesian Products of
Locally Convex Spaces; Studia Math. 46 (1975), 201-221
Universitat Oldenburg Fachbereich Mathematik
2900
Oldenburg
Fed. Rep. Germany
This Page Intentionally Left Blank
Furrctiowl Analysis. Holomorphy und Approximution Theory rr, G I . Zapata ( e d . ) 0Elsevier Science l'ulrlislirrs fl. V. (North-Holland), 1984
57
ON LIPSCHITZ CLASSES AND DERIVATIVE INEQUALITIES IN VARIOUS BANACH SPACES
2.
Ditzian
1. INTRODUCTION
In a series of papers see [l]
,
[2]
and r 3 ]
it was shown that
certain results on derivative inequalities, best approximation and convolution approximation can be extended from
C
(the space o f
continuous functions) to other Banach spaces for which translation is an isometry or contraction and for which translation is a continuous operator.
In this paper w e shall survey the results o f
those papers and extend them to some Banach spaces for which those This group of Banach spaces will
theorems were not applicable. include
L,,
B.V.
(functions of bounded variation) and duals to
Sobolev or Besov spaces.
2. THE LANDAU-KOLMOGOROV AND SCHOENBERG-CAVARETTA INEQUALITIES
In [4] Kolmogorov has shown that for
where
/Ig((3 sup lg(x)l
and
f E Cn(-m,a)
15 k
5:
n-1,
X
and calculated the best constants
K(n,k).
the result was proved earlier by Landau.
For
n = 2
and
k = 1
In [5] Schoenberg and
Cavaretta developed a method to calculate the best constants of (2.1) for
f E Cn(0,-).
It was shown in [l, p.1503 that if
Supported by NSERC grant A-4816 of Canada.
T(t)
58
2. DITZIAN
is a
Co
contraction semigroup on a Banach space
where
Af = lim t*0+
where
K(n,k)
(The
Tof-f t
in
B
B,
mentioned above are best possible in general, i.e.
for all spaces, but for a particular space
B
and semigroup
it is possible that smaller constants are valid.)
-
< t <
is a
m
is valied with
K(n,k)
Co
functions on IIf(*+t)ll
f(')(*),
(-,m)
o r on
Ilf(*)I
and
S
B,
then (2.2)
being Kolmogorovfs constants. B
for which
(0,m)
the strong derivative of
As an example
a Banach space of = Ilf(.)II
Ilf(.+t)li
= o(1)
Ilf(*+t)-f(.)Il
T(t)
Moreover, if
g r o u p of isometries on
of the use of the generalization one has for
or
f E D(An)
are those calculated by Schoenberg and Cavaretta.
K(n,k)
T(t)
and
t
-t
0,
then for
(2.1) is valid with the
f(*),
Kolmogorov o r the Schoenberg-Cavaretta constants respectively. One can also prove the following somewhat more general result.
For a Banach space
B
and a semigroup
(4
T(t)
we can define
(AWf,g) = (h,g) = lim T t) I f,g) for all g E B", t-t 0 for those f for which that limit exists we say f E D(Aw).
AW
by
If a Banach space is the dual of another Banach space (X*=B),
(w
= lim t+ 0 we say
then we can define f,g)
on
B;
T(t)(Aw)nf
(%,)nf
and
= (h,g) =
and if for
f
the limit exists,
D ( A ) c D(Au)
~(t),
satisfying
tends to
(Aw)nf
o
5
t <
and
f E D(Ai)
is a contraction semigroup
m
or
weakly or
fE
T(t)(AW
D ( C *) * )nf
in a weak* way, we have
-k (2.1)
X
X* = B.
Suppose f
(q,*f,g)
It can be noted that
D(A) c D(A~+) if THEOREM 2.1.
g 6 X,
for all
f E D(A,,).
by
Aw*
and
k 1-n
is such that tends to
ON LIPSCHITZ CLASSES AND DERIVATIVE INEQUALITIES
59
and k
-
k
(2.2)
with
K(n,k)
If
the Schoenberg-Cavaretta constants.
T(t)
< t<
-m
is in addition to the above a group
m
of isometries, then the inequalities (2.1) and (2.2) are valid with
the Kolmogorov constants.
K(n,k)
If
COROLLARY 2 . 2 . T(t)Anf
-+ Anf
T(t)
is that of Theorem 2.1,
f E D(An)
and
either weakly or weakly*, then n
(2.3)
with
K(n,k)
as in Theorem 2.1.
Note that this is slightly more general than the result in
REMARK 2 . 3 .
(a)
E B
Let
T*(t)
g
E B*,
for all
f
g E B*,
the assumption
(b)
f
E B
Let and
and
T'(t)
be defined by and if
T(t)(Aw)nf
-I
be defined by
g E X
(X*=B),
then the assumption
(T(t)f,g)
IIT*(t)g-gllB* (Aw)nf
weakly can be dropped.
IIT'(t)g-gllX
+ (Au*)nf
T(t)(Aw*)nf
for all
-+ 0
= (f,T'(t)g)
(T(t)f,g)
and if
= (f,T*(t)g)
for all
+ 0 for all g E X ,
in weak* fashion can be
dropped. Many examples of applications of this remark will be shown here. PROOF OF THEOREM 2.1. T(t)Auf .c
that for
f
= AwT(t)f .c
D(Ai
*
)
T o prove (2.1) and ( 2 . 2 ) for
f E D(Ai)
respectively.
and
we observe first
T(t)Ai*I.
w* (The right hand side can be defined
when the left hand side is not if we d o not assume
f
E D(A;*)
C < n,
respectively.) and
f E D(A,"*)
Obviously
implies
= AL T(t)f
f E ):A(.
f E D(Ak
*
)
f
E D(Ai)
implies & < n.
and
f E ):A(.
We choose
g
60
Z. DITZIAN
g E
such that
B"
F(t) = (T(t)f,g)
or
p.,
n.
4
where
with
CnfO,m)
Cn(-m,m)
= B
X"
when
when
= (T(t)A:f,g)
F(')(t)
Using
E X
is in
contraction and in Moreover,
g
T(t)
T(t)
or
with norm 1,
and
is a semigroup of
is a group of isometries.
= (T(t)A'
F(')(t)
f,g)
for
w*
[4] and [5], we obtain
K(n,k)
are the Kolmogorov constants in case
T(t),
-m
it<
m ,
is a group of isometries and the Schoenberg-Cavaretta constants for 0L t <
T(t),
a semigroup of contractions.
m,
completed now by choosing
gE
in
B"
X
or
The proof can be
such that
-
k k respectively, and observing for gE E B", IIAwf(l E 4 (A,f,g,) = = IFE ( k ) ( 0 ) ( 5 IFik)(t)( < K(n,k)((FE(n)(t)\(k/n L K(n,k)((Aif\(k'n ((f(1 l-k/n and the similar relation for
gE
E
X
and
A W"
In most applications
T(t)f(x)
.
= f(x+t)
in some sense and
we can summarize our result in the following theorem. THEOREM 2.4.
Let
B
be a Banach space and
B c S'(A)
(B
being
continuously imbedded in the Schwartz distribution over A ) and let A
be
[O,m),
f(t+*) A
=
or
(-m,m)
T = [-n,n]
be a contraction for
(-m,m)
or
A =
A = [O,m)
T. Then for
and periodic.
and a n isometry for
f(n)(*),
a weak derivative or a weak" derivative of f(n)(*)
Let T(t)f(-)=
a strong derivative, f(-),
and f(n)(t+*)
+
in weak or weak* mode, we have
(2.3)
where
K(n,k)
and
//f(t+.)li
for
A = [O,m)
are the Kolmogorov constants for
-)lit
= \If(
and
A = T
or
A=
(-m,m)
and are the Schoenberg-Cavaretta constants
llf(t+*)II
Ilf(.)I .
ON LIPSCHITZ CLASSES AND DERIVATIVE INEQUALITIES
Ilf(t+.)-f(.)Il,= ~ ( l ) t
If
PROOF.
f ( n ) ( t + . -+ ) f(n)(.), g E B*,
all
and if
we h a v e
-+
for a l l
0
f(n)(-)
-+
-i0
g E
for a l l
,
X,
X*
f(n)
weak* mode, w h e n e v e r
= B,
we h a v e
t
we h a v e -i
for
0,
weakly whenever f ( n ) ( - ) l l g ( t + - ) - g ( . ) l l X = o(l),
e x i s t s i n a weak or a s t r o n g s e n s e , a n d if
t
f E B,
Ilg(t+*)-g(*)l(D* = o(l),
f(ll)(t+.)
61
f ( ” ) ( t + . )-+ f ( ” ) ( . )
in
e x i s t s i n a weak* s e n s e and t h e r e f o r e T(t)f(.) = f(t+.),
t h i s i s a c o r o l l a r y o f Theorem 2 . 1 where
and
w h e r e we u s e Remark 2 . 3 .
I t i s w e l l known t h a t i n e q u a l i t y ( 2 . 3 ) i s v a l i d f o r
161
( w h e r e t h e i d e a seems t o o r i g i n a t e ) ,
on
R
on
R
11 ,
or
R+
[
or
R+
w i t h t h e norm
and
Lp(li)
L p ( R + ) [ 11 , O r l i c z s p a c e x +1 f ( x ) ’dx)’”) 13.
I
I
sup( x x I n e q u a l i t y ( 2 . 3 ) i s v a l i d a l s o f o r , f u n c t i o n i n t h e Sobolev space Sp(llfl/
I)fll
=
max Osxim
m I/fll =
C
Ilf(i)il
i=l
R
or
R+
P
Ilf(i)llp
or
and f o r f u n c t i o n s i n Besov or L i p s c h i t z s p a c e on
w i t h c e r t a i n norms
( t h e n o r m s f o r which t r a n s l a t i o n i s
i s o m e t r y or c o n t r a c t i o n ) . Using s t r o n g d e r i v a t i v e s and a n e a r l i e r r e s u l t [l, p.1501, we c a n s e e f o r
T = [-n,n]
and
-n
Kolmogorov i n e q u a l i t y i s v a l i d f o r
on
T
identified with C(T),
Lp(T)
that the
and O r l i c z s p a c e
a s w e l l a s f o r t h e S o b o l e v , B e s o v and L i p s c h i t z s p a c e on
T
T ( t ) f ( * ) = f ( * + t ) s a t i s f i e s t h e con-
w i t h a p p r o p r i a t e norm,
since
d i t i o n s imposed i n [l]
already.
The r e s u l t h e r e i m p l i e s K o l m o g o r o v ’ s i n e q u a l i t y ( w i t h K ( n , k ) b e i n g e i t h e r Kolmogorovls o r t h o s e of Schoenberg and C a v a r e t t a ) f o r many s p a c e s f o r w h i c h e a r l i e r r e s u l t s a r e n o t a p p l i c a b l e .
The f o l -
lowing spac e s w i t h t h e a p p r o p r i a t e d e r i v a t i v e s w i l l s a t i s f y ( 2 . 3 ) A.
La(R), Lm(T) derivatives i n
B.
B.V.(R),
Lm(R+)
or
B.V. ( T )
S’
w i t h weak* d e r i v a t i v e s
which a r e i n
or
B.V. (R’),
(and a c t u a l l y
La).
t h e f u n c t i o n s o f bounded v a -
r i a t i o n , b e i n g d u a l t o t h e space of continuous f u n c t i o n s .
The
62
2.
DITZIAN
weak* derivatives (which are also derivatives in belong to C.
B.V.)
S'
that
satisfy ( 2 . 3 ) .
R, T
The dual of the Sobolev space over
or
R+
with the
regular norm and weak* derivatives. D.
The dual of the Lipschitz or Besov spaces over
R, T,
R+
or
with norm induced by a norm of the Lipschitz or Besov space under which translation is either isometry or contraction and (Most of the norms given in the li-
with weak* derivatives.
terature for these spaces satisfy this requirement.) One should note that in order for ( 2 . 3 ) to be valid in the spaces mentioned in
nth
A, B , C and D the
weak* derivative must
exist in the given space. We remark that the constants of Kolmogorov and SchoenbergCavaretta always apply but are not necessarily the best.
The
constants in ( 2 . 3 ) dependon the norm and whatever constant is valid for one norm may not be valid even for an equivalent norm. The above discussion provides an upper bound for the constants in case the norm used satisfies
3.
Ilf(-+a)llB
5;
Ilf(*)ilB.
BEST TRIGONOMETRIC APPROXIMATION
B
For a Banach space
distribution on the circle with
n)
En(f,B)
inside
S'
T([-n,n]
(T), the Schartz space of
where
-n
is identified
we can define the best trigonometric approximation =
inf IIf-T 1) "B
where
TnE B
We define also
Ahf(*)
Tn
= f(*+h)
-
is a trigonometric polynomial. f(-) and
A E f = Ah(A;-lf).
Following earlier results, we have: THEOREM 3.1.
For a Banach space
is an isometry, that is
Ilf(.+a)ll
B c S' (T) for which translation =
Ilf(.)I
for all
a E T,
the
ON L I P S C H I T Z CLASSES AND DERIVATIVE INEQUALITIES
63
following a r e equivalent:
= O(ha)
(a)
I I A hrf l l B
(b)
En(f,B)
Moreover,
h + O+
n +
= O(n-a)
s u p / / Ar hfIIB
5
m.
c
M(r)hr
rlsh
C nr-lEn(f,B) B
and
(c)
converges,
n/21
kr-bk(f ,B).
= ~ ( 1 )f o r a l l
l]Al,fllB
I I T i r ) ( f ) ( l B = 0(nr-')
approximation t o PROOF.
f
f
E B,
(where
Tn
then i s the best trigonometric
i n B ) i s a l s o e q u i v a l e n t t o ( a ) and ( b ) .
T h i s t h e o r e m was e s s e n t i a l l y p r o v e d i n [ 2 ] .
and d e f i n i n g
F(x) = fug = ( f ( x + . ) , g ( * ) )
bounded f u n c t i o n , we s e e t h a t if e i t h e r
En(f,B)
and i f
exists as a strong derivative i n
f(r)
s M(r)CF
Pn(f('),B)
If
(n+l)'-%,(f,B),
n4h-l
= o(l),
Using
g
E
B*
which always y i e l d s a
l]AhfIIB
= o(1)
w h i c h a r e e q u i v a l e n t a s we s h a l l show,
or
F(x)
is
c o n t i n u o u s and t h e theorem i s d e r i v e d f r o m t h e a n a l o g u e f o r continuous functions a s i n [ 2 ] .
To p r o v e t h a t ( c ) a l s o i m p l i e s (a)
we f o l l o w d i r e c t l y S u n o u c h i t s r e s u l t
i s used and t h e r e f o r e IIAhflIB
= o(l),
En(f,B)
h + 0,
= o(1)
[7] but there
If
(/AhfllB
= o(1)
i s needed which f o l l o w s from
t h a t i s imposed h e r e .
but omitted erroneously.)
En(f,Lp)
= o(l),
( I n [2]
i t i s used
h + 0,
then
1[f(.tt)( ds t iM s define d ) ( s i2 nce
F n ( f ) = 2rrn
f(*+t) is
c o n t i n u o u s i n t h e norm t o p o l o g y ) a n d i s a t r i g o n o m e t r i c p o l y n o m i a l , and t h e r e f o r e
which t e n d s t o z e r o .
If
En(f,B)
= o(l),
n +
a,
we h a v e
64
Z. D I T Z I A N
h = l/n2
which tends to zero if
= o(1) I/Ahf/(
for all
f E EJ
I!Ahg// = o(1)
for all
g
f(x) = 0
function
Tn = 1
for
cannot be replaced by conditions like
E X
X" = B , and
(O,TT)
but not
I3 = D.V.(T)
as for
1
(-n,O)
for
I]Ahfll = o(1).
IIAhfllB
of the theorem do not require
the Tn = 0
has
A l l other parts
f o r all
= o(1)
f
E B
and
U = B.V.(T).
for instance are applicable to
4.
for example.
w
whose derivatives are equal to zero and therefore sa-
\ \ T i r ) l l = O(nr-a)
tisfy
-+
n
We may observe that in proving the equivalence of (c)
JIEMANK.
or
and
INVERSE RESULTS IN A P P R O X I M A T I O N THEORY
In [ 2 ] it was shown that direct and inverse theorems for a convolution approximation process that are valid for continuous functions o n
(R, I < + , T
A
or a Cartesian product of them) implies
that those theorems are also valid f o r Banach space
of general-
B
ized functions, for which translation is a contraction, that is
Ilf(.+a)l!
L
Ilf(*)II, and f o r which translation is also continuous,
llAhf(*)llB
that is,
= o(l),
h
-t
f
0+, for all
in
B.
We will
relax here the condition on continuity of all elements of the space.
all
a
Let
B
E
and
(I) (11) (111)
A
llAhflIB
be a Banach space such that
= o(l),
h + 0, for all
Ilf(.+a)llB
f E B
Ilf(*)IIB
for
or
(Ahf,g) = o(l),
h
0+, for all
f E B
and
g E B"
or
(Ahf,g) = 0(1),
h + 0+, for all
f E B
and
g E X
such
that
-t
X" = B.
Then we can define f o r any finite measure o n
f
4
f(*+x)dun('),
define for (I) since
topology, and for (11) and (111) by
f
A,
pn(*),
f+pn
=
is continuous in norm
(f+pn,g\
=
((f(t+*),g(-))dpn(t)
O N LIPSCHITZ CLASSES AND DERIVATIVE INEQUALITIES
f o r the appropriate THEOREM
4.1.
For
g.
B
in
(g
and
or in
B"
f+p,
= B.)
X"
X,
65
L (f) n
d e s c r i b e d above and
f
f+p n
t h e i n v e r s e theorem f o r continuous f u n c t i o n s
PROOF.
W e observe t h a t
Ln(f,
)
) + g = Ln(f*g,
where
g E
c a s e c o n d i t i o n s ( I ) o r (11) a r e s a t i s f i e d a n d w h e r e f+g = F E C(A),
i n c a s e of c o n d i t i o n (111). S i n c e of norm have
1
liA:fil E,
in
(X"=B)
X
w e choose
g
and a p p l y i n g t h e theorem f o r continuous f u n c t i o n s ,
we
= M,ha.
IIA;FIlc(*)
such t h a t
g E B"
Choosing
I(n,'f('),g,(.)>l
-c
2
I(AEf(*),g,(*))I =
of norm
gc
-
ilA,'fl/
c ,
l A E F (o ) l
1
in
or
B"
x
and r e c a l l i n g t h a t
r
5
~ ~ ~ h F ~ 4~ Mlha c ( A )f o r a l l
we c o m p l e t e t h e p r o o f .
REMARK.
Our r e s u l t now e x t e n d s i n v e r s e r e s u l t s t o s p a c e s l i k e
B.V.(A),
d u a l of S o b o l e v a n d Besov s p a c e s on
c o u r s e even t h e r e s u l t i n [ Z ] i s a p p l i c a b l e t o
and
A
Lp
L,(A).
spaces,
Of
Sobolev
a n d Besov s p a c e s , O r l i c z and o t h e r s .
5.
AN EQUIVALENT CONDITION ON DERIVATIVES In
[s]
we p r o v e d t h e e q u i v a l e n c e of some a s y m p t o t i c r e l a t i o n s
a n d we r e q u i r e d t h e r e t h a t
IIAhfl/B
= o(1)
for all
f
E B.
This last
c o n d i t i o n c a n b e r e l a x e d i n a way s i m i l a r t o t h a t u s e d i n e a r l i e r s e c t i o n s of t h i s p a p e r . Let such t h a t
B
b e a Banach s p a c e of d i s t r i b u t i o n o v e r
Ilf(-+a)ll =
Ilf(*)l\
and
R
or
T
66
Z.
( I ) IIAhfll
= o(1)
for a l l
(11) ( A h f , g ) = o ( 1 ) (111) ( A h f , g ) = o ( 1 )
Define
Anf
or
for a l l
f
E B
and
g E B*,
+
for a l l
f
E B
and
g E X
0
Gn E L1
and
by
Anf
or (X*=B).
= rf(t+-)Gn(t)dt J
= A n ( A nk - 1 f ) .
k Anf
4 , while
as i n section
E B,
f
h + 0
h
f E B
for
DITZIAN
W e have t h e f o l l o w i n g theorem:
5.1.
THEOREM
fIGn(Y)ldY
M,
I
f E B,
For
Anf
1IB.V.
IIG(r-l)
and
a s above and / G n ( y ) d y = 1,
Gn
f o r some
4 MrUir
f u n c t i o n could b e u n d e r s t o o d i n t h e 0
l y l B IGn(y)ldy
I
4
1 4 On/on+l
IIAEf/lB 5 M h'
[ I ( = d)
rk
rk
that
4
Ma!
for
then f o r
M,
= 0(anrkta)
(derivatives of
s e n s e ) and f o r some
S'
un = o(1) n +
satisfying
a > 0
f o r any i n t e g e r
k An(f,X)IIB
0,
r
m
the following a r e equivalent:
r,
> a.
r
f o r any i n t e g e r s
r, k
such
> a.
11 ( A n - I ) .e f l l B
= O(u:)
such t h a t
.t
f o r any i n t e g e r
.t > a/min(g ,I). Gn
i s even,
RK 5.2. 0
1,
> a/min(@,2) i s s u f f i c i e n t ) .
It is c l e a r t h a t f o r t h e
since (A)
i s much s t r o n g e r .
i s t h a t we do n o t have t o assume B
f
i n q u e s t on
llAhfIIB
= o(1)
The advantage i n t h i s theorem
IIAhfllB
= o(1)
on t h e whole
space
and t h e r e f o r e our theorem a p p l i e s now t o many s p a c e s f o r which
i t was n o t v a l i d b e f o r e .
For example B . V . ,
L,
and d u a l s of
Sobolev and Besov s p a c e s . PROOF.
A combination of
[s]
w i t h t h e simple i d e a used i n s e c t i o n
w i l l c o n s t i t u t e a proof o f o u r theorem.
4
ON LIPSCHITZ CLASSES AND DERIVATIVE INEQUALITIES
67
REFERENCES
1.
Z. DITZIAN,
Some remarks on inequalities of Landau and
Kolmogorov, Aequationes Math., 12, 1975, 145-151. 2.
Z. DITZIAN,
Some remarks on approximation theorems on various
Banach spaces, Jour. of Math. Anal. and Appl., Vo1.77, (2),
19809 567-576. 3.
Z. DITZIAN, Lipschitz classes and convolution approximation processes, Math. Proc. Camb. Phil. SOC., 1981, (go), 51-61.
4. A.N. KOLMOGOROV, On inequalities between the upper bounds of the successive derivatives of an arbitrary function on an infinite interval, 1939, Amer. Math. SOC. transl. 4, 1949, 233-243.
5.
J.J. SCHOENBERG and A. CAVARETTA, Solution of Landau's problem concerning higher derivatives on half line, Proceedings of the international conference on constructive function theory, Varna, May 19-25, 1970, 297-908.
6. E.M. STEIN, Functions of exponential type, Ann. of Math., 65, 1957, 582-592
7. G.I. SUNOUCHI,
Derivatives of trigonometric polynomials of best approximation, in "Abstract spaces and approximation", Proceedings of conference at Oberwolfach, 1968, (P.L. Butzer and B.Sz. Nagy Eds.), 233-241, Birkhtiuser, Base1 und Stuttgart, 1969.
Department of Mathematics The University of Alberta Edmonton, Canada T6G2G1
This Page Intentionally Left Blank
A U I r U H C A T I O N 5 C T 4 S 5 0 C I A T E D T O TIIF C O P Y PlIENOhrCNON I N TIIll s P l C L 01 GAUGE E I1'Ll)i
I-'r a n c i s c o .4n t o n i o D o r i a
We show t h a t g a u g e f i e l d c o p i e s a r e a s s o c i a t e d t o a s t r a t ified bifurcation t o b e t h e l o c u s of
s e t i n gauge f i e l d s p a c e .
Such a s e t i s n o t i c e d
o t h e r b i f u r c a t i o n phenomena i n g a u g e f i e l d t h e -
o r y b e s i d e s t h e copy phenomenon.
1. INTRODUCTION The phenomenon t h a t we a r e g o i n g t o d i s c u s s i n t h e p r e s e n t p a p e r h a s b e e n d i s c o v e r e d by two t h e o r e t i c a l p h y s i c i s t s , T . T . and C . N .
Yang,
Idu
i n s e a r c h for d i f f e r e n c e s b e t w e e n t h e s o - c a l l e d
A b e l i a n gauge t h e o r i e s and t h e i r non-Abelian
counterparts
[7]
.
Gauge f i e l d t h e o r i e s a r e p h y s i c a l i n t e r p r e t a t i o n s f o r t h e u s u a l t h e o r y of
connections
on a p r i n c i p a l f i b e r b u n d l e
u s u a l l y t a k e spacetime manifold (a 4-dimensional
[5].
Physicists
r e a l Hausdorff
smooth m a n i f o l d w i t h a n o w h e r e d e g e n e r a t e q u a d r a t i c f orm, t h e "metric t e n s o r " ) a s base space f o r t h e bundle, while t h e f i b e r i s i d e n t i f i e d with a f i n i t e - d i m e n s i o n a l semi-simple L i e group. general finite-dimensional
More
d i f f e r e n t i a b l e m a n i f o l d s a r e sometimes
used a s b a s e space f o r bundles of p h y s i c a l i n t e r e s t , s o t h a t o u r r e s u l t s w i l l n o t i n g e n e r a l d e p e n d on t h e b a s e m a n i f o l d ' s
dimen-
sion. The a b o v e d e s c r i p t i o n i s t h e m a t h e m a t i c a l s e t t i n g for t h e s o - c a l l e d Wu-Yang a m b i g u i t y or g a u g e f i e l d copy phenomenon.
Let us
b e g i v e n t h e e x p r e s s i o n for a c u r v a t u r e f o r m on s u c h a b u n d l e i n a l o c a l c o o r d i n a t e system:
FRANCISCO ANTONIO DORIA
F = (1/2)F U V (x)dx’
Here the components
F
dxv
A
are Lie-algebra valued objects.
YV
cp
the expression of the bundlels curvature form identity cross-section
U x El],
domain in the base manifold the bundle
P(M,G)
-
(1.1)
where
where
is
at a (local)
is an open trivializing
Curvature and connection forms on
M,
G,
U
F
a semi-simple Lie group as describ-
ed above, is the bundle’s fiber,
-
are related by Cartants structure
equation: cp = du + ( 1 / 2 ) r a A
&I.
(1.2)
(In a local coordinate system, at the identity cross-section,
where the
a cbc
are Lie-algebra structure constants,)
If we are given a curvature form connection form
a 7
is the Abelian group ness of
cp
9,
do we have a unique
The answer is no, in general,
U(1),
If the group
we can immediately check that unique-
can only be a local phenomenon provided that
simply connected.
M
is not
If the group is any non-Abelian semi-simple grcup
it is easy to show that there exists a curvature form ed by a Lie-algebra valued 2-form
F
cp
represent-
which can be obtained out of
an infinite family of connection forms which are
not related any-
where on spacetime by the so-called gauge transformations, that is by the natural action on the bundle
induced by the right action of
G
on the fiber.
M
be the four-plane with Cartesian coordinates and let the fiber
The example is quite simpler
let our spacetime
group be any non-Abelian Lie group (the other assumptions are not essential to the example.)
If
L(G)
is
GIs
Lie algebra, we
71
A BIFURCATION SET
choose as components
for the connection and curvature forms (at
the local identity cross-section),
It is now easy to check that
is also a connection form for
F,
whenever
h(x2)
2
function of the Cartesian coordinate (x ) . and
A
B
9
and
should commute [ 81
8'
gument has a rather more physical flavor:
j
= aVFMV +
#
0
fe,e']
C1
f 0,
cannot be gauge-related, for if it were s o , it is imme-
diate that
j'
Now if
is any
for
[A,,
B.
,FWv]
.
We then check that
.
The Wu and Yang ar-
they formed the current j = 0
for
A
and
As there is no gauge transformation that can make
the current vanish,
A
and
B
cannot be gauge-related.
The Wu-Yang ambiguity has remained a curiosity until now. However some recent work has opened the way for deeper undcrstanding of the phenomenon along physical and mathematical lines [l].
2.
MAIN RESULTS I N THE FIELD COPY PROBLEM Since our goal is to describe the geometry of copied curva-
tures and connections in the space of all curvatures and connectians, we review here the main characterizations for copied curvatures and connections.
FRANCISCO ANTONIO DORIA
72
Let
M
be a differentiable real n-manifold,
simple finite-dimensional Lie group, G-bundle over U C M
If
with projection
TT:
n.
-+ M
P(M,G)
a semi-
a principal
Suppose that over
a nonvoid open set, the curvature form
a'
cp
and
a
Under the above conditions
8
from two different connection forms
G
2
n-'(U),
can be derived
,
a 2 = a' + 8 .
We then conclude: PROPOSITION 2.1.
satisfies:
(2.1)
We now define the auxiliary connection f o r m
[&I.
This implies:
COROLLARY 2.2.
Condition (2.1) is equivalent to d(aO)e
where PROOF.
a 0 = a 1 + (u2) 0
d(ao)
zDef de
+
[ao
A
el
= 0,
denotes the covariant exterior operator w.r.t.
Substitute
a'
= ao
-
(1/2)9
(2.2) ao.
into (2.1).
Condition (2.1) implies also a well-known necessary condition for the existence of gauge copies: COROLLARY 2.3.
PROOF.
One calculates the derivative and then substitutes i n the
result Cartants structure equation and equation (2.1). (2.3) was erroneously considered by the author to be also a
A BIFURCATION SET
73
sufficient condition for the existence of connection ambiguities
f81. A counterexample is given elsewhere [ 121. More on that below. Equation (2.2) can be solved if we suppose that g
@ = Ldg,
where
is a Lie-algebra-valued equivariant function on the bundle.
Covariance considerations indicate that
IJ
= Ad(u),
action of a (possibly local) gauge transformation substitute
8' = Ldg
the adjoint
If we then
u.
into (2.2) we get (2.&a)
(2.4b) Equation (2.4a) can be rewritten as
[a1
A
d@] = -(1/2)[dB
A
dg]
.
(2.5)
If we delete the combined product symbols, we see that solving 2
(dp ) =
(2.1) or (2.2) is equivalent to solving the equation = (al)(dB).
We have two possibilities:
(i)
the infinitesimal,
continuous copies, given by (dp)2 = 0
iff
= 0,
(al)(dB)
and the (ii) discrete, paired copies,
a'
=
1
-
(1/2)dp,
Names are due to the following: EL
0,
5
above iff
A connection form
[a'
[a
A
A p]
a2-a1 =
Cp,
0
> 0,
p l = 0,
a '
is infinitesimally copied as
= 0, p = dp.
Equation (2.1) with
first order). 1
if we put
(2.5b)
we get
LEMMA 2.4.
PROOF.
1
( E )d@ = 0.
@ = ep
implies
d(al)p
= 0
(to the
And this last equation implies and is implied by
P = d@.
Solutions like (2.5b) are said to be discrete because
'8 = dp
is
74
FRANCISCO ANTONIO DORIA
a unique s o l u t i o n f o r ( 2 . 1 ) whenever
6
1
-1 = a
+
dg,
-1
[a
A
dg] = 0.
W e w i l l soon s e e t h a t " i n f i n i t e s i m a l " c o p i e s form a boundary s e t i n t h e s p a c e of a l l copied p o t e n t i a l s . We f i n a l l y n o t i c e t h a t combining
(2.3)
with
( 2 . 5 ) we s e e t h a t
copied curvatures should s a t i s f y
That i s , t h e ( a l g e b r a i c ) o p e r a t o r integrable nullspace.
[cp A
-1
s h o u l d have a n o n t r i v i a l ,
T h i s p r o p e r t y a l l o w s u s t o show t h a t c o p i e d
c u r v a t u r e s form a boundary s e t i n t h e s p a c e of a l l c u r v a t u r e s t h a t [cp A
satisfy
e]
= 0
f o r a nontrivial
on an open s e t i n t h e
8
bundle. A f i n a l r e s u l t w i l l be v e r y u s e f u l i n t h e n e x t s e c t i o n s :
w e w i l l need t h e f a c t t h a t PROPOSITION 2 . 5 .
n-'(U), group
U
H(cp)
cp
has gauge-equivalent d i f f e r e n t p o t e n t i a l s over
and open set i n
M,
i f f i t s Ambrose-Singer holonomy
has a n o n t r i v i a l c e n t r a l i z e r i n
For t h e prof see
[7].
on
G
n-'(U).
T h i s a l l o w s u s t o show t h a t " t r u e t '
c o p i e s a r e d e n s e i n t h e s p a c e of a l l c o p i e s , and t h a t ( l o c a l l y a t l e a s t ) g a u g e - e q u i v a l e n t c o p i e s b e l o n g t o a boundary s e t i n t h e s p a c e of c o p i e d c u r v a t u r e
7.
and c o n n e c t i o n forms.
DEGENERACIES I N C O N N E C T I O N AND CURVATURE SPACE
We a r e g o i n g t o d e s c r i b e some a s p e c t s of
t h e geometry of
c o n n e c t i o n and c u r v a t u r e s p a c e s t h a t have a t l e a s t one of t h e degeneracies l i s t e d i n t h e preceding section.
I n o r d e r t o summarize
t h e s e d e g e n e r a c i e s f o r t h e b e n e f i t of o u r e x p o s i t i o n , we n o t i c e that if
i s a ( p o s s i b l y l o c a l ) 1-form and
covariant derivative operator w . r . t .
a,
d(u)
the exterior
the condition
dz(a)8 = 0
75
A BIFURCATION SET
is equivalent to condition (2.3), or 0
be an ad-type tensorial form.
[cp A
e]
= 0,
provided that
We can thus list:
Covariant cohomology condition: 2
d (a)e = 0
iff
[cp A
e]
= 0,
Necessary condition for copies: [lp A
de] = 0 ,
Existence of copies:
+
d(al
(1/2)f3)9
= 0,
Discrete copies:
[a1
de] = -(1/2)[de
A
A
de],
(3.4)
Infinitesimal copies:
[a1
de] = 0 = -(1/2)[dp
A
A
de]
,
(3.5)
False copies : H(cp)
(3.6)
with nontrivial centralizer.
O u r objects are connections and curvatures for principal
fiber bundles
P(M,G)
with a real n-dimensional smooth manifold
as its base space and a fixed finite-dimensional semi-simple Lie group
G
as its fiber.
The geometry of curvature and connection
space is already a pretty well-known subject [5] and we will sketch here some of its main lines.
Curvature forms can be identified
with cross-sections of the bundle L(G)-valued
2-forms on
M.
all smooth cross-sections M,
nl(M,L(G)).
n2(M,L(G))
of Lie-algebra
Connection forms can be identified with of the bundle of L(G)-valued
1-forms on
Curvatures are ad-type objects; connection forms
will be s o provided that we fix and arbitrary connection (which can be the z e r o , o r vacuum, connection 0) and identify cross-sections of
n'(M,L(G)).
Near any point
xo
a
-
E M,
0
with the any L ( G ) -
76
FRANCISCO A N T O N I O D O R I A
v a l u e d 2-form
f
can b e s e e n a s t h e c u r v a t u r e of a p a r t i c u l a r con-
n e c t i o n form a v i a t h e c o n s t r u c t i o n
[4]
( i n a l o c a l c o o r d i n a t e system)
.
G l o b a l l y , w h i l e t h e map t h a t
sends a c o n n e c t i o n form o v e r i t s c u r v a t u r e i s p r e t t y w e l l - b e h a v e d , t h e i n v e r s e map i s f u l l of p a t h o l o g i e s [13]. W e w i l l consider here connection curvature
3 c C"(n2(M,L(G)))
G
5
C"(n'(M,L(G)))
s p a c e s t o be endowed w i t h a n a t u r a l
F r 6 c h e t s t r u c t u r e i f we endow t h e s e c r o s s - s e c t i o n Cm
topology.
and
spaces with t h e
T h i s r a t h e r weak s t r u c t u r e w i l l b e enough f o r o u r
f i r s t s e r i e s of r e s u l t s . We w i l l f i r s t r e s t r i c t our remarks t o t h e c a s e when a s p a c e t i m e , t h a t i s , a k-dimensional with a nondegenerate m e t r i c t e n s o r . Hodge
+!
operator w . r . t .
M
is
r e a l smooth m a n i f o l d endowed W e t h e n d e f i n e on
the
M
t h e s p a c e t i m e m e t r i c and t h e n check t h a t
(3.1) becomes a s p a c e t i m e - p a r a m e t r i z e d l i n e a r homogeneous system: [cp A
e]
= (Ad+lP)O = 0 .
(3.8)
T h i s system w i l l o b v i o u s l y have n o n t r i v i a l s o l u t i o n s p r o v i d e d t h a t det(Aduep) = 0
p
somewhere i n s p a c e t i m e .
globally s a t i s f i e s a property
t h e whole m a n i f o l d
P
We now s a y t h a t a c u r v a t u r e
iff
i s v e r i f i e d by
P
(or t h e whole bundle
M
P(M,G)).
cp
over
With t h i s
d e f i n i t i o n i n mind we a s s e r t :
P R O P O S I T I O N 3.1.
Curvatures t h a t g l o b a l l y s a t i s f y property
form a c l o s e d and nowhere d e n s e s e t i n PROOF.
Let
det Ad
Y
F
s i o n a l function matrix 2-form
3
i n the
be t h e d e t e r m i n a n t of Ad
*
F,
where
F
topology.
Cm
the finite-dimen-
i s t h e Lie-algebra valued
over s p a c e t i m e a s s o c i a t e d t o a c u r v a t u r e
t h a t globally s a t i s f y property
(3.1)
( 3 . 1 ) w e have
Cp.
d e t Ad
For curvatures
*
F
= 0
on t h e
A BIFURCATION SET
whole of' M.
77
And from the map
+
net: 3 -+ c ~ ( M )
F + * det Ad+Y we see that
t 3
Det-'(O)
3.
is closed and nowhere dense in
This immediately implies: COROLLARY 3 . 2 .
in
5
in the
Globally copied curvatures form a nowhere dense set Cm
topology.
The copy condition (3.3) implies (3.1), as shown i n Corol-
PROOF.
lary 2 . 3 . lu'hat about objects that satisfy one of the properties ( 3 . 1 ) -
(3.6) only locally, that is, over a nonvoid open set in in
F o r property
P(M,G)).
(or
M ?
(3.1) w.r.t. bundles over a spacetime
we can settle that question with: PROPOSITION 3.3.
Curvatures that satisfy property (3.1) locally
over spacetime form a closed and nowhere dense set i n
cm
5
i n the
a9
with the
topology.
PROOF.
We first form
topology induced by the map vanishes over an open set in property (3.1) locally over in
= a9 c Cm(M)
+Det(3)
+Det(-). M
M.
in the induced topology.
h(f)
a
g €
C
Det-'(h)
then Let
and endow f E a9,
F o r any
F E Det-'(f)
h(f)
if
f
satisfies
be a neighborhood of
I t is immediate that there exists
which is nowhere vanishing on
M.
Thus any
will never satisfy property (3.1) on
M.
G E Det-44
The conclusion
follows immediately. This result allows the solution of a question raised by
M. Halpern [Ill.
Halpern suggested that ambiguous curvatures and
connections should give extra contributions to the integrals i n Feynmann quantization techniques.
f
Despite the fact that we don't
78
FRANCISCO ANTONIO D O R I A
have a rigorous measure theoretical construction for a general Feynmann integral, we have several heuristic procedures for such calculations which try to characterize objec s similar to Bore1 sets on connection and curvature spaces.
A s ambiguous curvatures
and connections (which satisfy property (3.1 ) are nowhere dense in 3
and in
in a natural topology like the
G,
Cm
topology, we have
now reason to expect that they should be ignored during Feynmann integral calculations. The next question is:
are copied curvatures dense in the The answer
space of all curvatures that satisfy condition (3.1)? is no: PROPOSITION 3.4.
Copied curvatures are nowhere dense in the space
of all curvatures that satisfy (3.1). PROOF.
If cp
is a copied curvature then it satisfies ( 3 . 2 ) .
5 = [cp
sider all tensorial
e].
A
If
Cm
n'(M,L(G))
Con-
is the space
of all tensorial L(G)-valued 1-forms on the bundle, and 4 the space of all tensorial L(G)-valued 4-forms on Cm 0 (M,L(G)) the bundle, we can form the map d: 3
[CP where
d
nl(M,L(G))
X
A
-t
e]
COD
n4(M,L(G))
d[cpAe]=[CPAde].
is the standard exterior derivative.
differential identity we have that d"(0)
c 3 x Cmnl
dense set and s o is
p,(d-'(O))
c3,
0 1
pl: 5 x C
n
= {[cpAO],
d"(0)
I t is immediate that
Due to the Bianchi 0 a cocycle].
is a closed and nowhere
where
p1
denotes the
I31
= 0 - 1 another closed and nowhere dense subset in 3 x C n ,
projection
-t
3.
Condition
the corresponding restrictions to
d"(0)
rp A
and
p,d-'(O).
defines and so do Thus
globally copied curvatures are nowhere dense in the space of all curvatures that satisfy (3.1).
For locally copied curvatures we
79
A BIFURCATION SET
must follow a reasoning similar to the one used i n Proposition 3.3. The same technique can be applied to prove: PROPOSITION 3 . 5 .
Infinitesimally copied fields are nowhere dense
in the space of all copied fields. PROOF.
Consider the map f: G
X
Cmnl -+ Ci x Cmnl
and apply the same reasoning as i n Proposition 3.4.
We notice that
is the set of all infinitesimally copied curvatures.
f ' ( 0 )
What about "falseffcopies, that is, connection form ambiguities that can be (locally at least) eliminated modulo a gauge transformation?
This question is settled by
PROPOSITION 3 . 6 .
Curvatures with false copies are nowhere dense
in the space of all curvatures with potential ambiguities. PROOF.
Curvatures with false copies are stabilized by gauge trans-
formations that take values in the centralizer of Ambrose-Singer holonomy group generated by
cp.
H(cp),
the
We can thus apply
an adequate slice theorem to get via this symmetry a stratification wherefrom one sees that these symmetric curvatures are nowhere dense in the space of all copied curvatures.
3
the embedding of Lie algebras within
We could also reproduce in L(G)
associated to false-
ly copied curvatures. Propositions 3.4 the dimension of
M.
-
3 . 6 are valid without any restriction on dim M = 4 ,
But if we consider the case when
we can apply the same technique as in Propositions 3.1
-
3.3 to get
a stratification i n the set of all curvatures that obey (3.1) induced by the embedding of ideals in the space of all matrices Ad+F.
FRANC'ISCO ANTONIO D O R I A
4.
DIFFERENTIABLE V E R S I O N S O F O U R RESULTS
I n t h e p r e s e n t s e c t i o n we s u p p o s e t h a t a l l M-defined
objects
h a v e c o m p l e x - v a l u e d componentes t h a t b e l o n g t o a c o n v e n i e n t S o b o l e v space.
More p r e c i s e l y ,
if
U
c M
i s a n a r b i t r a r y open n o n v o i d s u b -
s e t , we w i l l s u p p o s e t h a t o u r o b j e c t s h a v e components i n one o f t h e Hilbert-Sobolev
H m ( U ) = I-12'm(U).
spaces
We t h u s h a v e a d i f f e r e n t i -
a b l e norm f o r o u r o b j e c t s and a v e r y s i m p l e d i f f e r e n t i a b l e s t r u c t u r e i n o u r f u n c t i o n s p a c e s , s o t h a t we c a n nov g i v e a more r e f i n e d v e r s i o n f o r our previous r e s u l t s . When we c o n s i d e r o b j e c t s t h a t are i n a S o b o l e v s p a c e we i m p l i c i t l y a d m i t t h a t o u r smooth ( i . e . t o h a v e compact s u p p o r t s .
C")
o b j e c t s a r e supposed
Such a s u p p o s i t i o n may i n t r o d u c e some
p r o b l e m s when we d e a l w i t h g l o b a l l y d e f i n e d smooth o b j e c t s on a noncompact s p a c e t i m e ; however we n o t i c e t h a t p h y s i c a l c a l c u l a t i o n s a r e a l w a y s done i n a p a r t i c u l a r c o o r d i n a t e domain, a l w a y s r e s t r i c t e d t o a compact r e g i o n , compact c l o s u r e .
A n o t h e r way of
and t h a t d o m a i n c a n b e
or t o a n e i g h b o r h o o d w i t h
looking a t t h i s r e s t r i c t i o n i s t o
s u p p o s e t h a t " p h y s i c a l " o b j e c t s become ( a p p r o x i m a t e l y ,
a t least)
z e r o beyond a c e r t a i n r a n g e ( s u c h a s u p p o s i t i o n i s commonly encount e r e d i n t h e d i s c u s s i o n of some r e s u l t s i n c l a s s i c a l f i e l d t h e o r y ) . Anyway t h e g a u g e f i e l d copy p r o b l e m i s an e s s e n t i a l l y l o c a l phenomenon. With t h o s e r e m a r k s i n mind,
we c a n o b t a i n t h e smooth v e r s i o n s
of our previous r e s u l t s : PROPOSITION
4.1.
Let
U c M
h a v e compact c l o s u r e and c o n s i d e r t h e
c l a s s of a l l c u r v a t u r e s on t h e b u n d l e t h a t do
( 3 . 1 ) anywhere on
U.
not
s a t i s f y condition
T h a t s e t i s a n open s u b m a n i f o l d i n
3.
A BIFURCATION SET
PROOF.
81
Consider the map
and consider its inverse (the norm we use is any finite-dimensional norm composed with the Sobolev norm),
h-l(W+-[O]
smoothness we have here an open submanifold i n
) c 3x0'.
Due to
3.
Appropriate modifications can be made in the other results. A sample is: PROPOSITION
4.2.
Let
U
be as above and consider the class of all
curvatures that have a discrete connection ambiguity all over They form an open, dense submanifold of
so,
U.
the space of all
copied curvatures. PROOF.
Consider the map
and act as i n the preceding proposition.
5. INTERPRETATION AND CONCLUSION Stratified sets first appeared in the study of bifurcation problems in Geometry [lb].
We have here a rather complex stratified
system, which depends i n part on symmetry properties of the systems (the embedding of,Ambrose-Singer holonomy algebras.)
Stratifica-
tions similar to this last one lead in General Relativity to the classification of spacetime geometries that are unstable i n the linear approximation [3]
.
A similar phenomenon leads to the li-
nearization instability of gauge fields uncoupled to any gravita-
82
FRANCISCO ANTONIO DORIA
tional field 1151. Fields that satisfy condition (3.1) ( o u r first stratum) can be shown to be associated to a nonvanishing torsion tensor that satisfies the same set of Bianchi identities.
Such a degeneracy
can be related to well-known "inconsistencies" in higher-spin field theory [2]
.
Fields that possess infinitesimal copies can be shown
to generate a very interesting version of the Higgs mechanism [l] where the gauge field can be shown to generate
,
a field that sa-
tisfies the standard electromagnetic wave equation [lo].
Finally
fields with false copies are shown to imply Nambu's condition for the existence of nontrivial topological effects such as magnetic monopoles and vortices.
This class of fields exhibits also an in-
consistency that appears when one tries to add a gauge-like interaction to spin-0 fields; here this inconsistency is shown to be a symmetry-breaking condition. We do not have a clear interpretation for the coupled sets of nonequivalent potentials that form discrete copies systems, despite the fact that they were one of the first examples of copies
to be found [ 6 )
.
6 . ACKNOWLEDGMENTS The author wishes to thank Professor G . Zapata f o r his kind invitation to expose these ideas at the 1981 Holomorphy and Functional Analysis Symposium in Rio de Janeiro.
He also thanks
Professor Leopoldo Nachbin for his constant interest and encouragement.
A BIFURCATION SET
83
REFERENCES 1.
A.F. AMARAL, F.A. DORIA and M. GLEISER, Higgs fields as Bargmann-Wigner fields and classical symmetry breaking,
J. Math, Phys. 24 (1983), 1888-1890. 2.
A.F. AMARAL,
The Teitler lagrangian and its interactions,
D.Sc. Thesis, Rio de Janeiro (1983) (in Portuguese).
3.
J.M. ARMS,
Linearization instability of gauge fields,
,
443-453.
J. Math, Phys. 20 (1979)
4. C.G. BOLLINI, J.J. GIAMBIAGI and J. TIOMNO, Gauge field Phys. Lett. 83 B (1979), 185-187.
copies,
5.
Y.M.
CHO,
Higher-dimensional unification of gravitational J. Math. Phys. 16 (1975), 2029-2035.
and gauge theories,
6.
S. DESER and F. WILCZEK,
potentials,
'7.
F.A. DORIA,
Non-uniqueness of gauge field
Phys. Lett. 65 B (1976), 391-393.
The geometry of gauge field copies,
Commun.
Math. Phys. 79 (l98l), 435-456.
8. F.A. DORIA, copies:
Quasi-abelian and fully non-abelian gauge field A classification,
J. Math. Phys. 22 (1981),
294'3-2951.
9. F.A. DORIA and A.F.
AMARAL,
gauge field copies,
Linearization instability implies
Preprint , Universidade Federal do
Rio de Janeiro, 1983. 10.
M. GLEISER,
Gauge field copies and the Higgs mechanism,
M.Sc. Thesis, Rio de Janeiro (1982) (in Portuguese). 11.
M.B. HALPERN, Gauge field copies in the temporal gauge, Nucl. Phys. B 139 (1978), 477-489.
12.
M.A. MOSTOW and S. SHNIDER,
Counterexamples to some results
on the existence of field copies,
Preprint, Univ. North
Carolina, 1982,
13.
M.A. MOSTOW and S. SHNIDER,
Does a generic connection depend
continuously on its curvature? Carolina, 1982.
Preprint, Univ. North
84
14.
FRANCISCO ANTONIO DORIA
R. THOM,
L a estabilite topologique des applications poly-
nomiales,
Llenseignement mathematique Vol. 8 ( 1 9 6 0 ) ,
24-33.
Interdisciplinary Graduate Research Program Department o f Theory of Communication Universidade Federal do R i o de Janeiro Av. Pasteur 250 22290
R i o d e Janeiro, RJ, Brazil
1~'unctionalA ndysis, Holoniorphy awd .4pproxitnation Theory 11, C.I. Zopato (cd.) @ Ekevier Scierice P u b l i s h s B. I< (North-Holland), I984
O N THE ANGLE: OF DISSIPATIVITY O F O R D I N A R Y
AND PARTIAL DIFFERENTIAL OPER4TORS"
H.O.
Fattorini
1. INTRODUCTION
Let
A
E.
Banach s p a c e
of
u
where
b e a d e n s e l y d e f i n e d , c l o s e d o p e r a t o r i n a complex
c o n s i s t i n g of a l l
(u*,u)
operator
A
u*
d e n o t e by
@(u)
i n t h e d u a l space
the duality s e t such t h a t
E*
d e n o t e s t h e v a l u e of t h e f u n c t i o n a l
u*
at
u.
The
is dissipative if
Re(u*,Au) If
u E E
F o r each
(u E D ( A ) ,
0
5
u* E @(u)).
(1.2)
we h a v e
(XI f o r some
1 >
0,
then
A
-
A)D(A)
= E
i s called m-dissipative
s t r o n g l y c o n t i n u o u s c o n t r a c t i o n semigroup
The c o n v e r s e i s a s w e l l t r u e . assumed f o r a s i n g l e e l e m e n t replace
(1.2)
(1.3)
{ S(t); t
We n o t e a l s o t h a t
u*
of
and g e n e r a t e s a 2
(1.2)
O]
,
need o n l y be
@ ( u ) ; equivalently,
we may
by Re(8 (u),Au)
0
(u E D ( A ) )
*
(1.5)
T h i s work was s u p p o r t e d i n p a r t by t h e N a t i o n a l S c i e n c e F o u n d a t i o n , U.S.A.
u n d e r g r a n t MCS
79-03163.
. . FATTORINI
86
H 0
8
where with
8:
i s a d u a l i t y map, t h a t i s , an a r b i t r a r y map
e(u) E @(u) (u E D(A)). space, (1.3)
i s a Hilbert
-t
E"
We n o t e a l s o i n p a s s i n g t h a t i f
i s e q u i v a l e n t t o maximality o f
c l a s s of d i s s i p a t i v e o p e r a t o r s . theory see f o r instance
E
A
E
i n the
F o r a l l n e c e s s a r y f a c t s on t h e
[k],
I n c e r t a i n q u e s t i o n s of c o n t r o l t h e o r y ( r e l a t e d t o t h e com-
(I
p u t a t i o n of t h e i n v e r s e
- aS(t))'l)
d e c i d e whether t h e semigroup
I arg 5 I
s cp
(cp
>
More g e n e r a l l y , cp
whether t h e r e e x i s t s
>
and
0
IJJ
= ~ ( c p ) such t h a t
such t h a t ( 1 . 6 )
cp 2 0
a s t h e supremum of a l l
cp
2
0
e*irp(A
8,
the supre-
w = w(cp)
> 0.
may be c h a r a c t e r i z e d
cp(A)
such t h a t
( 1 . 5 ) w i t h r e s p e c t t o some d u a l i t y map cp.
A,
h o l d s f o r some
E q u i v a l e n t l y , t h e a n g l e of d i s s i p a t i v i t y
g e n e r a l of
(1.4)
i t i s o f t e n enough t o i n q u i r e
t h e a n g l e of d i s s i p a t i v i t y o f
cp(A),
mum of a l l t h e
can be e x t e n d e d t o a s e c t o r
i n t h e complex p l a n e i n such a way t h a t
0)
i s preserved there.
We d e n o t e by
S ( * )
i t i s o f importance t o
with
- wI) w
satisfies
depending i n
Some obvious m a n i p u l a t i o n s show t h a t t h i s r e q u i r e m e n t
transl at e s t o Re(e(u),Au) 5 fbIm(8(u),Au)
f o r some d u a l i t y map
8
and
+ wllul12
(1.7)
(u E E )
8 = t g cp.
The o b j e c t of t h e p r e s e n t p a p e r i s t h e c o m p u t a t i o n of t h e a n g l e of d i s s i p a t i v i t y of second o r d e r u n i f o r m l y e l l i p t i c o p e r a t o r s .
m A =
m
m
C ajk(x)DjDk + C b j ( x ) D j + j=1 k = l j=1 C
( a j k ( x ) = a k j ( x ) , x = (xl
R
,...,x m ) ,
of m-dimensional E u c l i d e a n s p a c e
tion
Ap(e)
in
LP(n)
(1 5 p <
m )
i n a bounded domain
D j = a/axj)
Elm,
or
(1.8)
C(X)
or r a t h e r , of t h e r e s t r i c A(@)
in
C(G)
of
A
de-
87
ON THE ANGLE OF DISSIPATIVITY
fined by a boundary condition
of one of the following types:
, .
Here
=
DW+>
(1)
(11)
Y(X)U(X),
I' is the boundary of n
...,wm)
Dw
all
u
fi
r).
=EX j a ~k . (x)wj},
r;
the outer normal vector on
when the Dirichlet of all continuous
C(5)
is replaced by the subspace
that vanish on
E
indicates the derivative
boundary condition (11) is used the space functions in
(X
,.
and
in the direction of the conormal vector v = (wl,
o
=
U(X>
consisting of
C,(n')
r.
The results are as follows. assumptions on the coefficients of
Under the standard smoothness A,
on
y
and on
r
the angle
of dissipativity turns out to be independent of the operator and the boundary condition the space 'P(Ap(B))
E = LP(n)
p
(1 < p <
and only depends on the space. m)
A
In
the angle of dissipativity
is
+ m when p -+ l,m we may 'PP surmise that the angle of dissipativity is zero in the spaces L1(n)
(Theorem 2.1 and Section 5 ) .
and
C(5).
A(B)
- wI
Since
This is in fact true; moreover, if will not be dissipative for any
UI
additional assumptions ((6.1) for the space the space
C(n')),
6
is of type (I),
unless
L1(n)
B
satisfies
and (6.8) for
although these restrictions can be bypassed
through a renorming of the space (Section 6).
These results are
presented in detail i n Sections 5 and 6. We treat separately the case
m = 1
in Sections 2 and 4 ;
here the results are slightly more precise while many of the technical complications disappear.
Finally, we include in Section 3
an application of the one-dimensional results to the estimation of the norm of certain multiplier operators.
88
2.
H - 0 . FATTORINI
Lp, 1 < p <
ORDINARY DIFFERENTIAL OPERATORS I N
Let
. A
m
.
be the formal differential operator
+
A0U(X) = a(x)u”(x)
+
b(x)u‘(x)
c(x)u(x).
(2.1)
O u r standing assumptions on the (real) coefficients are:
twice continuously differentiable, entiable,
c(*)
is continuous in
Po
We denote by
b(*)
n
a(*)
is
is continuously differ-
= Ex; 0
x s I,].
L
x = 0
a boundary condition at
of one of
the two types
(I)
8,
and by
u ’ ( 0 ) = you(o)
a boundary condition at
The coefficients
y o , yc
The operator
(11)
u(0) = 0
x =
.t,
of one of the two types
p <
a)
are real.
Ap(Po,BL)
AP(@,,BL)u
= AOu
Ap(BO,P,,)
consisting of all
(1
4
in the complex space u
is defined by the domain of
Lp(O,.f,),
E W2’p(0,.t,)
that satisfy the cor-
responding boundary condition at each endpoint.
w2 ’P(0,t)
consists of all
derivatives
u ’ , u”
(2.2)
u E Lp(O,t)
belong to
Lp(O,.L)
Here the space
such that the distributional as well.
We shall show i n
( o r , rather, a translate) Ap(@ O ,p L ) fits into the theory of Section 1. That (1.3) holds means i n this
the rest of this section that
case that there exists a sufficiently large each
f E Lp
there exists
u E D(AP(@,,BL))
>
X
if
function of the boundary value problem
(XI-A)u
Po,
B,
such that, for
with
This can be shown by elementary means:
to the boundary conditions
0
G(x,5)
= f
is the Green corresponding
(which function will exist for
ON THE ANGLE O F DISSIPATIVITY
sufficiently large
1)
89
then
.e.
u(x) =
[
G(x,S)f(5)dS
For d e t a i l s s e e [ 3 ] . W e check now
(1.7) for
1 4 p
t h a t t h e o n l y d u a l i t y map from
Lp
<
a.
into
T o t h i s e n d , we r e c a l l (Lp
*
= LP’
(p’-l+p-l= 1
is
Accordingly, f u n c t i o n ) and
if
u = u1
p z 2,
+
iu2
e(u)
f
0
i s smooth ( s a y , a Schwartz t e s t
i s continuously d i f f e r e n t i a b l e with
On t h e o t h e r h a n d , we have
Assume b o t h boundary c o n d i t i o n s a r e of t y p e ( I ) . Then we have
We t r a n s f o r m t h e sum of t h e f i r s t t h r e e i n t e g r a l s on t h e r i g h t - h a n d s i d e using t h e following r e s u l t :
given a constant
a > -1,
H.0. FATTORINI
90
1zI2 + a((Rez)2 z E
for every
i 6(Rez)(Imz))
2
0
if and only if
C
To prove this we begin by observing that ( 2 . 7 ) is homogeneous in z , thus we may assume that z = eiV, reducing it to the trigonometric 2 identity 1 + ~ ( C O Srp 6cosrpsinc~)2 0 o r , equivalently (setting $ = 2rp)
,
function if
2
+ a ( 1 + cosJI f 6 sin$)
2
= cos$ f 6sin$
g($)
- a ( ( 1 + b2)'l2
the maximum of hold if
2
-
-
g
is
lal((1
2
0.
Since the minimum of the
-(1+62)1/2,
equals
1) z 0, which is ( 2 . 9 ) .
(1+6 )
+ 6 2)1'2
so
+
that if
1) 2 0 ,
(2.8) will hold On the other hand,
-1 < a < 0, (2.8) w i l l
which is again ( 2 . 9 ) .
In
view of the homogeneity of (2.8) it is obvious that if ( 2 . 9 ) is strict there exists z E
V
> 0 (depending on 6 )
such that, for every
c,
+
1zI2
We use (2.10)
z =
for
*
a((Rez)2
2
V1zl2.
(2.10)
Gu':
-(p-2) i 6(p-2)
s
b(Rez)(Imz))
2
(
al~l~-~(Re(&~'))~dxi
al~l~-~Re(;u')Im(;u')dx
(2.11)
'0
'0
where
I
f = Iiiu' 2
+ (~-2){(Re(;u'))~ 2
for some
v >
0
(depending on
k 6Re(uu')Im(Gu')]
vliiu' I 2 6 )
2
(2.12)
if
(2.13) We must now estimate the other terms on the right-hand side
91
ON THE ANGLE O F DISSIPATIVITY
of
(2.7).
T o this end, consider a real valued continuously differ-
entiable function
p
+ ppluIp-2 Re(;u/)
0 4 x 4 &.
in
we obtain, f o r any
E
7
= p' lulp +
(pluIp)'
Since
0,
lulPdx
(2.14)
where we have applied the inequality = 2(E
2lul Iu'
-1
IUI)(EIU'I)
We use (2.14) for any
(we may take
p
p
+
E21U'12
4
E
-2 IU
2
(2.15)
such that
linear) to estimate the first two terms on the
right-hand side of (2.7); for the fourth integral we u s e again (2.14), in both cases with the constant of
(2.7)
v
E
in (2.10)).
sufficiently small (in function of We can then bound the right-hand side
by an expression o f the type
JO
for some constants
JO
w = w(6)
and
c = c(6)
7
0.
Upon dividing by
/lulip-2 we obtain we(u),Ap(BO,BL)U)
4
T o extend (2.16) to any imation argument. when
Po,
BL
*-(B
( u ) , A ~ ( B ~ , B ~ ) u+)
u E D(Ap(BO,Bc))
wllull 2
(2.16)
we use an obvious approx-
Inequality (2.16) is obtained in the same way
(or both) are of type (11).
In the case
1< p c 2
the function
e(u)
nay not be contin-
u o u s l y differentiable; however a simple argument based on the Taylor
formula shows that if
u
is a polynomial (or, m o r e generally,
an
92
. FATTORINI
H .O
e(u)
analytic function) then
is absolutely continuous and the com-
putations can be justified in the same way.
Details are omitted.
We have completed half of the proof of the following result: THEOREM 2.1. Ap(BO,BL)
1< p <
Let
in
LP(O,L)
is given by
= rp,
rp(AP(B0’BL))
If
Then the angle of dissipativity of
m.
= arc tgE(&
-
p
1}1/2
(2.17)
is the (analytic) semigroup generated by
S ( - )
rp,
for every
0 < cp < rpp
there exists
UJ
= w(cp)
Ap(B0,B,)
then
such that (1.6)
holds. All we have shown s o far is that
rp(Ap(po,Bd))
2
cpp.
To
obtain the opposite inequality we must prove that (2.16) cannot
6 = tg ep
hold if
with
rp >
v,.
We sketch the argument for
boundary conditions of type (I).
Assume that (2.18)
Then we can find a complex number 1Zl2
Let
q
z
(say, of modulus 1) such that
-
+ (~-2)((Rez)~ 6(Rez)(Imz))
be a smooth real valued function in
= -p < 0 . 0
S
x
5
L.
Then the
function u(x) = ezq(x) belongs to
D(Ap(eo,B,)) q’(0) =
We have
Yo/%
satisfies the boundary conditions q‘(L) =
f
(2.20)
YL/Z.
= z ~ ’ ( x ) e ~ ( ~ ~ ~ ) q=( ~z$(x) )
u(x)u‘(x)
Accordingly, if
q
if
(2.19)
with
JI
real.
is the function in (2.11)’ f = -p$
2
= -plul
2
lu’
I2
.
(2.21)
Making use of this equality and estimating the rest of the terms i n ( 2 . 7 ) in a way similar to that used i n Theorem 2.1 we obtain an
ON THE ANGLE OF DISSIPATIVITY
93
inequality of the form
- 6 I m ( e (u),Ap(BO,BC)U)
Re(0 (u) ,Ap(BO,BL)U) 2
CII
UII 2-p
r,“
I2dx
-
CIl~11~.
(2.22)
Assume that (2.16) holds as well for the same value of
c
6.
Then
we obtain from (2.22) that
lulp-21u’I2dx
C’
5
’,I
lulPdx
q
for all functions of the form (2.19) where boundary condition (2.20). f o r instance taking
rl
(2.23)
satisfies the
But (2.23) is easily seen to be false,
to be rapidly oscillating function.
This
completes the proof of Theorem 2.1.
If .A
REMARK 2.2.
AOu(x)
is written in variational form,
+
b(x)u’
a(.),
b(*)
= (a(x)u’ (x))‘
we only need to require that
(x)
3.
0
(2.24)
C(X)U(X)
(resp.
uously differentiable (resp. continuous) i n observation will apply i n Section
+
Z
be contin-
.(a))
x
5
4,.
The same
4.
AN APPLICATION: COMPUTATION OF THE NORM OF CERTAIN MULTIPLIER OPERATORS
We limit ourselves to the following example. A 0 u = u”
operator ditions
in the interval
u ’ ( 0 ) = u ‘ ( r ) = 0.
0
is the multiplier operator
(for
-c
defined for
Re
ancosnx)
5 >
0;
x s rr
Then the semigroup
Ap(Bo,B,)
u(x)
L
in the space
LP(O,n).
the alternative formula
Consider the
with boundary conS(C)
generated by
Note that
S(C)
is
H. 0 . FATTORINI
94
can be used, where
u
is extended 2n-periodically to
i n such a way that
u
is even about
x = 0
-m
x = rr.
and
-
< x <
It follows
from (3.1) that the norm of S ( c ) i n L2(0,n) is I l S ( c ) l / 2 = 2 max[e’n n z 01 = 1. On the other hand, the norm of S ( c ) in
can be estimated from ( 3 . 2 ) :
C[O,n]
=4mK
(3.3)
thus we obtain the following estimate for the n o r m of LP(o,~), 2
5
p
Ils(c)ll,
(lc
in
using interpolation:
ia,
5
S(5)
I/ReS) (p-2)/2p > 1
(3.4)
(Rec > 0)
A far more precise estimate can be obtained from Theorem 2.1 o r ,
Noting that in this
rather, from a close examination of (2.7). a = 1, b =
case
c = y
0
= 0
= YI,
of ( 2 . 7 ) is non-positive for
6
5
we see that the righ hand side tg (pp
(~p,
given by (2.17)).
A ccord ingly,
Ils(c)II, = in the sector
(arg 5 1
(that
5 (pp
(3.5)
1 z 1
IIS(C)llp
is obvious).
On
the other hand, it follows from the necessity part of Theorem 2.1 that ( 3 . 5 ) does not extend to any sector
in other words there exists
15,1
1< p
5
> 1). P 2
5 =
5
I
5
(in fact, a sequence
(lcpl
with
Cp
5,
rp > c p P ;
with
such that IlS(c)II P > 1 The same results can be achieved in the range
+ 0) in the ray
(llS(cn)ll
5
larg
arg
(p
> ep,)
(for instance, by using duality).
ON THE ANGLE OF DISSIPATIVITY
4. ORDINARY DIFFERENTIAL OPERATORS I N
L1
AND
95
C
We consider again the formal differential operator A 0u(x)
+ b(x)u’(x)
= a(x)u”(x)
+
(4 1)
C(X)U(X)
under the assumptions on the coefficients used i n $ 2 ) . Al(Po,B,,)
in
has already been defined there for boundary
L1(O,C)
conditions of any type. i n the space
C[O,L]
The definition of the operator
of continuous functions i n
ed with its usual supremum norm) is D(A)
The operator
A(po,pL)u
u
consisting of all functions
0
L
x
A(B0,B,) 1,
S
(endow-
with domain
= A 0u
twice continuously differ-
entiable satisfying the boundary condition at each end.
Note,
however, that if the boundary condition at zero is of type (11), D(A)
will not be dense in
E = C[O,L] u ( 0 ) = 0.
by its subspace
E;
this is remedied replacing consisting of all
Co[O,&]
When the boundary condition at
L
u
with
is of type (11)
(resp. when both conditions are of type (11)) the corresponding subspace is defined by
CLIO,L]
u(L) = 0
defined by
u ( 0 ) = u(L)
(resp.
Co,LIO,&]
= 0).
The first difficulty we encounter here is that will not be dissipative for any
w
THEOFEM
4.1. (a)
WI
unless the boundary conditions
(if of type (I)) are adequately restricted. with the operator
A1(BO,p,)-
The same problem exists
A(pO,pI,). Assume the boundary condition at
0
is of type
(I). Then the inequality yoa(0)
-
a’(0)
is necessary for dissipativity i n
+
b(0)
L1(O,l,)
5
0
(4.2)
of any operator
A ( p ,p ) using the boundary condition p o at x = 0. If the 1 0 L at x = 9 is of type (I) the corresponding boundary condition p,
96
FATTORINI
H.O.
inequality is
-
Y!,a(&)
+ b(b)
a'(&)
5;
(4.3)
0
If (4.2) and (4.3) hold (or if the corresponding boundary conditions are of type (11)) then
A1(pO,BL)
for sufficiently large
w.
(b)
-
WI is m-dissipative in
L1(O,l,)
Assume the boundary condition at 0
is of type (I). Then the inequality
Yo
2
is necessary for dissipativity i n LIO,&])
cO
(1
(4.4)
PROOF.
Let
(L1)* = L"
A(Bo,BL)
at
where
5;
1
u E L
,
u(x)
- wI
is m-dissipative in
u f 0.
5
x <
of
in
u
(4.6)
= llUlI1l~(X)
= 0; at those
an element of
0
@(u)
consists of all the functions of the form
x
where
necessary for dissipativity of any
x < a
C[O,&]
w.
Then the duality set
u(x)
(which will be irrelevant in what follows).
in
is of type
(4.5)
= 0
is arbitrary save by the restriction that
4
x = &
0
sufficiently large
U*(X)
0
Bb
and ( 4 . 5 ) hold (or if the corresponding boundary conditions
( c O [ o , b ], c ~ , ~ [ o ,) L ]for
u
C,[O,b],
the corresponding inequality is
are of type (11)) then
u*
(Co[O,&],
using the boundary condition
If the boundary condition
Y& If
C[O,&]
A(Po,Bk)
of any operator
x = 0.
at
PO
(4.4)
0
D(Al(po,pc)
and zero for
a
Iu*(x)I
5
((uII1
T o show that ( 4 . 2 )
Al(flo,@.e)
let
0
is
< U < &,
which is positive in the interval
x > a.
and we have
the definition of
Then any
u* E @ ( u ) equals
IIuII1
ON THE ANGLE: O F DISSIPATIVITY
[ -
+r
+
u(0) = 1
and
= -(Yoa(0)
+
((au’)’
a’(0)
97
(b-a‘)u’ + cu)dx
b(O))u(O)
+
(4.7)
(a”-b/+c)udx
-
If yoa(0)
a‘(0) + b(0) < 0, the right hand side of
made positive taking
shows the necessity of (4.2);
C[O,l]
space
sufficiently small.
the argument for
This
(4.3) is identical.
(4.4). Recall that the dual space
We prove the necessity of of
a
(4.7) can be
can be identified linearly and metrically with the
C[O,l]
of all finite Bore1 measures defined in
0 5
x
5
1
endowed with the total variation norm, application of a functional p E C
u E C
to an element
given by
If the boundary condition at space is
~ ( ( 0 ) =) 0.
Co[0,4,]
of
4,
is
= (x;
C[O,&]
u(&) = 0
u ( 0 ) = u(G) = 0.
u E C
w
with
or where the two boundary conditions
The duality set
@ ( u )E
c
of an element
consists of all measures supported by the set
I u(x) I
yo < 0
= IIull]
such that
I+
m(u)
=
is a positive measure and
II UII ‘
w e can obviously construct a real element of
D(A(@,,@,)) that
then the relevant
consisting of all
llr-111 =
If
u(0) = 0
Similar comments apply to the case where the boundary
condition at are
is
0
(4.8)
udll
whose dual can be identified through (4.8) to
Co[O,G],
the subspace
[
=
(Ll,u>
having a single positive maximum at
u ( 0 ) = 1,
fixed later.
u”(0) =
Then
a
where
O(u) = ( 6 } ,
6
a
x = 0
and such
is arbitrary and will be the Dirac delta and we have
98
H . 0 . FATTORINI
a.
which c a n b e made p o s i t i v e by j u d i c i o u s c h o i c e of
The s t a t e m e n t s c o n c e r n i n g m - d i s s i p a t i v i t y of t h e o p e r a t o r s
-
A1(Bo,pZ,)
w I
and
A(BO,~{,)
- wI
can be r ead o f f t h e f o llo win g
(4.3),
two more g e n e r a l r e s u l t s where we show t h a t c o n d i t i o n s ( 4 . 2 ) ,
( 4 . 4 ) , ( 4 . 5 ) c a n i n f a c t b e d i s c a r d e d if one a r e n o r m i n g of t h e s p a c e s Let
1 5 p
0 5 x 5 L.
<
L’(0,C)
p
a,
and
C[O,L].
a continuous p o s i t i v e function i n
C o n s i d e r t h e norm
.e (
IIuIIp =
in
Lp(O,g). we w r i t e
Lp;
II*lIp
Clearly
UP I 4 X )
11 *(Ip
i s e q u i v a l e n t t o t h e o r i g i n a l norm of
t o indicate that
Lp(O,.f,)p
c a n be i d e n t i f i e d w i t h
Lp‘ ( O , ) , ) ,
u s u a l norm, a n e l e m e n t
u* E Lp’(O,L)
t h e formula
p
by
ep(u) = e ( p u ) ,
p = 1
0
(see (2.6)).
u E L’(0,d) L1(O,t)
p’-l
[
+
i s equipped w i t h
The d u a l s p a c e L p ( O , L ) *
P
p-’
= 1 endowed w i t h i t s
a c t i n g on
Lp(O,g)p
u*updx
through
(4.10)
t h e r e e x i s t s o n l y one d u a l i t y map
If
of
Lp
.e
( U * , U ) ~= 1
(4.9)
IpP(x)pdx)
r a t h e r t h a n w i t h i t s o r i g i n a l norm.
>
i s w i l l i n g t o perform
P
8
P
: Lp
Lp‘
given
t h e d u a l i t y map c o r r e s p o n d i n g t o t h e c a s e For
p = 1
t h e d u a l i t y s e t of a n e l e m e n t
up
c o i n c i d e s w i t h t h e d u a l i t y s e t of (see
-t
(4.6)).
We t a k e now
u
smooth and p e r f o r m t h e
c u s t o m a r y i n t e g r a t i o n s by p a r t s , a s s u m i n g t h a t tinuously d i f f e r e n t i a b l e a s well:
a s an element
p
i s t w i c e con-
ON THE ANGLE OF DISSIPATIVITY
{ (apP-’p‘)‘
+
It is obvious that
1 (a”-b’+pc)PP P
+ (a’-b)pP-lp’] [ulpdx.
(4.11)
~ ‘ ( 4 , ) can be chosen at will, hence
and
p’(0)
99
we may do s o in such a way that the quantities between curly brackets in the first two terms on the right-hand side of (4.11) are nonpositive, say, for
1< p
2.
5
Since the first two integrals
together contribute a nonpositive amount, we can bound (4.11) by an W‘Ilullp 5 cullu\l~where
expression of the form p.
Consider now the space
L1(O,L)p.
tion (4.10) the duality set u* E L m ( O , ~ ) with
all where
U(X)
f 0 and
limits in (4.11) as
Op(u)
u* x) =
Iu*(x p + 1
I
h
w
Again under the identificaof an element
II ulIp I u(x)
IIuI(
P
does not depend on
I -%x)
u
=
elsewhere.
consists of
II UPll I 4 x ) l - 1 w
We can then take
and obtain an inequality of the form (4.12)
in
L1,
The inequality is extended to arbitrary
by means of the usual approximation argument.
u E D(A1(BO,Bl,))
Now that A1(BO,~l,)-wI
has been shown to be dissipative, m-dissipativity is established by
100
H.O.
FATTORINI
The c a s e where one ( o r b o t h )
u s i n g Green f u n c t i o n s a s i n S e c t i o n 2 .
of t h e boundary c o n d i t i o n s a r e of t y p e (11) i s t r e a t e d i n a n e n t i r e l y s i m i l a r way; n a t u r a l l y ,
t h e u s e of t h e weight f u n c t i o n i s un-
necessary i n the l a s t case. We have completed t h e proof THEOREM 4 . 2 .
Let
Then t h e o p e r a t o r
s(.)
Here
in
IIS(t)\lp
be a p o s i t i v e t w i c e c o n t i n u o u s l y d i f f e r e n t i -
0 5 x 4 t,
able function i n
group
p
such t h a t
g e n e r a t e s a s t r o n g l y c o n t i n u o u s semi-
A1(BO,BL)
such t h a t , f o r some
L1(O,L)p
i n d i c a t e s t h e norm o f
Assumption (4.13) ( r e s p .
L1(0,4,)p.
of
boundary c o n d i t i o n a t
0
(resp. a t
> 0,
W
a s an o p e r a t o r i n
S(t)
( 4 . 1 4 ) ) does not apply i f t h e 4,)
i s of t y p e (11).
To prove a s i m i l a r r e s u l t f o r t h e o p e r a t o r space
C
A
we renorm t h e
o r t h e c o r r e s p o n d i n g s u b s p a c e by means of
(4.15) where
in
p
is a positive,
0 5 x 5 4,.
twice continuously d i f f e r e n t i a b l e f u n c t i o n
The u s e of t h e weight f u n c t i o n
p
i s a g a i n un-
n e c e s s a r y when b o t h boundary c o n d i t i o n s a r e of t y p e (11): below i n d e t a i l t h e c a s e where
Po
and
lfmixedfl c a s e b e i n g e s s e n t i a l l y s i m i l a r .
B4,
we t r e a t
a r e of t y p e ( I ) , t h e
Choose
p
i n s u c h a way
that ~ ' ( 0 +)
if
yo
<
0
rOp(o)
2 0
( r e s p . yc > 0 ) .
( r e s p . ~ ' ( 4 , )+ Y ~ P ( G ) 4 0 ) The d u a l o f
C[O,G]
(4.16)
equipped with
ON THE ANGLE OF DISSIPATIVITY
11 . / I p
C[O,L],
can again be identified with
u E C[O,L]
acting on functions
(Ll,u) =
QI
lip))
=
Op(u)
sets
i,“
U(X)P
up)
I
mp(u)
p E Z[O,L]
(4.17)
(X)P(dX) *
p E
c
C*
as a n element of
is
and the identification of the duality
Op(u)
is the same as before;
with support i n
(or
Ip (dx)
an element
through the formula
Accordingly, the norm of a measure still
101
= ( x ; lu(x)p(x)l
is a positive measure i n
= Ilull,]
mp(u)
p E Z
consists of all
with
same comments apply of course to the spaces
Co,
and such that upp lip11 =
I)UI/~.
C L , C0,&
the corresponding measures are required to vanish at
The
where
0, 4 ,
and
0
4 . We now show that large enough.
u(t) - ,!Y
then
A(oo,p,)
- WI
u’ (0) = y o u ( G ) ,
Observe first that if up
is m-dissipative for u’(C)
w =
satisfies the boundary conditions
where
Using elementary calculus we show that for any u E D ( A ( B O y D )),
L
u
#
0
the set
Y 0 , P ’’4 YP
> 0,
mp(u)
does not contain either endpoint if both
s o that
(4.20) On the other hand, if either
Y0,P
Or
Y&,p
vanish,
m,(u)
may
contain the corresponding endpoint but we can prove again that (4.20) holds. Writing q = p 2 we have ( l u p 1 2 ) ’ = 2 ( u1u 1 ’ + u2 u 2 ’)q + 2 2 2 1 + 2(u; +u;Z)q + + (u1+u2)q’, ( I u P 1 2 ) ” = 2 ( u 1u”+u2u’;)q 2 2 + 4(u1u;+u2u;)q‘ + (u1+u2)q”. Hence, it follows from (4.20) that
102
. FATTORINI
H .O
(4.21)
for some constant pative.
(XI
That
which shows that
W,
-
A(po,bt))u
= v
A(Po,Pg)
- wI
has a solution
u
is dissifor all
v
is once again shown by means of Green functions. The following result, that settles completely the question of angles of dissipativity in
LL
and
C
is a simple consequence
of the identification of angles of dissipativity in
L2
in Theorem
2 . 1 and of the theory of interpolation of operators between
L2
and between
3 . (a)
- L~I p,,~,)) in
C
L2
and
C.
Let ( 4 . 2 ) and ( 4 . 3 ) be satisfied. is m-dissipative in
= 0.
L1 and
(b)
(Co,Cg,C,,t)
L~
for
co
Then
sufficiently large
The same conclusion h o l d s for if
(4.4) and (4.5) hold.
ON THE ANGLE OF DISSIPATIVITY
5.
ELLIPTIC PARTIAL DIFFERENTIAL OPERATORS IN
1 <
LP,
<
m
Since the conclusions and some of the arguments are the same as those for the one-dimensional case we only sketch the details. T o simplify the notation we write m
A.
=
m C Dj(a. (x)Dk) j=1 k=l Jk
C
The domain
+
A.
in variational form,
m C bj(x)Dj j=1
is bounded and of class
C(2).
+
(5.1)
C(X)
We assume that the
-
and that and the b are continuously differentiable in R a jk j c is continuuus in 5. If the boundary condition is of type (I) we suppose also that boundary in
r.
y
is continuously differentiable on the
Finally, we assume that
. A
is uniformly elliptic
0 , that is, that there exists a constant n > 0
The domain of the operator
Ap(p)
such that
is the subspace of
W19p(Q)
consisting of functions that satisfy the boundary condition
r
(in the sense of Sobolevfs imbedding theorems); here
p
at
WIPp(n)
is the Sobolev space of all functions with partial derivatives of order
52
in
LP(0).
F o r a proof that the equation
(XI ha3 a solution instance [ 11
.
u E D(A
P
(p))
-
A,(~))U for every
=
(5.3)
f
f E LP(n)
see for
T h e dissipativity computation (2.7) becomes
104
H.0 . FATTORINI
T 5
(p-2)
f
+ 1
I
I u I P-4(CCa
i,
;Dku))
dx
f
-
(pc
jkRe (;Dju)Im(
C Djbj]lulPdx
(5.4)
Jn TJe have r
1
-(p-2)
IuI P'4{CCa
I
'L3 i 6 ( p- 2 ) \n
-
(
jkRe(iiDju)Re(?iDku))dx
I u I P-4( CC a
kRe (
ju ) Im ( uDku) ] d x
IuIP-2(CCajkDjuDk;)dx
n
where
+
fjk = (CDju)(uDk;)
+ Let matrix
*
(p-2)(Re(cDju)Re(GDku) zl,.
Z = ( z zjk =
.. ,zm
) jk
8Re(GDju)Im(GDku)],
be arbitrary complex numbers.
(5.6) Consider the
with elements
zjZk + a ( (Rez .)(Rezk) J
f
6 (Rez .) ( I m z k ) ] J
which is nonnegative if 1 + 5
2
a+2 (a')
(5.7)
105
ON THE ANGLE O F DISSIPATIVITY
in view of ( 2 . 8 )
and following comments; it follows then that
positive definite. see that i f 5
(5.7)
V1z121c12
Z is
A s in the one dimensional case it is easy to
Z
is strict then
c1 ,...,<,,
for all complex
CXzjkcjck
will satisfy
,...,
z1
zm.
2
This is seen
to imply that CCajkfjk
2
vlul2
We estimate next the other terms in field
p = (pl, ...,pm)
such that
(p,?)
+ p I uI p’2Cp
(5.4).
(5.8)
Consider a real vector
continuously differentiable in
= y
normal vector on
c ~ D2 L ~
r.
-
p-lb
Then
jRe(;Dju).
on
r,
where
div(lulPp)
v
-
n
and
denotes the outer
= lul’div
+
p
By virtue of the divergence theorem,
$2 9
where we have used the inequality
This is also used in an obvious way to estimate the fourth volume integral in
(5.4).
Taking
E
> 0 sufficiently small i n the result-
ing inequality and in (5.9) and dividing by
IIullp-2
we obtain an
inequality of the form Re(e (u),Ap(8 ) u >
4
which is then extended to
*6Im(e ( u ) ,A~(B)u, + D(Ap(B))
wllull
2
(5.11)
in the way indicated before.
106
H.O.
FATTORINI
As in the one dimensional case, (5.10) shows that cP(Ap(B))
2
vp9
cp(Ap(B))
5
(pp
where
vp
is given by (2.17).
The proof that
is essentially similar to that for the case
m = 1
and is therefore omitted.
6. ELLIPTIC DIFFERENTIAL OPERATORS IN
L1
The assumptions on the operator
A
are the same as in the previous section. E = L1(n). Condition (4.2) becomes
AND
We treat first the case
A,(@)
On the other hand, if (6.1) is satisfied then
w
large enough.
fined as the set of all f
( = A,(@)u)
where
Ab
(if
p
. d
p
is
DVu = y u
then
F o r a proof that, for
8' X
is
proof that
A,(p)
-
is
wI
D(A1(B))
is de-
n
-
Dk) Cb .Dj + c, and jk J satisfying the adjoint
is of type (11) then Dv"v = y ' v
with
y'
p'
= p;
v
if
+ C b.v.).
= y
J J
large enough,
(XI has a solution
The domain
CCDj(a
is an arbitrary smooth function in
p'
A1(p)
Ao.
such that
is the formal adjoint
boundary condition
for any
such that there exists
u E L1(n)
L1(n)
in
n
and on the domain
and (6.1) is necessary for dissipativity of
m-dissipative for
C
u E D(Al(@))
- A,(@))u f o r any
= f f E L1(n)
see
[a].
The
is dissipative follows from the computation
below (the higher dimensional counterpart of (4.11)) where we show i n fact that condition (6.1) can be bypassed through a renorming of the space
L1(n).
As in the case
m = 1
we define
ON THE ANGLE OF DISSIPATIVITY
UP
The space
LP(R)
(6.4)
equipped with this norm will be denoted
L:(n).
The identification of the dual space is the same as that in Section
4; we use the same notation for duality maps and for appli-
cation of functionals to elements of
Assuming that
L:(n).
p
is
twice continuously differentiable we obtain
-
(p-2)
l ~ l ~ - ~ [ Z Z a ~ ~ R e ( u D j u ) R e ( cpPdx D~u)]
I ulP-2[CZajkRe(&Ju)Dkp]
-
+;[ -
[n
R[
(pc
- ZDjb j) 1 U I
pP-'dx
'pPdx
R
(Cbj D j p )
I uI 'pP-'dx
(6.5)
We transform now the third volume integral keeping in mind that p(ulP-2Re(cDju) = Dj( lulp) the vector
U = (U..) J
and using the divergence theorem for
of components
U. = ~ u ~ p p p za.. - ~~~p J Jk
(1
5
j
L
m).
Once this is done, the right hand side of (6.5) can be written
FATTORINI
H.O.
108
I
-
(p-2)
[
IU I
P-4(CCa jkRe (;Dju)Re
We can now choose
DVp
(CDku)] p 'dx
at the boundary in such a way that
quantity between curly brackets in the surface integral in is nonpositive.
Noting that the first two volume integrals combine
to yield a nonpositive amount we can bound the right hand side of
(6.5) by an expression of the form depend on
p.
where
Wl)ullp
w
does not
Using a limiting argument as in the one-dimensional
case we obtain
where the expression between brackets indicates application of the functional the norm of
u* E O ( u ) C Li(Q)
E Li(n)
A,(B)u
and
1) *Ilp
is
The customary approximation argument shows
Li(0).
that (6.7) holds in
D(A1($)).
of course unnecessary when
The case
to
E = C(E)
Renorming of the space
L'(f2)
is
is the Dirichlet boundary condition.
fj
if the Dirichlet boundary con-
(C,(Z)
dition is used) is handled in a similar way.
Condition
(4.5) is
now Y(X)
5
0
(X
E
r)
(6.8)
which condition is necessary for dissipativity of
If (6.8) is satisfied then C(n')
(C,(fi))
if
w
A(B)
is large enough.
defined as the set of all
u
- WI
A($)
for any
is m-dissipative in
The domain
that belong to
W2'P(n)
D(A(e))
is
for every
109
ON THE ANGLE OF DISSIPATIVITY
p
5
1,
satisfy the boundary condition
AOu E C ( 5 )
The fact that, for
(Cr(fi)).
(XI has a solution
u E D(A(B))
-
r
on
@
1
and are such that
large enough,
(6.9)
A ( ~ ) ) u= f
f E C(n)
for arbitrary
particular case of results in [ 5 ] .
The proof that
is a
(C,(fi))
A(B)
pative if (6.8) holds corresponds to the particular case
is dissip
= 1
in
the computation below, which shows that even i f (6.8) does not hold,
- wI
A(p)
will be dissipative f o r
placement of the norm of
C(5)
w
large enough after the re-
by the equivalent n o r m
(6.10) where
p
6;
is a continuous positive function in
the identifica-
tion of the duals is achieved along the lines of Section 4.
If
is of type (I) and condition (6.8) is not satisfied we select
@
p
twice continuously differentiable and such that
(6.11) u
Let on
Y
be a smooth function satisfying the boundary condition and
m p ( u ) = Ex; Iu(x)Ip(x)
is supported by
mp(u)
sitive measure i n up
m,(u).
= Ilull,]
and is such that
so
upp
$
that any
p E 0 (u)
(or
is a po-
up)
P
-,
Since
Dvu(x) = y(x)u(x)
at the boundary,
satisfies the boundary condition
(6.12) where
(6.13) The argument employed i n Secion 4 shows that the boundary
r
if
y(x) < 0
everywhere on
mp(u)
I'
so
does not meet that
(6.14)
..
H 0 FATTORINI
110
where
indicates the Hessian matrix o f
#(x;g)
other hand, i f points of Writing
r
y(x) = 0
for some
r,
at
x.
On the
may contain
mp(u)
but we can prove in the same way that (6.14) holds.
q = p2
and
+ iu2
u = u1
D j l ~ p= ( ~2(ulDju1 + u2Dju2)q jk = 2(ulD D u1
+
x E
g
+
j k
u2D D u2)q
2(ulDJul + u2DJu2)D ’ kq
+
=
real we obtain
+ (u1+u2)DJq, 2 2 ‘
2(ulD ku1 + u2D ku2)Djq
(6.14) that if
11 u/I-2(ulDjul
-
1
5
IIu((-~~uI
On the other hand, again with
Accordingly, if
ul, u2
DjDklup12 = ’ k ’ k + 2(DJu1D u1 + DJu2D u2)q +
Accordingly, it follows from Re(u-lDju) =
with
+
(ul 2
+ u2)D 2 jDkq.
x E mF(u),
+ u2Dju2) 2 -1 j q D q =
x E mp(u),
-
1
-1 j D r(.
we have
r ~ .E Q p ( u ) ,
(6.15)
for some constant, showing (after the usual approximation argument) that
A(@)
- WI
is dissipative.
Theorem 4.3 has an obvious analogue here: A1(P)
- WI
is m-dissipative for
A similar observation holds for
w
when (6.1) holds,
large enough but
cp(Al(@))
= 0.
A(a).
We point out finally that a ”multiplicative” renorming of the spaces
LP(n)
like that used in
angle of dissipativity of
A,(@).
L1(n)
does not change the
ON THE ANGLE OF DISSIPATIVITY
111
REFERENCES
1.
S. AGMON,
On the eigenfunctions and the eigenvalues of general
elliptic boundary value problems, Comm. Pure Appl. Math. 15
( 1 9 6 ~ )119-142. ~ 2.
H. B d Z I S and W.A. STRAUSS, equations in
3.
Ll,
Semi-linear second-order elliptic
J. Math. SOC. Japan 25 (1973), 565-590.
E.A. CODDINGTON and N. LEVINSON,
Theory of Ordinary Differ-
ential Equations, McGraw-Hill, New York, 1955.
4. A. PAZY,
Semi-groups of Linear Operators and Applications to
Partial Differential Equations, Univ. of Maryland Lecture Notes #lo, College Park, 1974.
5.
B. STEWART,
Generation of analytic semigroups by strongly
elliptic operators, Trans. Amer. Math. SOC. 199 (1974), 141-162.
Departments of Mathematics and System Science University of California
Los Angeles, California 90024
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Functional Anolysis, Holoniorphy and Approximation Theory 11, G.I. Zapnta (ed.) 0 Elsevier Science Publishers R. V. (Nortll-Holland), 1984
113
TWO EQUIVALENT DEFINITIONS OF THE DENSITY NUMBERS FOR A PLURISUBHARMONIC FUNCTION I N A TOPOLOGICAL VECTOR SPACE
Pierre Lelong
1. INTRODUCTION
In the following C;
field
E
will be a vector space over the complex
G C E,
in a domain
plurisubharmonic functions
we denote by
The topology is not supposed at
f.
I
the beginning to be locally convex; neighborhoods of the origin
?,W c W
ed if and only if the topology of
E~ = E/N
for
E
+
< f(x)
have
f (x+N) < f(x)
x'E
x + N;
projection
C ,
> 0, for
E
+
therefore TT:
x'E
= f(x)
f(x')
E + El,
1).
L
we write
space
f
reduces to
is disk-
We may suppose
E P(G)
f
of the origin such x
+
W.
W + N = W,
Then by
we
is upper bounded for for
x'- x E N.
f(x) = fl0n(x)
plurisubharmonic function defined i n the domain given function
W
If it is not, we pass to
W
f (x' )
and
E
111
the upper semicontinuity of
there exists an open neighborhood f(x')
(we recall that
h € C,
for
6. BY
denotes a basis of disked
E
in
0
to be Hausdorff.
N =
the class of
P(G)
f l E P(G1)
Using the
and G1
fl
is a
= n(G) ;
the
defined in the Hausdorff
El. We denote by
Do
the compact disk
IuI
4
1
in
= x + Doy its linear image in E. A set A C G X*Y such pluripolar in G if there exists f € P ( G )
D
AC
(1) The set
A
A'
=
[X
E G ; f(x)
=
C
and by
is called
--I.
is called a cone with vertex the origin if and only if
114
PIERRE LELONG
?,A = A
1 f 0.
1 E C,
for all
if there exists a neighborhood pluripolar in
A cone is called pluripolar in U
of its vertex such
x + W,
Equivalently:
gx
- = [x E
q
=
f(x)
f(x) f r),
U
f
is
which is supposed to be
we define in r
E
> 0 such
the set
f =
-
on D 1. X,rY
is the union of the complex lines through
which contain a disk
.r
W E C,
and
gx = [y E E ; there exists
(2)
n
(see [ 4,b] ).
U
For a plurisubharmonic function defined in
A
E
Dx,y
-=I;
G ; f(x) =
,
y f 0
which is in the pluripolar set
if such lines d o not exist, and if
is reduced to the origin of
-m,
gx
-m;
we have by translation
and its translated
x
TI
-
x
[ gx
n
E;
it is empty if
W] c ( q - x )
are pluripolar, then
n
W;
gx
the set is a pluri-
polar cone.
In function
the density number
Cn
f
in
x
v(x,f)
of a plurisubharmonic
appears to be like a multiplicity and character-
izes the concentration of positive laplacian measure point
x;
Af
near the
which is called the Lelong number (see [ 3 ]
v(x,f),
and
[ 6 ] ) is an invariant of the one to one holomorphic mappings (see
[ 6 ] ) and it has a geometrical (for complex analytic geometry) meaning.
I recall here for the convenience of the reader three (equivalent) definitions of
(I) current (11)
w (x,f).
V(x,f)
n i
is the Lelong number of the closed and positive
1 aaf =
n
c dd f
is the regular density (in real dimension 2n-2) 1 of the positive measure a = - A f , defined by the quotient 2rr
a(x,r)
w(x,f)
is the mass of
volume of
B(0,r)
in
a Rm.
in the ball
B(x,r)
and
~ ~ ( ris) the
TWO EQUIVALENT DEFINITIONS OF THE DENSITY NUMBERS
(111) A direct calculation of value
X(x,r,f)
of
f
v(x,f)
on the sphere
115
is possible using the mean
S(x,r),
v(x,f) = lim (log r)-' r=O
0 < r < 1:
for
X(x,r,f).
None of the definitions (I), (11), (111) is available if
is
I succeed in [&,a]
E.
defined in an infinite dimensional space
f
in
giving a suitable definition using the property of the traces of
f
L
on the finite dimensional affine subspaces
through
flL x,
specially the complex lines
L = L = X,Y
(4) If
fIL
[Z
E E ; z = X+UY, u E C].
is not the constant
the restriction of a
TIL
(or, equivalently
-m
is subharmonic and for
set for the direction
dependent of
y
given by
inf
y
v (x,f
L
IL).
x r
By definition, function
v(x,y,f)
cp(u) = f(x+uy)
then
with exception
we get a value in-
for the direction
y:
2rr
v(x,y,f) = lim (log r ) r=O
(5)
$ ' gx),
More precise we define
f
the tangential density number of
L,
of
fIL
y
f(x+re
i0
y)d6.
is the density number of the subharmonic in
u = 0.
Then the definition given in
[&,a] was: DEFINITION 1. G C E,
the density number in v(x,f)
(6) and
Given a function
v(x,y,f)
f
plurisubharmonic in a domain
x E G
is defined by:
= inf v(x,y,f) Y
for
has the value (5).
I give here a different definition. tion of C . Kiselman (see [ 3 ] and [&,c]) spaces of
y E E-10)
(log r)-'
M(x,r,f)
for
I t aroses from a ques-
on the limit in normed
r I 0
and
PIERRE LELONG
116
M ( x , r , f ) = sup f ( x + x ’ )
I)x’II s r
for
X’
for
f E P(E)
and
b e i n g a Banach s p a c e ( s e e l a t e r Theorem 2 ) .
E
Given a domain
DEFINITION 2.
and a d i s k e d n e i g h b o r h o o d i s u p p e r bounded i n
(7)
i n a complex t o p o l o g i c a l s p a c e E ,
G of
W
x+W C G .
f E
and
0,
P(G);
0 < r < 1
Then we d e f i n e f o r
~ ~ ( x , r ,= f )s u p f ( x + x ‘ )
for
we s u p p o s e f
x’E r W
and v W ( x , f ) = l i m ( l o g r)-1 ~ ~ ( x , r , f ) .
(8)
r=O
We p r o v e i n t h i s p a p e r
= vw(x,f).
\I(X,f)
A consequence i s :
i f we d e n o t e by
e d n e i g h b o r h o o d s of vW(x,f)
such
0
t h e family of t h e disk-
i s u p p e r bounded i n
f W E
d o e s n o t depend of
4
@kC
W € @:
given
A
we c o n s i d e r t h e cone and
y(A) obtain
MA(x,r,f)
i s n o t Ittoo v(x,f)
v(x,y,f)
then
u hA, x
instead
A C W,
and d e f i n e f o r
MW(x,r,f)
i n (7) i f
The d e f i n i t i o n o f t h e “ s m a l l n e s s t 1 t o
s p a c e , i t i s s u f f i c i e n t for
t o calculate
y(A) =
w i l l depend of t h e s p a c e
then the r e s t r i c t i o n f o r
W,
f o r a disked s e t
W E :@:
It i s possible t o use
+
@;.
More p r e c i s e r e s u l t s w i l l b e g i v e n : and
x
y(A)
y E y(A)
for
v(x,f) = inf v(x,y,f) for Y i s not p l u r i p o l a r i n E .
E.
If
E
i s a Fr6chet
t o be a n o t p l u r i p o l a r c o n e ; and
y E y(A), y E y(A),
u
\r
0
of
f(x+uy) enables
and t h e n we w r i t e if
y(A)
i s a c o n e which
TWO EQUIVALENT DEFINITIONS OF THE DENSITY NUMBERS
117
2. THE TANGENTIAL DENSITY NUMBERS
Given D
X,Y
= x
+
f
G C E
plurisubharmonic in a domain
Doy C G ,
cn
we define
and a disk
f(x+re i9y)de
L(x,y,r,f) = 2,
m(x,y,r,f) = sup f(x+uy)
for
IuI 5
(x,y) -+ .P,
By a classical result, the functions
r
and
. (x,y)
are defined locally and are plurisubharmonic functions of if the compact disk 0 5
r < 1.
D
Then for
-t
m
(x,y)
is moving in G and r is fixed X,Y D c G we define two tangential density X,Y
numbers
If
f(x) f
y E E,
-m,
v(x,y,f) = v(x,y,f) = 0
we obtain by (9):
for all
as an obvious consequence of
It is convenient to change the signs in order to calculate
v
and
v1
functions.
by increasing limits of plurisubharmonic and negative For given
x E G,
of the disked neighborhoods is bounded above for of
f.
For
W E @: .
x'E and
Then we obtain for
let us consider the family
W
W;
of
f
PI;
f(x')
4
such
0
and
C
@x
f(x+x')
by the upper semicontinuity
@
a
x+W C G
PI:
for
0 < r < 1:
XI-
x E W,
we define
118
PIERRE LELONG
Given a domain
PROPOSITION 1.
G
C
E, 5 E G ,
and
f
harmonic in
G,
there exist two disked neighborhoods
of
-v
and
such
0
y E W' of
are defined by (10) for
as limits of the increasing functions
r,
r
for
functions of
\
y E E
For fixed
0.
(x,y)
Moreover for fixed for
-vl
for x,
x
r,
-5
0
W E P,
the upper bound of
< r < 1, we obtain Q (x,y)
functions of
E W,
for
W'E
and are negative.
for
Q1
v(x,y,f) ),
E C, x f 0 .
such
W + W'E
@'
and
s
Then in (lo),
as negative plurisubharmonic y E
E W,
@
5 + W + W'.
in
f
and
-5
x
Ql(x,y,r)
vl(x,y,f).
F o r the proof, we take
for
y E W'
-5
W'
are plurisubharmonic
V(x,Xy,f) = v(x,y,f)
which satisfies
a
E W,
and
Q(x,y,r),
Q,
and
W
there exists a continuation of
The same properties hold for
denote by
Q
x
plurisub-
w'.
The graph of the function
v = log r + ~(x,y,r,f-a)= v ' ( v ) is increasing and convex for defined by
v'2 C ( v ) ,
the origin
v = v' = 0 .
increases as
v
v
-a 4
-a
4
< v 0,
4
0.
In R 2 (v,v')
the set
is a convex set and contains
This has for consequence that
is decreasing and tends to
-m.
$(x,y,r)
Then (10) defines
-v(x,y,f)
by an increasing limit of negative plurisubharmonic
functions
Q(x,y,r)
x
in
x
-5
E W,
y E W'.
Moreover for
E 6,
Writing
r'
= 1 1I-'r,
we obtain by (11):
= lim (log r' =O
1 -1 7 ) .t,(x,y,r' ,f-a)
1
#
0,
119
TWO EQUIVALENT DEFINITIONS OF THE DENSITY NUMBERS
v(x,Xy,f) = V(x,y,f)
and
Now for
m > 1, m E !N
is given by
Then
v(x,y,f)
#
),
and y E W‘.
0
y E mW’ ,
~ ( x , y , f ) for
It is obvious that (10)
-
0 < r < 1 m
y E mW‘ and
E C,
),
the definition of
-1y,f). = w(x,m
V(x,y,f)
remains true for
is proved for
.
x E G,
is defined locally for
u
y E
mW’ = E.
It
m
is given as a limit of an increasing sequence of negative plurisubharmonic functions and The same holds for
REMARK 1.
y = 0, and
For
= -w1(x,O,f) =
-V
=
x E
-0).
X E
for
C,
),
#
-
-m.
0.
v1(x,y,f).
y E gx,
For
= v(x,y,f)
w(x,hy,f)
7
For
we obtain
-w(x,y,f)
= [x E G, f(x) y = 0, x
=
:,
-1,
= -vl(x,y,f)
-v
we have
x,O,f) =
-v(x,O f) =
we obtain
(X,O,f) = 0 . 1
3 . EQUALITY
THE TWO TANGENTIAL NUMBERS
OF
Now we prove Given a plurisubharmonic function
PROPOSITION 2.
f
in a domain
G c E , for all x E G, y E E , the equality, for the definition, see[10]:
(14)
V(X,Y,f)
holds.
If
#
y
density number
0,
= Vl(X,Y,f)
the value (14) will be called the tangential
v(x,y,f)
of
f
in
x E G
for the direction
By the preceding remarks, (14) is proved for for W E @:
for
We have only to prove (14) for
y E gx.
and
IuI
~ ( u =) f(x+uy)
< 1.
x E G,
y = 0 y
y.
and
E W,
and
being a subharmonic function of u E C,
We put
It is sufficient to prove that the quotient by
log rl
of the
PIERRE LELONG
120
m(r)
difference
was given in
n = 3
For
and
(c,
and
& ( r ) tends to z e r o f o r
0; such a result
\r
a direct calculation gives a more precise result.
R,
1.1
subharmonic for
0 < R < 1, then f o r m(r)
and
&(r)
r
.
by V. Avanissian [l]
Cn
(I) Given a function ep(u),
LEMMA 1. u E
-
r
< 1,
in the interval 0 < r < R,
defined by (15): 2Cr R-r
p(R)
is the mass of
1
1
p =
UI
in the disk
5
R
and
211
1 C = 2n
lcp(Re
ia
)Ida.
'0
(11) Moreover for
r
I
0:
- t ( r ) ]=
lim [ m ( r ) r=O
0.
The proof of (16) u s e s the classical Riesz decomposition of
cp
in
IuI < R:
ep(u)
+
= HR(U)
l o g lu-al
dCL(a) [a I
HR(u)
is harmonic for
equal to
HR(0).
1 -1
Then f o r
0 5 m(r)
+6"
< R;
for
C
HR
0 < r < R:
- t ( r )S
W(t)log(r+t)
the mean value
-
sup HR(reie)
I"
dM(t)sup(log
-
HR(0)
+
r , log t).
0
F o r the first difference we have the bound
IHR(reie)
- HR(0)I
2n
s R 2r - r 2n
[
R-r
= 2Cr
Icp(Reie)lde
j0
Otherwise the two integrals in
i,'
dP(t)
log
(17) are bounded by
+ 2
r+t
c
dp(t)
log
.
is
TWO E Q U I V A L E N T D E F I N I T I O N S O F THE D E N S I T Y NUMBERS
An upper bound of t h e i n t e g r a l i n ( 1 8 ) i s given by p ( R ) W e o b t a i n f o r given
E > 0
o s m(r)
(19)
and
0
- & ( r )s
E
< r < r log 2
+
121
log(1
+ r ). 0
< R r[R-ro
r
which proves ( 1 7 ) Then by ( 1 0 ) and ( 1 6 ) t h e P r o p o s i t i o n 2 i s proved f o r
4.
EQUIVALENCE O F THE D E F I N I T I O N S 1 AND 2.
F i r s t we g i v e p r o p e r t i e s of t h e f u n c t i o n
i t i s of t h e t a n g e n t i a l numbers variable i n
y
x E G , y E E.
v
and
v1
F o r a r e a l valued f u n c t i o n
h(y)
hx(y) = l i m inf h ( y ' ) Y1' Y
and
x E G,
and
we denote h*(y) = l i m sup h ( y ' ) YI' Y
h.
THEOREM 1. ( I ) The lower r e g u l a r i z a t i o n of
l i m inf W(x,y',f) Y y E E.
f o r fixed
E.
t h e t w o r e g u l a r i z a t i o n s of
Y" of
y E E + v(x,y,f),
h a s a f i n i t e p o s i t i v e v a l u e which does n o t depend
Let us denote
(20)
it i s
W(x,y,f)
i t s value.
W(x,f)
w(x,y,f) 2 W(x,f)
for a l l
Then y E E
and
(11) F o r g i v e n
cone g :
gk
x,
of v e r t e x
c o n t a i n s t h e cone
The cone
g:
(111) If
the s e t 0;
g:
= 6
[y E E , w(x,y,f) i f and only i f
f(x)
#
is a
-m;
gx.
i s a p l u r i p o l a r cone i n E
> W(x,f)]
E
i s a F r 6 c h e t space o r i f
or E
g:
= E.
i s a B a i r e space and
122
P I E D LELONG
i s a continuous plurisubharmonic function (it i s
f
tinuous) then
#
gk
i s a p l u r i p o l a r cone i n
E
h(y) = -v(x,y,f)
Let us w r i t e
X
f o r fixed
X #
h(Xy) = h ( y )
Then
i s a plurisubharmonic f u n c t i o n i n
h*(y)
C,
€
i s con-
E.
p o s i t i o n 1,
for
ef
x;
and
0
by t h e Proh(y)
0.
5
because i t i s
E
t h e upper r e g u l a r i z a t i o n of t h e u p p e r e n v e l o p e i n ( 1 0 ) of $(x,y,r) h*(y) h"
h*
0,
5
which i s a p l u r i s u b h a r m o n i c f u n c t i o n s of
0,
5
i s u p p e r bounded i n
y € E
for
and f i x e d
x.
y.
By
and t h e n , i t i s a c o n s t a n t
E,
W e have
h(y)
h* = V ( x , f ) ,
G
which
proves ( 2 0 ) ;
( 2 1 ) i s a consequence of t h e d e f i n i t i o n of t h e r e g u -
larization.
If
f(x)
#
V ( x , f ) = 0.
and t h e r e f o r e
f(x+uy)
4
f(x) =
by
G
-m.
has a f i n i t e value; there e x i s t s
IuI < 1 ;
for
-CU
= inf v(x,y,f)
y
v(x,y,f) <
then
for a l l
>
[ x E G; v ( x , f )
The s e t
tained i n the s e t defined i n v(x,f) 2 0
V(x,y,f) = 0
we have
-m,
i s con-
03
x € G,
For e a c h
#
0
such v(x,f) =
and
m
y;
has a f i n i t e value.
Y
To prove (11): g:
by
~ ( x , y , X f )= w ( x , y , f )
f(x)
#
i n which and
For
-m.
f
v(x,y,f)
y E gx,
0 E gk
and
f(x) =
if
y
#
To end t h e proof
By ( I ) , we have
then
-m
Therefore
-OD.
y E g:,
gx c gk
and
S(Y)
and
s,(y)
= sup SJY)
0
5
4
0.
n V(x,f),
we have i n s*(y)
E
0.
E
and
0,
By t h e
0.
8
gk =
if D
X1Y
v(x,y,f) =
+a,,
i s proved.
of (11) and (111),we w r i t e f o r
sn(y) E P(E)
By t h e d e f i n i t i o n of
and
X #
C,
there e x i s t s a disk
0,
has the constant value
> v(x,f);
1 E
i s a cone of v e r t e x
= [ y E E ; w(x,y,f) > y(x,f)]
Remark 1, we have seen:
for a l l
rn + 0:
TWO EQUIVALENT DEFINITIONS O F THE DENSITY NUMBERS
Now we have t w o p o s s i b l e s i t u a t i o n s : a)
Yo
Suppose t h e r e e x i s t s
such
S(Y0)
= sup s n ( y o ) = 0.
Then
p a s s i n g t o a subsequence i f n e c e s s a r y , we may P suppose C l s n ( y o ) l < m . Then Vp(y) = C s n ( y ) < 0 i s a d e c r e a s n 1 Then i n g sequence o f p l u r i s u b h a r m o n i c w i t h f i n i t e l i m i t i n y o .
by
sn(yo)
V(y) = s(y)
<
0,
S
C sn(y)
i s plurisubharmonic
n 0,
V(y) =
and
If and
g:
yo
which makes t h e proof
5)
~ " ( 2 )=
k-everywhere s p a c e , and
on
s*
Now, values i n
are closed. 9
gk
= @
#
E;
supp
y
g.
u
1-1
C
P.
E
E G,
there exist
of s u p p o r t i n
P
such
(dil(a)s(6+a) s(y) = 0
we o b t a i n
if
A s a consequence:
E
s ( y ) = s*(y)
v ( x , y , f ) = w(x,f)
we have
a n d , by (11))t h e cone
suppose
2
( a c t u a l l y p o l y c y l i n d e r s of
P
s n ( 5 + a ) IZ 0 ,
w(x,y,f),
for
E
i s a F r 6 c h e t s p a c e , we use a
i s a constant, the s e t
Coming back t o set in
and
0,
5
everywhere i n E
of (11) complete.
and a p r o b a b i l i t y measure
.*(t:) Then by
< s*(y)
i n e a c h neighborhood of
compact d i s k e d and convex s e t s center
s(y)
E
To o b t a i n (111), f i r s t i f r e s u l t of Coeur6 [ 21 :
we have
E.
does n o t e x i s t , t h e n
= E,
y E , : g
For
G.
As a consequence
-a.
i s a p l u r i p o l a r cone i n
in
i s a Frdchet i s dense i n E . i n a dense
is pluripolar i n
gk
f
i s a B a i r e s p a c e and
i s continuous with
The s e t s
The same p r o p e r t y h o l d s f o r
and a s a consequence, yo E E ,
p o l a r cone i n
E.
yo
#
gk
gk
=
e x i s t s and
Now we prove:
eq gi
e
=
nn
e
n,q' i s meager i n 9
E.
BY ( 2 2 1 , E.
Then
i s a meager and p l u r i -
PIEFtRE LELONG
124
THEOREM 2 .
I n a complex t o p o l o g i c a l v e c t o r s p a c e
t h e two den-
E
s i t y numbers g i v e n by t h e D e f i n i t i o n s 1 and 2 c o i n c i d e f o r a f u n c tion
it i s f o r
which i s p l u r i s u b h a r m o n i c ;
f
a neighborhood
x+W
of
u p p e r bounded i n
f
x: w(x,f) =
WW(X,f).
0 < r
For t h e p r o o f , w e w r i t e f o r
< 1:
v W ( x , f ) = l i m ( l o g r)-’ M W ( x , r , f ) r=0 M ( x , r , f ) = sup m ( x , y , r , f ) W Y
(22) If
a
i s a n u p p e r bound of
g r a p h of
log r
+
m(x,y,r,f)
By ( 2 2 ) , and f o r
f
in
= sup
x+W,
e
0 < r < 1
y E W.
for
by t h e c o n v e x i t y o f t h e
f(x+reiey),
(and
we o b t a i n :
log r < 0 ) :
Then (24)
v W ( x , f ) = 1 i m ( l o g r)
Conversely,
for
D i v i d i n g by
-1
~ ~ ( x , r , 2f )v ( x , f ) .
r=O 0
< r < 1, y E W,
l o g r < 0,
W E
@;,
(24) gives:
and t a k i n g t h e l i m i t f o r
r
\.
Or
The c o m p a r i s o n o f ( 2 4 ) a n d ( 2 5 ) makes t h e p r o o f c o m p l e t e .
TWO EQUIVALENT DEFINITIONS O F THE DENSITY NUMBERS
5.
125
CONSEQUENCES AND EXAMPLES, F i r s t we g i v e a p p l i c a t i o n s t o t h e c a l c u l u s of
COROLLARY 1.
If
i s p l u r i s u b h a r m o n i c i n a domain
f
complex t o p o l o g i c a l v e c t o r s p a c e x E G
v(x,f).
E,
v(x,f)
G
of a
can be c a l c u l a t e d i n
by
if t h e cone
yx =
ux
hw
of v e r t e x
0
in
E
h a s a n o t empty i n t e -
rior. A s consequence:
x+Wc G,
if
t o calculate
f(x+uy)
for
y E yx,
i s a d i s k e d neighborhood of
W
v(x,f),
IuI
we h a v e t o u s e o n l y t h e v a l u e s of
0,
\r
0 , and
+x
if
empty; t h e c o n d i t i o n i s s a t i s f i e d i f
i s supposed t o be n o t
y
i s a b s o r b i n g f o r a n open
set. A s a c o n s e q u e n c e o f t h e Theorem 1 we s t a t e :
COROLLARY 2 . and
E
If
f E P(G)
i s a Fr6chet space o r i f
i s continuous,
v(x,f)
i s g i v e n by ( 2 6 ) i f
s u p p o s e d n o t t o b e a p l u r i p o l a r cone i n
E.
COROLLARY 3 .
E,
By t h e same h y p o t h e s i s on
integer
p z 1
sion
(for
p
PROOF.
g:. then
For
For
an a f f i n e space p=l
p = 1
p > 1,
v(x,flL1)
take the l i n e
through
take
L'
x,
and f o r
= v(x,y0)
B
yo E L
v(x,f
is
of f i n i t e dimenv(x,f) =
n o t i n t h e p l u r i p o l a r cone L1 @ g:
containing a l i n e
Lp
yx
there e x i s t s f o r each
we o b t a i n a complex l i n e ) s u c h
= v(x,f)
v ( x , f ) = v(x,flL1)
Lp
i s a Baire space
E
1
2
,
yo
f 0,
v(x,f)
and
through we o b t a i n
x,
126
PIERRE LELONG
COROLLARY 5 .
If
i s a Banach s p a c e and
E
f u n c t i o n w i t h f i n i t e u p p e r bound i n the b a l l
r < R,
r,
/Izll L
f
a plurisubharmonic
in
if
E,
y E y
w(x,y,f)
y
which a r e n o t p l u r i p o l a r and meager
( 2 7 ) t h e number
i s supposed t o be c o n t i n u o u s ; i n
f
then
we have t h e bound:
w(O,f) = i n f w ( ~ , y , f ) f o r Y and ( 2 7 ) h o l d s for a l l cones
I/z// < R ,
\ \ f ( ( R i n the b a l l
c a n be c a l c u l a t e d u s i n g
&(O,y,r,f)
m(O,y,r,f)
or
for
Y E Y. REMARK 2.
If
morphic i n u = 0
W E %,
x+W,
of
f = l o g IFI,
i s a B a i r e s p a c e , and
E
F(uy)
w(O,y,f)
f o r fixed
y.
for
F
holo-
i s t h e m u l t i p l i c i t y of t h e zero
y
The c o n d i t i o n f o r
t o be n o t
p l u r i p o l a r and meager c a n be r e p l a c e d by t h e c o n d i t i o n t h a t
y
n o t c o n t a i n e d i n an a l g e b r a i c cone d e f i n e d by one e q u a t i o n i t i s [ y E E ; Pw(y) = degree F
01 ;
= v(x,f);
V
i s an homogeneous polynomial of
Pv
y
of
i t i s t h e f i r s t t e r m of t h e T a y l o r s e r i e s o f
a t t h e o r i g i n ; a more p r e c i s e r e s u l t i s a v a i l a b l e w i t h a much
more p r e c i s e h y p o t h e s i s .
cn,
In
v(x,f)
i s an u p p e r semi-continuous
N ( c , f ) = [ x E G ; w ( x , f ) 5. c ] ,
and t h e s e t s
ed (by a theorem of Y.T. analytic s e t ) ,
We prove:
THEOREM 3 .
E
w(x,f)
If
Siu, c f . [ 6 ] ,
N(c,f)
c > 0,
B
a
are clos-
i s proved t o be an
f
i s a n upper s e m i - c o n t i n -
x.
We suppose t h e t o p o l o g y i s g i v e n by a f a m i l y norms;
x,
i s a l o c a l l y convex s p a c e , t h e d e n s i t y number
of a p l u r i s u b h a r m o n i c f u n c t i o n
uous f u n c t i o n of
for
f u n c t i o n of
i s the unit b a l l
p,(x)
subharmonic and upper bounded i n
x
{pa}
< 1.
W e suppose
+
and w r i t e f o r
B
a
f
of semii s pluri0 < r
< 1:
TWO EQUIVALENT DEFINITIONS OF THE DENSITY NUMBERS
~ ~ ( ~ , r =, f sup ) f(x+y) Y
log r -+ M
The function tinuous.
pa(x'-x) < q,
Bu(x,r-q)
For
Mu(x,r-q,f) E
>
Then for
p,(x'-x)
U
q
such
< q,
Mu
(x,.)
-t
way as in
Cn
M (x,r+q,f) U
(r'-rl < q,
x
1
using the limit (8).
-
Mu(x,r-q,f)s~.
we have proved
< 2e. r > ro > 0 ;
r > 0;
for given
v(x,f)
-t
Mu(x,r+q,f).
is continuous for
is a continuous function of x
5
Mu(x,rif)
Mu(x,r,f)
upper semi-continuity of
U
0 S
and
-
B (x,r+q)
C
Ma (x' ,r,f)
5
lMu(x' ,r' ,f) The function
U
B (x',r)
C
there exists
0,
E; p (x'-x) < r] , we have for
0 < q < r:
and
(28)
is increasing, convex; it is con-
U
Bu(x,r) = [x'E
Writing
< r.
P,(Y)
for
127
then the
can be proved by the same We state the following conse-
quence. COROLLARY 6.
If
E
is a complex vector space with locally convex
topologyi the density sets N(c,f) = [x E G; v(x,f) of a plurisubharmonic function polar closed sets in REMARK 3. N(c,f)
f
2
c > 0
c],
in a domain
Gc E
are pluri-
G.
By Corollary 6 a new problem arises:
analytic sets like in
Cn
for
c > 0 7
are the sets We conjecture that
the result of Y.T. Siu (cf. [ 6 ] ) remains true in Banach spaces having the approximation property (see [ 7 ] ) .
But such a result in
an infinite dimensional space would not have the same precise geometrical consequences as in
Cn,
o r in finite dimensional mani-
folds, if no further information is available on the codimension of
N(c,f)
(see for example 1 5 1 , p.33).
PIERRE L S L O N G
128
4.
REMARK
3 remain t r u e
W i l l Theorem
a l o c a l l y convex s p a c e ?
Cn
In
if
i s n o t s u p p o s e d t o be
E
t h e u p p e r s e m i - c o n t i n u i t y of v ( x , f )
was a c o n s e q u e n c e of t h i s e l e m e n t a r y p r o p e r t y : and a n open n e i g h b o r h o o d W
borhood the basis
E
i s i n f i n i t e dimensional,
of t h e n e i g h b o r h o o d s of
0
d o e s n o t c o n t a i n any r e -
such
0
l a t i v e l y compact
r > 0,
t h e r e e x i s t s a disked neighIf
of
I
K,
of
UI
K
g i v e n a compact
K+W C W .
we h a v e t o t a k e t h e
W;
sup
of
on
f
w h i c h i s n o t r e l a t i v e l y compact a n d t o w r i t e
rW,
MW(x,r,f) =
for y E r W , r > 0. I n d e e d t h e p r o o f of t h e Theorem Y 3 was o b t a i n e d u s i n g t h e c o n t i n u i t y of M W ( x , r , f ) f o r f i x e d x
= sup f ( x + y )
r
and v a r i a b l e
>
0
(such a p r o p e r t y remains t r u e i f
E
i s not
of t h e t o p o l o g y which i s t h e
l o c a l l y c o n v e x ) and a p r o p e r t y ( P ) following:
(P)
-
Given a d i s k e d n e i g h b o r h o o d
W
of t h e o r i g i n , and
t h e r e e x i s t s a d i s k e d neighborhood
w +
(29)
harmonic f u n c t i o n s
f
x
in
0,
such
0
(l+Tl)W,
W‘C
p r o p e r t y ( P ) , t h e n t h e d e n s i t y number of
v(x,f)
E
has t h e
of t h e p l u r i s u b -
i s a n upper semi-continuous f u n c t i o n
x.
PROOF. y E W
log r
By t h e c o n v e x i t y of f(x+y) s a ,
and
function
q
(29) there e x i s t s x E W’
for
r + MW(x,r,f)
there exists
X I -
of
I f t h e t o p o l o g y o f t h e complex s p a c e
PROPOSITION 2 .
of
W’
>
q
such W‘
0
y
E
-b
W,
M (x,r,f) = sup f(x+uy), W lu(sr
i s continuous. S
M (x,r+qyf)
such
we o b t a i n f o r x’+ W C x
a
W + W’C (l+q)r
For g i v e n
- Ma(x,r-q,f)
(l+q)W;
x
r
>
0,
x, r, 0 < r
<
C .
+ (l+q)W
the
< 1
Then b y
therefore f o r
< 1:
+ w’+ w c
for
TWO EQUIVALENT DEFINITIONS OF THE DENSITY NUMBERS
By (29), given
0 <
q,
< 1
and a disked neighborhood
W" of
there exists a disked neighborhood
(31)
MW[x,(l-q)r,f]
or
~ ~ ( x , r - q , fs)M~(x' ,r,f) 2
F r o m ( 3 0 ) and (32) we obtain:
perty (P), then of
MW(x,r,f)
of
0
such
0
s MW(x' ,r,f) if
MW(x',r,f)
W
+ VC w
(1-q)w
(32)
129
if
x'-
x E w"
x'-
- E.
MW(x,r,f)
if the topology of r > 0
is for
E W"
x
has the pro-
E
a continuous function
x.
From the continuity of
If
a in
f
x
MW
and (8) we then can deduce Proposition2. 1 -1 MW(x,r,f-a). -v(x,f) = lim(1og F)
+ W , we obtain
The quotient is an increasing limit of continuous functions, for r
I
0; the Proposition 2 is proved.
REMARK 5 .
E
Examples of non Baire spaces
subharmonic functions
f
polar and is all the space example was given by
C.
such i n
E,
x
with continuous pluri-
the cone
can be given.
gk
is not pluri-
For instance (this
Kiselman in an unpublished letter) let us
consider the space E = $ CN E
i s
x = Exn}
the space of the sequences
the exception of a finite set for =
1: j E N;
A basis We
@
xj
supp x =
We define
n.
xn = 0 with
f 01.. of the neighborhoods of
= [x; lxj) < ej],
c j > 0, j E
is plurisubharmonic in u E 6,
such that
E
v(O,y,f)
N.
if we take
we obtain at the origin
will be the
0
Then c
7
0.
For
= inf cj,
for
j E supp y
j E IN.
j
loglxjl
fy(u) = f(yu),
x = 0:
= v(O,f Y ) = inf c j v(O,f)
j
f(x) = sup c
PIERRE LELONG
130
If s
c
-
2-j,
j -
6.
for
y E E
and
.
= Csup j ; j E SUPP Y]
REMARK
> v(0,f) = 0
v ( O , y , f ) = 2-'
then
C a l c u l u s of
w(x,y,f)
and
v(x,f)
E = @ En
if
More g e n e r a l l y , l e t u s c o n s i d e r a s e q u e n c e l o c a l l y convex complete s p a c e s ; i n
M = nE n ny
En
.
of complex
E
we d e f i n e
by
E = $ E n C M and t a k e on denote
E
E + En
p,:
E
d e f i n e d by
We s u p p o s e on e a c h monic f u n c t i o n
Let u s d e f i n e Given
xo E E ,
jn: En
+ E;
= 0,
p,(x)
i s the
T
i s t h e c l o s e d sub-
jn(En)
m f n
and
p,(x)
= x
= [ x n E E n ; Un(xn) = --]
En.
C
we d e f i n e a p o s i t i v e i n t e g e r s u c h p n ( x o ) = 0 for n
Now we s u p p o s e t h a t t h e o r i g i n b e l o n g s t o L
G i v e n a complex l i n e
in
qn
0
> s(x )]. En
in
f o r each
E:
x = x o + u y ,
u E C
P j b ) = Pj(XO) + U P j ( Y ) . x E L,
t h e number s(x)
s(x)
i s bounded
s s u p ~ s ( x O ) , s ( y ) ]= s ( L ) .
L e t us d e f i n e l i k e i n t h e p r e c e d e n t example f (x)
(33) Then
pn(x) = 0
i f m=n.
and we d e n o t e :
s ( x o ) = [ i n f n , n E IN,
For
We
we h a v e d e f i n e d a c o n t i n u o u s p l u r i s u b h a r -
En
Un(xn),
:n
xn = p n ( x ) ;
t h e p r o j e c t i o n s and
t o p o l o g y of t h e mappings space o f
T.
t h e l o c a l l y convex d i r e c t s u m t o p o l o g y
and
= s u p Unopn(x) = sup v n ( x ) n n Unopn(x) =
-m
for
n
.
> s(L)
and
n.
TWO EQUIVALENT DEFINITIONS OF THE DENSITY NUMBERS
f I L = sup U j o p j ( x o + u y ) f o r 1 5 j 5 s(L) j tion ( o r z -m) d e f i n e d by t h e formula
i s a subharmonic f u n c -
N o w we suppose t h a t t h e p o i n t
xo
belongs t o
xz = p n ( x o ) E :n
or
Un(xz)
Then i f
L
f o r each
n
i s a complex l i n e t h r o u g h
v(xo7y,f) = v(xo,flL)
(33
131
-
nrln
n.
f o r each
m
xo,
it is
of d i r e c t i o n
y = [yj}.
= inf v[x;,pj(y),ujl J
and we have t o t a k e i n
1 s j
s(y).
(33) the
inf
The d e n s i t y number
f o r given
v(xo7y,f)
y E $En
and
is
(34) for
y E E = $En,
I n the
En
and
1 5
s(y).
a r e closed subspaces o f
t h e space
E,
E
i s not a
i s a continuous plurisubharmonic function.
Baire space;
f(x)
From ( 3 3 ) and
( 3 4 ) we g e t
(35)
j S
v ( x o , f ) = inf v ( x o , y , f ) = inf v[pj(xo),uj1. Y
The t a n g e n t i a l d e n s i t y number
j (xo,y,f),
g i v e n by
(33) i s the
inf
o f a f i n i t e s e t of p o s i t i v e numbers; we can choose i n ( 3 5 ) t h e continuous plurisubharmonic functions v(xo,f) = c
> 0
and
v(xo,y,f) > c
Uj €
P(Ej),
for a l l
i n order t o obtain
y E E.
PIERRE LELONG
132
BIBLIOGRAPHY
1.
AVANISSIAN, V . , Fonctions plurisousharmoniques et fonctions doublement sousharmoniques, Ann. E .N. S , , t. 78, p. 101-161.
2.
C O E d , G., Fonctions plurisousharmoniques sur les espaces vectoriels topologiques, Ann. Inst. Fourier, 1970, p.361-432.
3. KISELMAN, Ch.,
a
Stabilite du nombre de Lelong par restriction une sous-variet6, Lecture Notes Springer ne 919,
P. 324-9379 (1980)
4. L E M N G , P.,
a/ Plurisubharmonic functions in topological vector spaces, Polar sets and problems of measure. Lecture Notes, no 364, 1973, p. 58-69. b/ Fonctions plurisousharmoniques et ensembles polaires sur
une aog8bre de fonctions holomorphes,
Lecture Notes, no 116,
1969, pa 1-20. c/ Calcul du nombre densit6 v(x,f)
et lemme de Schwarz pour
les fonctions plurisousharmoniques dans un espace vectoriel topologique,
.
Lecture Notes Springer nP 919, p. 167-177,
(1980)
d/ Integration sur un ensemble analytique complexe,
Bull.
SOC. Math. de France, t. 85, p. 239-262, 1957.
5.
RAMIS, J.-P., Sous-ensembles analytiques dcune variet6 banachique complexe, Ergebnisse der Math., t. 53, Springer,
1970
6.
SIU, Y.T., Analyticity of sets associated to Lelong numbers, Inv. Math., 6. 27, p. 53-156, 1974.
7 - NOVERRAZ, Ph.,
Pseudo-convexit6, convexit6 polynomiale et domaines dlholomorphie en dimension infinie, North Holland,
Math. Studies, vol. 3 (1973).
Dgpartement de Mathematiques Universite de Paris VI
4 Place Jussieu 75230
Paris
CEDEX 0 5
Functionnl Anolysis, Holoniorplry orid Appruxiniotion Theory 11,G.I. Zupata ( E d . ) 0 Elsevier Science Publislrers B. V. (Nurtli-Holland), 1984
133
CHEBYSHEV CENTERS OF COMPACT SETS WITH RESPECT T O STONE-WEIERSTRASS SUBSPACES
Jaroslav Mach
Let C(S,X)
S
be a compact Hausdorff space,
X
a Banach space,
S
the Banach space of all continuous functions on
X
values in
with
In this note two
equipped with the supremum norm.
results concerning Chebyshev centers of compact subsets of with respect to a Stone-Weierstrass subspace of
C(S,X)
C(S,X)
are es-
In particular, a formula for the relative Chebyshev
tablished.
radius in terms of the Chebyshev radius of the corresponding set valued map is given.
I t is shown further that the proximinality
of all Stone-Weierstrass subspaces implies the existence of relative Chebyshev centers for all compact subsets of
C(S,X).
The proximinality of Stone-Weierstrass subspaces has been studied by many authors.
Mazur (unpublished, c.f.,
e.g., [ 6 ] )
proved that any Stone-Weierstrass subspace is proximinal if the real line (a subspace led proximinal if every ximation
x
in
G,
G
of a normed linear space
yE Y
X
i.e., if there is an x € G).
xo € G
is cal-
such that
The question for which
every Stone-Weierstrass subspace of
proximinal is due to Pelczynski
C(S,X)
X
and an L1-predual space, respectively.
In
is uniformly convex [2]
those Banach spaces
for which any Stone-Weierstrass subspace is proximinal were
characterized.
is
[4] and Olech [ 3 ] . Olech [ 3 ] and
Blatter [I] showed that this is true if
X
is
possesses a n element of best appro-
/(y-xo)S [/y-x((holds for every Banach spaces
Y
X
134
JAROSLAV MACH
We will employ the following notations and definitions.
Let
E X,
x
center
r > 0.
B(x,r)
and radius
x
pact subsets of
X.
C(X)
r.
T
V
g E C(T,X).
G
of
V
C(S,X)
C(S,X)
is said to be a
if there is a compact
and a continuous surjection
is the set of all functions
some
Let
f
S
'p:
T
-t
such that
f = goCp
having the form
be a set-valued mapping from
@
171 ) if for every
borhood
U
of
E S
so
and every
sup
<
dis(x,@(so))
X
.
(cf. 151
there is a neigh-
0
s € U
such that for every
so
>
E
for into 2
S
is said to be upper Hausdorff semicontinuous (u.H.s.c.)
and
with
will denote the class of all com-
A subspace
Stone-Weierstrass subspace of Hausdorff space
X
will denote the closed ball in
we have
E .
xE@( s ) is lower semicontinuous if the set
@
for any open set [ s :
@ ( s )fl H
f € C(S,X)
#
iP
G.
61
( s :
n
G(s)
#
G
@) is open
is upper semicontinuous if the set
H.
is closed for any closed set
is said to be a best approximation of
A function
@
in
C(S,X)
if the number dist(f,G) = sup SES
is equal to
inf dist(g,C)
g E C(S,X).
Let
F
sup
IIx-f(s)l/
X€@(S)
where the infimum is taken over all
be a bounded subset of
X,
G
a subspace of X.
The number
rG(F) = inf sup IIx-yll xEG yEF is called the Chebyshev radius of x
E G
if
F
with respect to
is said to be a Chebyshev center of
IIx-yI(
denoted by
2
rG(F) cG(F).
note the number
for all
y E F.
F
G.
A point
with respect to
G
The set of all such x will be X For a set-valued map I: S -t 2 , r@ will de-
sup sEs
.)(a,
135
CHEBYSHEV CENTERS OF COMPACT SETS
THEOREM 1.
Let
cp
fined by
V
be a Stone-Weierstrass subspace of
T.
and
F
Let
be a compact subset of
r (F) =
V
where
@:
PROOF.
T
n
tu E {t: @(t)
E cp
iP
-1
H f 0}
n
(t,)
U.S.C.
fa E F s
U
s
-+
and
@(t) fl H f 0 .
g E
C(T,X)
sup
we show that the set
tu
sup
-+
t.
f
U
H.
Let
Then there are
fu(sa) E H.
such that
so
f(s) E H , For any
is
be such that
generality assume that and
Then
r@
is closed for any closed set
H f 0)
and
C(S,X).
de-
is the set-valued map
To prove that
(t € T: @(t)
S&
C(X)
-+
C(S,X)
-+ f.
Without l o s s of
Then clearly s E cp”(t)
we have
IIx-g(t))) = dist(g,@).
tET x€@ (t) It was proved in
[2]
that dist(g,@) = r@.
inf
& c (T,X) It follows
THEOREM 2 .
Let
of
is proximinal.
C(S,X)
X
be such that every Stone-Weierstrass subspace Then
cV(F)
#
0
for every compact sub-
set .F
of
PROOF.
By Theorem 2 of [ 2 ] , the proximinality of every Stone-
C(S,X)
and every Stone-Weierstrass subspace
Weierstrass subspace of Hausdorff space
C(S,X)
T, any u.H.s.c.
V.
implies that for any compact map
@:
T
-+
C(X)
has a best
136
JAR0SLAV MACH
approximation
g
c(T,x).
in
Let
Then inf (h,@) = sup Ilf-go(pll = dist(g,$) = f€F h€ C (T,X ) inf hEC(T,X)
sup
Ilf-hocpll
= rV(F).
f€F
I t follows The following corollary is a consequence o f Blatterls result and Theorem 2. COROLLARY 1.
Let
be an L1-predual space.
X
for any compact subset subspace
F
of
C(S,X)
Then
cV(F) f Q
and any Stone-Weierstrass
V.
In [8], a bounded subset of an L1-predual space has been constructed whose set of Chebyshev centers is empty.
This shows that
Corollary 1 does not hold if compact subsets are replaced by bounded subsets.
It was shown in [2] that if
X
is a locally uniformly convex dual
Banach space then every Stone-Weierstrass subspace of proximinal.
is
The next corollary follows from this and Theorem 2.
COROLLARY 2.
Let
space.
cV(F)
Then
C(S,X)
X
be a dual locally uniformly convex Banach
#
Q
for any compact subset
any Stone-Weierstrass subspace
V.
F
of
C(S,X)
and
CHEBYSHEV CENTERS OF COMPACT SETS
137
REFERENCES
1.
J. BLATTER, Grothendieck spaces in approximation theory,
Mem.
Amer. Math. SOC. 120 (1972). 2.
J. MACH,
On the proximinality of Stone-Weierstrass subspaces,
Pacific J. Math. 99 (1982), 97-104. 3.
C. OLECH, Approximation of set-valued functions by continuous functions, Colloq. Math. 19 (1968), 285-293.
4. A. PELCZYNSKI,
Linear extensions, linear averagings and their
applications to linear topological classification of spaces of continuous functions, Dissert. Math. (Rozprawy Math.)
58, Warszawa 1968.
5.
W.
POLLUL, Topologien auf Mengen von Teilmengen und Stetigkeit von mengenwertigen metrischen Projektionen, Diplomarbeit, Bonn 1967.
6.
Z.
SEMADENI,
Banach spaces of continuous functions,
Monografje Matematyczne 55, Warszawa 1971.
7. I. SINGER, The theory of best approximation and functional analysis, Reg. conference ser. appl. math. 13, SIAM, Philadelphia 1974. 8.
D.
AMIR, J. MACH, K. SAATKAMP,
Existence of Chebyshev centers,
best n-nets and best compact approximants, Trans. Amer. Math. SOC. 2 7 1 (1982), 513-524.
Institut fiir Angewandte Mathematik der Universitgt Bonn Wegelerstr. 6 5300 Bonn
This work was done while the author was visiting the Texas A&M University at College Station.
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O N THE: FOURIER-BOREL TRANSFORMATION AND SPACES O F ENTIRE FUNCTIONS I N A N O W , n SPACE
Mdrio C . Matos ( D e d i c a t e d t o t h e memory o f S i l v i o Machado)
1. INTRODUCTION We i n t r o d u c e h e r e t h e s p a c e s o f e n t i r e f u n c t i o n s i n a normed s p a c e which a r e of o r d e r ( r e s p e c t i v e l y , n u c l e a r o r d e r ) ( r e s p e c t i v e l y , nuclear t y p e )
k E [l,+m]
Here
and
s t r i c t l y l e s s than
A E
(O,+-].
k E [l,+m]
and
A E [O,+m).
and t y p e
A.
The c o r r e s p o n d i n g s p a c e s
i n w h i c h t h e t y p e i s a l l o w e d t o be a l s o e q u a l t o when
k
A a r e introduced
These spaces have n a t u r a l t o -
p o l o g i e s and t h e y a r e t h e i n f i n i t e d i m e n s i o n a l a n a l o g o u s of t h e spaces considered i n Martineau [l]. Fourier-Bore1
I n t h i s p a p e r we s t u d y t h e
t r a n s f o r m a t i o n i n t h e s e s p a c e s a n d we a r e a b l e t o show
t h a t t h e s e t r a n s f o r m a t i o n s i d e n t i f y a l g e b r a i c a l l y and t o p o l o g i c a l l y t h e s t r o n g d u a l s of t h e a b o v e s p a c e s w i t h o t h e r s p a c e s of t h e same
I n a s e c o n d p a p e r , t o a p p e a r e l s e w h e r e , we p r o v e e x i s t e n c e
kind.
and a p p r o x i m a t i o n t h e o r e m s for c o n v o l u t i o n e q u a t i o n s i n t h e s e s p a c e s . The n o t a t i o n s we u s e d a r e t h o s e u s e d by N a c h b i n [l] a n d Gupta
[ 11,
Hence,
E
if
i s a complex normed s p a c e ,
s p a c e of a l l e n t i r e f u n c t i o n s i n a l l j-homogeneous norm
I/ *I/
and
E,
P(%)
continuous polynomials i n
PN('E)
#(E)
is the vector
t h e Banach s p a c e o f
E
with the n a t u r a l
t h e Banach s p a c e of a l l j-homogeneous
t i n u o u s p o l y n o m i a l s of n u c l e a r t y p e w i t h t h e n u c l e a r norm for a l l
j E N.
II.IIN
con-
MARIO C .
140
MATOS
S P A C E S O F ENTIRE FUNCTIONS I N NORMED SPACES
2.
I n t h i s section 2 . 1 DEFINITION.
p > 0
If
f E B(E)
s p a c e of a l l
d e n o t e s a complex normed s p a c e .
E
we d e n o t e
Bp(E)
t h e complex v e c t o r
such t h a t m
11 * ] I p .
normed by all
2.2
f E H(E)
>
0
t h e complex v e c t o r space of
znf(0) E pN(%)
p
For each
n E N
f o r each
> 0 , t h e normed s p a c e s
Bp(E)
and
and
a r e complete. m
(fn)n,l
If
PROOF.
a
such t h a t
PROPOSITION.
nN,p(E)
BN,p(E)
We d e n o t e
there i s
i s a Cauchy sequence i n
n
a
E IN,
(E)
m
element
P j E P (’E)
for a l l
rn
f o r every
such t h a t
.
for a l l
,
and n 2 n It f o l l o w s t h a t (djf,(O))OD is a a E n= 1 and i t c o n v e r g e s t o an Cauchy sequence i n t h e Banach s p a c e P ( j E ) 2 n
2
n
E
.
.
Hence, u s i n g ( 3 ) , w e have:
If w e prove t h a t m
f(x) =
C
1 3 Pj(x)
j=O
d e f i n e s an element of
#(E),
then w e get:
(X
E E)
ON THE FOURIER-BOREL TRANSFORMATION
Therefore
to
f.
f
141
and ( 4 ) i m p l i e s t h e c o n v e r g e n c e of
E Wp (E)
f E #(E)
I n o r d e r t o prove t h a t
(fn)m
n= 1
we n o t e t h a t
Hence
f E 1(E).
and
2 . 3 DEFINITION. v e c t o r space topology.
aN,p
A s i m i l a r p r o o f may be u s e d f o r A E
If
u
Bp(E)
we d e n o t e
(O,+m)
Exp;(E)
*
t h e complex
w i t h t h e l o c a l l y convex i n d u c t i v e l i m i t
P
1 ExPN,A(E) =
da~,,(E) is P
1 Exp0,*(E)
s i d e r t h e complex v e c t o r s p a c e s
1 E X P ~ , ~ , ~ (= E )
n
and
Wp(E)
P >A BN,p(E)
P 'A F i n a l l y , we c o n s i d e r
=
=
with the projective l i m i t topologies.
1 Expm(E) =
u B (E) p>o p
and
1 ExpN,=(E) =
u
@ N , p ( E ) w i t h t h e l o c a l l y convex i n d u c t i v e l i m i t t o p o l o g i e s P >O 1 1 1 a n d s e t Expo(E) = E X ~ ~ , ~ = ( E ) Bp(E) and ExpN,,(E) =
n
P>O
=
xpN
,0 ,0 ( E )
2 . 4 REMARK.
= oo ,n
BN,P (E)
with the projective l i m i t topologies.
It i s p o s s i b l e t o show:
1 E X P ~ ( E )= [ f E # ( E ) ;
1 -
l i m sup
II;jf(o)ll
j
< A],
Q
A E
(o,+~I
Q
A E
ro,+-).
j-bm
1 -
E x p1o , A ( ~ )= ~f
E
On t h e o t h e r h a n d , i f all
j E N,
we h a v e
# ( E ) ; l i m s u p l l ~ j f ( o ) l ls j A], j+m f E #(E)
i s such t h a t
;'f(O)
E PN(JE)
for
142
M A R I O C.
MATOS
~ ~ , ~ ( iEf ), and o n l y i f , f E E x 1
(a)
l i m sup I l ~ j f ( O ) l l ~
A E (o,+-]
for a l l
T h e r e f o r e , i t i s n a t u r a l t o c a l l t h e e l e m e n t s of
(respect-
Exp;(E) 1 E x ~ ~ , ~ ( E e) n) t i r e f u n c t i o n s of e x p o n e n t i a l t y p e
ively,
A.
i v e l y , of n u c l e a r e x p o n e n t i a l t y p e ) s t r i c t l y l e s s
A
we d r o p out " s t r i c t l y l e s s t h a n
A = +=
For
(I.
1 E X P ~ , ~ ( E )( r e s p e c t i v e l y ,
The e l e m e n t s of
(respect-
1 E x P ,O ~ ,A ( E l ) a r e
c a l l e d e n t i r e f u n c t i o n s of e x p o n e n t i a l t y p e ( r e s p e c t i v e l y , e x p o n e n t i a l t y p e ) l e s s t h a n or e q u a l t o
2.5 PROPOSITION, ( a ) A E
all
(b)
A E
Exp;(E)
and
nuclear
A.
ExpN,,(E) 1
are
DF
spaces f o r
(o,+-1. 1 ExpO,A(E)
and
1 ExPN,O,A (E)
a r e Fr6chet spaces f o r every
to,+-).
PROOF.
If
A E
[O,+@)
t h e proof i s s t r a i g h t f o r w a r d f o r p a r t ( b ) .
The m e t r i z a b i l i t y f o l l o w s from t h e f a c t t h a t we can g e t a sequence m
(an)n,l
of p o s i t i v e r e a l numbers,
s t r i c t l y d e c r e a s i n g and such t h a t
l i m an = A. If A E (O,+m] p a r t ( a ) f o l l o w s from t h e f a c t t h a t n+m 1 and E x ~ ~ , ~ ( aEr )e t h e i n d u c t i v e l i m i t of a sequence of Exp;(E) Banach s p a c e s .
It i s enough t o t a k e a sequence
s i t i v e r e a l numbers,
m
(bn)n,l
s t r i c t l y increasing,
converging t o
03
BN,bn(E)
t o c o n s i d e r t h e Banach s p a c e s
bn
(E)
and
of po-
A
and then
0
Now w e c o n s t r u c t s i m i l a r s p a c e s f o r e n t i r e f u n c t i o n s of f i n i t e order.
2 . 6 DEFINITION. ively,
k @N,p(E))
If
p
> 0
and
k > 1 we d e n o t e
t h e complex v e c t o r s p a c e of a l l
B;(E)
(respect-
f E #(E)
such
ON THE FOURIER-BOREL TRANSFORMATION
143
that
(respectively,
h j f ( 0 ) E PN(jE)
such t h a t
2.7 PROPOSITION.
The normed s p a c e s
for a l l
mE(E)
E IN
j
63'N , P ( E )
and
and
a r e com-
plete. The p r o o f
of t h i s r e s u l t f o l l o w s t h e p a t t e r n of
2 . 8 DEFINITION.
A
E (O,+m]
and
k
> 1 we d e n o t e
( E ) ) t h e complex v e c t o r s p a c e Expk N,A
(respectively, (respectivel y
If
t h e proof
,
u
B:,p(E))
If
A E
2.2.
Exp%(E)
u
BF(E)
P< A
w i t h t h e l o c a l l y convex i n d u c t i v e
P<* l i m i t topology.
of
LO,+-)
and
k
>
1
t h e complex v e c t o r
endowed w i t h t h e p r o j e c t i v e l i m i t t o p o l o g i e s .
I n order t o simplify
k k k t h e n o t a t i o n s , sometimes w e w r i t e : E x p m ( E ) = Exp ( E ) , EXP~,~(= E) k k k k ( E ) = Expk (E) (includ= ExpN(E) 9 ExpO,O(E) = 9 ExPN,O,O N,O k = 1).
ing the case
2 . 9 REMARK.
I t i s p o s s i b l e t o show t h a t
1
k
= ( f E # ( E ) ; l i mj - t ms u p
Exp0,,(E)
where
k
> 1
A E
and
i n t h e second c a s e . a j f ( 0 ) E pN(jE)
(O,+m]
j E
N,
if
we h a v e :
-1
((7 J. d J f ( 0 ) l l j
i n t h e f i r s t c a s e and
O n t h e o t h e r hand,
for a l l
j F 1 1 '
(E)
f E #(E)
5
A]
A E LO,+=)
i s such t h a t
144
(a)
MARIO C .
MATOS
f E E x k~ ~ , ~ ( i E f) , and o n l y i f ,
l i m sup
1 -
(&)”
1 -
~ l I ~ j f ( O ) l l<j A N
j-tm
for
> 1 and
k
A E
(O,+m],
T h e r e f o r e , i t i s n a t u r a l t o c a l l t h e e l e m e n t s of ively,
k E x ~ ~ , ~ ( E e) n) t i r e f u n c t i o n s of o r d e r
less than
k
and t y p e s t r i c t l y
( r e s p e c t i v e l y , e n t i r e f u n c t i o n s of n u c l e a r o r d e r
A
and n u c l e a r t y p e s t r i c t l y l e s s t h a n I l s t r i c t l y l e s s than
x P ~0, ,A
(respect-
Exp:(A)
A
’I.
When
A).
The e l e m e n t s of
A = +m
Expo, k A (E )
a r e c a l l e d e n t i r e f u n c t i o n s of o r d e r
(E))
nuclear order
k)
and t y p e l e s s t h a n o r e q u a l t o
n u c l e a r t y p e l e s s t h a n or e q u a l t o
2.10 REMARK.
k A
k
we d r o p o u t ( r e s p e c t i v e ly, (respectively, (respectively,
A).
We know t h a t
Using t h i s f a c t i t i s e a s y t o s e e t h a t , if we a l l o w f i n i t i o n s 2 . 6 and 2 . 8 ,
k = 1
i n De-
t h e s p a c e s we g e t c o i n c i d e a l g e b r a i c a l l y with
t h e s p a c e s of D e f i n i t i o n 2 . 1 and 2 . 3 and a r e i s o m o r p h i c t o p o l o g i c a l ly. 2 . 1 1 PROPOSITION.
and
ExpN,,(E) k (b)
If
k
(a)
If
k
> 1 and
A E
(O,tm],
then
Exp;(E)
a r e DF-spaces.
> 1 and
A E LO,+=),
then
Expk (E) O,A
and
E X k~ ~ , ~ , ~ a(r eE F) r 6 c h e t s p a c e s .
The proof f o l l o w s aa we have i n d i c a t e d i n P r o p o s i t i o n 2 . 5 . I n o r d e r t o c o n s i d e r s i m i l a r s p a c e s of f u n c t i o n s of i n f i n i t e o r d e r , we i n t r o d u c e some new d e f i n i t i o n s .
ON THE FOURIER-BOREL TRANSFORMATION
t
Ti
f E
v e c t o r s p a c e of a l l
145
51(Bh(0)) s u c h t h a t l i m sup \ \ & , G j f ( O ) l l T < j+m J. A endowed w i t h t h e l o c a l l y convex t o p o l o g y g e n e r a t e d by t h e
h A
f a m i l y of
seminorms
where
9
t h e complex v e c t o r s p a c e of a l l f E # ( B
We d e n o t e
HNb(B
such t h a t
a j f ( 0 ) E PN(jE)
(0))
A
for
and l i m s u p l l1 - ; i d J f ( 0 ) \5 \J 5 A ,
E N
j
i( 0 ) )
A
j+m
.
J *
endowed w i t h t h e l o c a l l y convex t o p o l o g y g e n e r a t e d by t h e f a m i l y of
These l o c a l l y convex s p a c e s a r e F r e c h e t s p a c e s ( s e e Gupta [ l ] and
[ 21 and Matos [ 1 3 ) . These s p a c e s a r e d e n o t e d r e s p e c t i v e l y E Xm ~ ~ , ~ ( and E ) ExpGIO,,(E). W e a l s o w r i t e Exp;(E) = E Xm ~ ~ , ~ ( E ) and
m
m
E X ~ ~ , ~ = ( E Exp ) N , 0,O(E) *
2.13 DEFINITION. ively
gNb(flk(0))
of Frechet spaces
HNb (B 1( 0 ) ) A
=
If
P
A E
Hb(6i(o))
we d e n o t e
(respectA t h e a l g e b r a i c and l o c a l l y convex i n d u c t i v e l i m i t (O,+m]
gb(B1(0))
p E
for
(0,A)
(respectively,
P
for
p E
(0,A)).
We remark t h a t ,
except i n the case
t h e s e s p a c e s may n o t b e t h e s p a c e o f germs of holomorphic
+m,
f u n c t i o n s i n a neighborhood o f
gl(0)
( f o r i n f i n i t e dimensional
A
normed s p a c e s ) . done as f o l l o w s . ively,
A n e q u i v a l e n t way o f d e f i n i n g t h e s e s p a c e s may b e
We c o n s i d e r t h e Banach s p a c e
N ~ ( B & ( O ) ) of a l l
f E
I! such t h a t
aJf(0) E SN(jE) m
C
j=O
P
~ ( B- ~ ( o )( r) e s p e c t i v e l y , f E # ( B P
&( 0 ) )
j E N)
f o r every
1
;ljf(Ol
j II j! p
Um(Bl(0)) ( r e s p e c t -
I1
satisfying
<
respectively,
endowed w i t h t h e norm
pp
( r e s p e c t i v e l y,
PN,J'
(See
( 7 ) and ( 8 )
146
M&IO
above).
MATOS
C.
Hence we have t h e l o c a l l y convex i n d u c t i v e l i m i t s
Now we u s e t h e f o l l o w i n g n o t a t i o n s : E x ~ ~ , ~ (= EHNb(B1(0)) ) OD
EXP-(E)
Exp;(E)
A E
for e a c h
T = E X P ~ ( E )= H b ( r O i )
= Hb(sl(0))
W e a l s o denote
(O,+=]. co
=
E X P ~ , ~ ( E=) EXP;(E)
and
and
A
HNb(roi).
I n o r d e r t o m o t i v a t e t h e s e n o t a t i o n s we n o t e t h a t
II *Ilm , p
OD
Hence we may t h i n k
as
Pp
and
m
= j1
pN,p
,p
'llN,m
and t h e s e
n o t a t i o n s w i l l a l s o be u s e d .
2 . 1 4 PROPOSITION. ( a ) Exp;(E) (b)
and If
If
A E
t h e l o c a l l y convex s p a c e s
DF.
E X P ~ , ~ ( E )a r e A E [0,+=)
(O,+-]
t h e l o c a l l y convex s p a c e s
Exp;,*(E)
and
E X ~ ~ , ~ , * ( Ea r) e F r g c h e t .
PROOF.
A s we remarked above ( b ) i s proved i n Gupta [
i n Matos [l].
11
and [ 23
and
I n o r d e r t o prove ( a ) i t i s enough t o s e e t h a t t h e s e
l o c a l l y convex s p a c e s a r e t h e i n d u c t i v e l i m i t s o f sequences of I n o r d e r t o g e t t h i s i t i s enough t o t a k e an i n -
Banach s p a c e s .
m
( P ~ ) ~ of= p~ o s i t i v e numbers such t h a t
c r e a s i n g sequence
l i m p n = A.
0
n-r-
2 . 1 5 PROPOSITION. ( a ) Taylor s e r i e s a t
0
If
tl,+-l
k E
o f e a c h element o f
and
k
Exp;(E)
k E x p N I A ( E ) )c o n v e r g e s t o i t i n t h e t o p o l o g y of (b)
If
k E [ l , + = ]and
A
E
[O,+-),
E
(O,+-],
then the
(respectively,
the space.
then the Taylor s e r i e s a t
147
ON THE FOURIER-BOREL TRANSFORMATION
0
o f each element of
Expk
0 9-4
(respectively,
(E)
E X P ~ ,( E~ l ), ~
c o n v e r g e s t o i t i n t h e t o p o l o g y of t h e s p a c e .
PROOF.
It i s e n o u g h t o n o t e t h a t for e a c h
in the c o r r e c t space
f
w e have:
2.16 PROPOSITION. belongs t o (b)
and t o (d)
k E
If
cp
k = 1
E
E‘
A E
to,+-),
A E (O,+m]
1
and
cp
E
llcpll
cp
E
then
then
ecp
belongs t o
E’.
ecp b e l o n g s t o
IIcpIl
such that
E’
,
cp E E ’ .
ecp
then
then
A E [O,+m),
such t h a t
A E (O,+-]
and
k E x ~ ~ , ~ ( fEo )r every
and
E x ~ ~ , ~ ( fEo r) a l l
If
E (I,+=]
k E X P ~ , ~ , ~ ( for E ) all
k = 1 and
If
k
and t o
(1,+a]
and t o
for all
PROOF.
Exp;(E)
If
Exp0,,(E) k
(c)
(a)
elp
E
Exp;(E)
< A.
1 1 E ~ P ~ , ~ ( E ) , E ~ P ~ , ~ , ~ ( E )
A.
I t i s e n o u g h t o u s e t h e d e f i n i t i o n s of t h e s p a c e s a n d t o
n o t e that
11
(O))) =
IIpll j =
11
(‘)]IN
’
0
148
-10
MATOS
C.
2 . 1 7 PROPOSITION. (1) The v e c t o r subspace g e n e r a t e d by a l l cp E E'
(2)
,
i s dense i n
(a)
k E x ~ ~ , ~ ( iEf ) k E
(b)
E X P ~ , ~ , ~ ( E if)
('1
1 ExPN(E)*
k E
1 E x p N I A ( E ) if
A E
PROOF.
1
ecpl cp E E'
ponding g e n e r a t e d v e c t o r s u b s p a c e .
p , ( j ~ )E
cpn E 8 For
s
PN(jE)
a
j E IN.
for a l l
for a l l
E C,
for a l l
n E N
a
f
and
IX
0,
j
llqll
A,
is
L A,
is
5
cp E E'
, llcpll
(o,+-). 8
I n a l l t h r e e c a s e s we d e n o t e
i s dense i n
,
((I,+-).
A E
E X P ~ , ~ , * ( E ) if
(O,+-1.
A E [O,+m).
(1,+0)] and
The . v e c t o r s u b s p a c e g e n e r a t e d by ,'e dense i n
n.
A E
(1,+=]and
The v e c t o r subspace g e n e r a t e d by dense i n
(3)
ecp,
t h e c l o s u r e of t h e c o r r e s -
By 2.15 and by f a c t t h a t P f ( j E )
E N,
i t i s enough t o show t h a t
T h e r e f o r e we o n l y have t o p r o v e t h a t cp E E ' .
We show t h i s by i n d u c t i o n i n
s u f f i c i e n t l y s m a l l we have
e
convergence i n t h e s e n s e of t h e t o p o l o g y of t h o s e s p a c e s .
From h e r e we g e t
E 8
Now, i f we suppose t h a t
for all (pJ
E 8
E E' for
Hence
i n a l l possible cases.
j S n-1,
we have
ON THE FOURIER-BOREL TRANSFORMATION
149
(ii)
(iii)
3.
BOUNDED SUBSETS
In this paragraph we give a characterization of the bounded subsets of
Ex~ ) k ~ , ~ ( Eand
Exp;(E)
for
k E Cl,+m] and A E ( O , + m ]
for k E [l,+-] and A E LO,+-). k and of E x P ~ ,(E) ~ , ~and Exp0,*(E) These characterizations will be used in the proof that the FourierBore1 transformations are topological isomorphisms. The family of all sequences
aj
2
0
1 r m [aj]J S A
such that
m
a = (aj) j=o
will be denoted by
j-w
3.1 PROPOSITION. norms
'N,k,a
For
and
k
pk,a
of real numbers
E (I,+=), A E (O,+m], defined by
SA.
,
150
M a 1 0 C . MATOS
k ExpN,,(E)
a r e continuous i n k = 1,
A E
and
(O,+-]
and i n
E 8
a
d e f i n e d by
a r e continuous i n
k =
+-,
A
E
Exp$,*(E) and
(O,+m]
Exp;(E)
t h e seminorms
B
and i n
a E S1
Exp;(E)
; ajll+llN Zjf
(f) = P ~ , m , a
and
'N,l,a
respectively.
t h e seminorms
-A
d e f i n e d by
respectively.
'N,-
,a
and
For P1,a
For p- ,a
0
j=O
j=O
a r e continuous i n
PROOF. C(p)
If
p
E
such t h a t
Ex~;,~(E)
(O,A),
aj
then
1
S C ( ~ ) T
and i n 1
1
> o A
.
Exp;(E)
Since
f o r every
P
k E (I,+-),
For
k =
+=
A E
(o,+mI
and
A E
we have:
(O,+m]
we o b t a i n :
respectively.
,
there i s S1 A j E N. Thus for
a
E
151
ON THE FOURIER-BOREL TRANSFORMATION
These l a s t s i x i n e q u a l i t i t e s imply t h e r e s u l t s of our p r o p o s i t i o n .
0 PROPOSITION.
'3.2
k E (l,+=),
For
k
E x ~ ~ , ~ ( (Er e) s p e c t i v e l y ,
p E (0,A)
there i s
respectively
If
k = 1,
Exp;(E)
If
k =
Exp;(E),
A E
(O,+m]
0
of
i s bounded i f and o n l y i f
Exp;(E))
such t h a t
,
A E
(O,+-1,
a subset
0
of
1 ExpN,,(E),
i s bounded i f , and o n l y i f , t h e r e i s
a ,
a subset
A E (O,+m],
a subset
0
of
that
, 1 j-m
f€O
(0,A)
ExpGIA(E),
i s bounded i f , and o n l y i f , t h e r e i s
r e s pe c t i v e l y
p E
respectively,
p €
such t h a t
respectively,
(0,A)
such
152
M A R I O C. MATOS
If one of the conditions ( 1 7 ) , ( 1 8 ) , ( 1 9 ) ,
PROOF. (a)
(22) holds it follows immediately that either in
k ExpN,,(E)
k ExpA(E),
or in
6
( 2 0 ) , (21),
is contained and bounded
since either
or
for every
p‘E
(p,A),
when
p
is a continuous seminorm such that
either P(f) 5 C ( P ‘
)l fl N,K,P’
(+
f
E Expk
)llfllk,p’
(v
f
E ~xpk(~)).
N,A
(E))
or p(f) (b)
‘
‘(P’
63
We suppose that
is bounded in
k Exp
N,A
(E)
(respectively,
Since these spaces are inductive limits of a sequence of
Exp:(E)).
k
DF spaces of the type lows that (E)
a
%Pn
(E)
(respectively,
k (E)), ‘pn
it fol-
is contained in closure of a bounded subset of
k (E)). In order to get our result it ‘pn is enough to show that the closure (for the topology of Exp[,*(E) k [respectively, Expk(E)]) of a closed ball of BN,p ( E ) (respect(respectively,
%Pn
ively
63k(E)) P
for
radius) of some p1 E
(0,A).
p E
%P1
(0,A)
(E)
is contained i n a ball (of finite 1
In fact, i f this is true we have
( respective~y,
Then, for
k E
(respectively,
(l,+m),
(E))
(respectively,
A E
(o,+=]
for some
ON THE FOURIER-BOREL TRANSFORMATION
(17) (respectively, (18)) follows.
Thus
153
The f o r m u l a s (19), ( 2 0 ) ,
( 2 1 ) and ( 2 2 ) f o l l o w i n a n a l o g o u s way. Hence,
k E
let
Ef
8 = If
g
a net
A E
(l,+m),
'
k
'N,P(~);
belongs t o t h e closure ( g i ) iE I
t o p o l o g y of
in
8
such t h a t
3.1
in
l i m gi
= g
and
13.
IlfllN,k,p
of
(0,A)
k ExpNIA(E), there i s
i n t h e sense of t h e
i EI
k ExpN,,(E).
By P r o p o s i t i o n
p E
(O,+m],
Hence
'N,k,a
i s a c o n t i n u o u s seminorm i n
1 ExPN,A(E)
where
and
a
-
j -
p-j
,
ak
= 0
se
k
f
j.
H e n c e , by ( 2 3 ) , we h a v e
T h u s we may w r i t e
Consequently
T h e r e f o r e , if
p1 E
(p , A ) ,
there i s
c(P 1)
2
0
such t h a t
9
154
M A R I O C. MATOS
Hence
p2
for
E
center
(pl,A).
i s contained i n t h e closed b a l l of
Thus
and r a d i u s
0
in
3.3 COROLLARY.
k E
For
p E
there i s
R k ~ ( E ),
(0,A)
3.4 PROPOSITION.
For
k
ively,
63
of
0
a subset
i s bounded i f , and o n l y i f , i s c o n t a i n e d and bounded i n
k E (I,+-), k
A 6 [O,+-),
Exp0,,(E),
a subset
63
of
i s bounded i f , and o n l y if,
,
respectively
k = 1
(O,+m],
The o t h e r c a s e s
@:(E)).
E X P ~ , ~ , ~ ( E r) e, s p e c t i v e l y ,
If
E
A
Exp;(E))
such t h a t
(~ r e s p e c t i v e l y ,
and
[l,+m]
E x k~ ~ , ~ ( E ( r )e s p e c t i v e l y ,
(E).
N,P2
p2
0
a r e proved a n a l a g o u s l y .
gk
and
A E [O,+-),
a subset
of
0
E X1 P ~ , ~ , ~ ( E r)e,s p e c t -
1
E X ~ ~ , ~ ( Ei )s , bounded i f , and o n l y i f ,
,
respectively
If
k = +-
,
and
respectively,
A
E
[O,+-),
a subset
63
of
m
E X ~ ~ , ~ , ~ ( E ) ,
E ~ p m g , ~ ( E ) ,i s bounded i f , and o n l y i f ,
ON THE F O U R I E R - B O R E L
155
TRANSFORMATION
respectively ,
PROOF.
4
Straightforward.
FOURIER-BOREL
4.1 DEFINITION. k
ExpNtA(E)
(for
k E
(l,+m]
(for FT
TRANSFORMATIONS
i s i n any o f t h e c o n t i n u o u s d u a l s o f E x p (E), N
If
T
k E
(l,+m]
A E [O,+a)),
and
FT(cp) = T ( e V )
i s d e f i n e d by
the continuous dual o f
Bore1 t r a n s f o r m
cp E E’
A = 0
For 1
llcpll
such t h a t
E + i ; T = 1.
For
W e define
cp E E ’ . A
E
transform
If
the Fourier-
(O,+m))
FT(cp) = T ( e V )
i s g i v e n by
is in
T
for all
c A.
=
k = 1,
h(k) =
k‘E
A =
k‘=
k-l
A E
for
we set
+m,
If
(O,+m).
A-l
= 0.
a s t h a t n u m b e r such t h a t
(l,+m)
w e set k
1
=
A-l
and, i f
+m
(1,+-) w e d e f i n e
1
k’= 1.
A-l
the Fourier-Bore1
for all
A s usual w e set
w e denote
k E
T
k
E ~ P ~ , ~ , ~ ( E )
and
(O,+a])
E x ~ ~ , ~ ( E ( f )o r
of
FT
4.2 NOTATIONS.
A E
and
+a
for
and f o r k E
k
(I,+=).
=
+a,
w e put
Since
k
l i m X(k) = 1 = l i m l ( k ) , k+ 1
(k-1) w e set
h(1) =
X(+m)
= 1.
k-Ko
4 . 3 THEOREM.
The F o u r i e r - B o r e 1
isomorphism between:
and
A E
and
A E [O,+a).
(O,+=].
transformation
F
i s a vector-space
156
M A R I O C.
PROOF.
By t h e d e f i n i t i o n of
F
clear that linear
F
i s an i n j e c t i o n .
MA.TOS
and by P r o p o s i t i o n 2 . 1 7 ,
It i s a l s o q u i t e c l e a r t h a t
F
is
. F i r s t we c o n s i d e r c a s e (1) w i t h
I n t h i s case, is
it i s
C(p)
> 0
f o r every
if
T E [ E x pkN , A ( E ) ] ‘ ,
k E (l,+m)
then f o r a l l
A
and p E
E
(0,A)
(O,+m]
there
such t h a t
f E E x k~ ~ , ~ ( E ) . Hence, f o r
P E PN(jE),
w e have
IT(P)
If w e s e t
T . = T ~ P ~ ( J E )w , e h a v e , by a r e s u l t of Gupta ( s e e [l]) J
BTj E P ( j E ’ )
eTj(ep) = T j ( c p J )
with
IIBT jll for all
for a l l
for a l l
P < A.
E E’ ,
p E
Hence we may w r i t e
By ( 3 0 ) w e have
(0,A).
Hence
for a l l
cp
E
E‘
and
ON THE FOURIER-BOREL TRANSFORMATION
157
Since
i t f o l l o w s f r o m ( 3 2 ) t h a t t h e r a d i u s of c o n v e r g e n c e o f (31) i s Therefore
FT
E
Exp
k‘
(E’
0 , ( A h ( k ))
Now we c o n s i d e r
H
E
Exp
1*
k’ 0 , (Ah(k)
p
< A,
Let
Tj
p
> 0,
E
[P,(jE)]’
= IlSjH(O)II
For
there i s
be such t h a t
( s e e Gupta [ 1 1 ) .
f E ExpN,,(E) k
> 0
C(p)
1
(E’ )
.
BT. = S J H ( O ) J
Hence
m
Hence
Hence, f o r e a c h
such t h a t
we d e f i n e
TH(f) =
+m.
1 C T J. ( -J . j=O
ZJf(0)).
and
l/Tjl/ =
MARIO C .
158
’ c ( P ) C ( e ) l l f / / N,K
ITH(f)I
for all
MATOS
k f E E x ~ ~ , ~ ( E a) l , l
p < A,
p 7 0
k TH E [ E x p N , A ( E ) ] ’ .
This implies t h a t
p
’l+C e > 0.
and a l l
It i s v e r y e a s y t o show t h a t
FTH = H . Now we p r o v e t h e c a s e (1) w i t h If
T E [ExpN,,(E)]‘,
for e a c h
k = 1 p
A E
and
E (O,A),
.
(O,+-]
there i s
c(p) > 0
such t h a t
for all
If
f E Exp (E). N,A
T j = T I P N (’E)
BTj E p ( j E ’ )
,
Hence, f o r
P E pN(jE),
,
by a r e s u l t of Gupta [l] pT.(cp) = T . ( c p j ) J J
such t h a t
we h a v e
we have
for a l l
Cp
E E’
BY (36)
\lBTjl/ = llTjll.
W e have 0)
Wcp)
, IIcpIl
=
1
c j! T
m
( c ~ ~ =)
j=O
c j=o
1
p ~ ~ ( l p )
for a l l
~p
E E’
for a l l
p
E
(0,A).
Hence t h e r a d i u s of c o n v e r g e n c e of
for a l l
p E
(0,A).
Thus i t i s
C A
( s e e P r o p o s i t i o n 2.17).
2 A.
By
It f o l l o w s t h a t
( 3 7 ) w e have
(37) i s z p
ON THE I”OUR1ER-BOREL TRANSFORMATION
H E Exp”
if
Now,
l(E’)
= Wb(BA(0)),
159
we have
0 -
’A
B y t h e r e s u l t of G u p t a mentioned e a r l i e r t h e r e i s BT. = i J H ( 0 ) J
such t h a t
f E ExpNPA(E)
and
= IIijH(0)II.
llTjll
T j E [PN(JE)]’
For a l l
we d e f i n e
W e have
BY
( 3 9 ) , if
p E
(0,A)
1
--
there is
IliJH(0)II
c(p)
S c(p)
Hence
(40)
and (41) imply
Hence
TH E
[ExpNVA(E)]’.
(0,A)
k =
T E [Ex~;,~(E)]’ t h e r e is
1 -
Y
j E N.
A s usual i s e a s y t o s e e t h a t
Now, we prove c a s e (1) w i t h
p E
such t h a t
P j
j!
Let
> 0
c(p)
>
+m
and
A
= cBm(Bl(0))]‘. 0
A
such t h a t
E
FTH = H.
(O,+m]. Thus, f o r a l l
160
MARIO C.
j E
P E P,(jE),
for all
T
If
B T E~ P ( j E ' )
G u p t a [ I ] we h a v e
BTj(cp) = Tj(cpn)
IN.
for a l l
MATOS
= TIPN(jE),
j
by a r e s u l t
IIBTjll = IITjll
such t h a t
of
and
cp f E ' .
W e have co
(FT)(rp) =
c
1
jr
W r p E E'
BTj(")
j=O
a n d , by ( 2 3 1 ,
1 l i m s u p [lBTjll j+m
j
S
i P
+ p
E
(0,A).
Hence
In c l e a r t h a t
Hence
FT E E x p
l(E'). 0
Now,
-'
'A
let
H E Exp
It f o l l o w s t h a t
l(E'). 0 -
'A
Let
T j E [P,(jE)]'
BT. = : j H ( O ) J
and
IITjll =
T h i s i s p o s s i b l e by the r e s u l t of Gupta mentioned
= IlAjH(0)II. earlier.
be such that
W e define TH(f) =
Tj(+) Ajf 0
(44)
j=O
for all
f
E
Exp;,*(E).
Thus, f o r
p
E
(O,A),
3 c(p) 7 0
such
that
for a l l
j E N.
T h i s f o l l o w s from
(43).
Now,
from ( 4 4 ) , w e g e t
O N THE FOURIER-BOREL TRANSFORMATION
ITH(f)l
that
E
p
for all
(0,A).
Thus
‘ ‘7 m
‘ TH E
I d l f ~O I
l
N
p
j=O
m
( E x p N , * ( E ) ) ’ - I t i s e a s y t o prove
FTH = H . Now we p r o v e c a s e ( 2 ) w i t h
c(p)
> 0
E (l,+m)
pTj E P ( j E ’ ) ,
and
A E [O,+m).
Thus t h e r e a r e
p > A
such t h a t
P E pN(jE),
for all
k
T E [ E x p N , O , A ( E ) ] ’ be g i v e n .
Let and
161
j
E
By G u p t a l s r e s u l t , mentioned e a r l i e r ,
N.
BTj((p) = T j ( ( p J )
and
IIPTjll = llTjll.
T h u s , by
(45),
and
Hence
FT
E
Exp
k’
(E’).
A 1(k)
Now l e t
H E Exp
k’
(E‘)
ah(k) and
c(p)
> 0
such t h a t
be g i v e n .
Hence t h e r e a r e p
E
A
162
MARIO
MATOS
C.
By a G u p t a t s r e s u l t m e n t i o n e d e a r l i e r t h e r e i s that
T . = s''(djH(0)) J
f o r every
f
E
/ITjll = IldJH(0)ll.
and
k (E). Exp N,O,A
We u s e ( 4 6 ) and
T j E [PN('E)]'
such
Now, we d e f i n e
I/Tjll = IldJH(0)ll
and
we o b t a i n
(47)
= c
If e > 0
i s such t h a t
p>
l+S
ITH(f)I
for all
f
NOW
E
k (E). Exp N,O,A
A
' '(P)'(') Hence
-
w e get
TH
IlfllN,k E
' l+C
[ E x P N , O , A( E ) ] '
t h e r e i s only t h e c a s e ( 2 ) w i t h
k = +a
and FTH = H. and A E c 0 , + O D ) .
I n t h i s case (El = HNb(B1(0))' A The p r o o f s w e r e d o n e i n G u p t a E l ]
a n d Matos [l] and [ 2 ] .
Q.E .D.
ON THE FOURIER-BOREL TRANSFORMATION
4.4
REMARK.
cp E E'
and
A s we saw i n P r o p o s i t i o n
llcpll
d e f i n i t i o n f o r i t s Fourier-Bore1
= T(e')
cp E E '
for a l l
t h a t we c a n d e f i n e
FT
with
eV E Exp N,O ,A(E)
T E [ E x p N , O , A( E ) ]
Hence if
A.
6
2.17,
transform
FT
'
,
if
the natural FT(cp) =
would be
However i t c a n b e p r o v e d
IIrp/( L A.
for a l l
163
cp E E '
with
)IcpII
s PT
,
PT > A ,
i n such a v e r y t h a t i t a g r e e s w i t h t h e p r ev io u s d e f i n i t i o n f o r
Ilcpll
cp E E ' t
E3
A.
4 . 5 DEFINITION.
If
T E [ExpN,O,A(E)]',
Bore1 tra nsf orm
FT
by
for a l l
cp E E'
T . = TIpN(jE) J
such t h a t and
PROOF.
If
(49) converges a b s o l u t e l y .
BTj E F ' ( j E ' )
A s we w r o t e p r e v i o u s l y
Thus
i s g i v e n by
IIBTjll = llBTjll
f E E X ~ ~ , ~ , ~ ( Hence E ) .
Here
BTj(cp) = Tj(cpn).
by a G u p t a l s r e s u l t .
T E [ExpN,O,A(~)]'there are
such t h a t
f o r all
we d e f i n e i t s F o u r i e r -
p 7
o
and
C(P)
> 0
164
MARIO C .
It f o l l o w s t h a t c o n v e r g e n c e of
s i n c e ( 5 0 ) s a y s t h a t t h e r a d i u s of
FT E H b ( B p ( 0 ) ) FT
is
2
MATOS
p. Q.E.D.
4.7
THEOREM.
The F o u r i e r - B o r e 1 t r a n s f o r m a t i o n
isomorphism between
(E)]’
and
Exp;(E‘)
-
rExpN, o , A PROOF.
F
for A E [O,+m),
A
4 . 6 and t h e d e f i n i t i o n o f
By P r o p o s i t i o n
i s a v e c t o r space
Expy(E’)
-
A FT E Exp;(E’)
clear that
T E [Exp
for all
-
A
T E [ExpN,O,A(E)]’ i s such that all 0
cp E E ’ ,
IlcPll
= IIBTj/l = llTjll
and Hence
< p
f o r some
for a l l
P E PN(jE),
Thus
Therefore, f o r every
j E N.
T j E [ P N ( ’E)]’
H E Exp;(E’)
-A
H E ab(Bp(0)).
such t h a t
T h u s , by
j E IN.
(see
C
> 0,
If
(FT)(cp) = 0
4.6).
T(P) = 0
for
Hence for a l l
j E N
T = 0.
= Ub(BA(0)).
Therefore
there i s
c(e)
such t h a t
such that
pT
.
= ajH(0)
J
and
I(Tjll = I I a J H ( O ) I l ,
( 5 1 ) , we h a v e
a
C
T j ( b :jf(O))
j=O f
Hence t h e r e i s
By a G u p t a f s r e s u l t m e n t i o n e d e a r l i e r t h e r e i s
TH(f) =
for a l l
.
i t f o l l o w s from P r o p o s i t i o n 2.15 t h a t
Now we c o n s i d e r
for all
> A
p
then
(E)]’
i s an i n j e c t i o n .
F
p > A
FT = 0 ,
N,O,A
it is
E Exp
N,O,A
(E).
We use (52) t o o b t a i n
ON THE FOURIER-BOREL TRANSFORMATION
for a l l
E
f
p> A. l+e
p
E X ~ ~ , ~ , * ( E )a ,l l
Thus
TH
E
>
[ E x p N , O , A( E ) ] ’
and a l l
A
.
165
e > 0
such t h a t
It i s e a s y t o s e e t h a t FTH = H. Q.E.D.
4 . 8 THEOREM.
The F o u r i e r - B o r e 1
transformation
i s a topological
F‘
isomorphism between:
and
E
A
(O,+m].
k [ExpN,A(E)]
Here
k CEXPN,*(E)I‘ a n d PROOF.
k rExpN,o,A(E)];
and
CExPN,O,A( E ) ] ’
k ExpN,,(E)
Since
denote the duals
with the strong topologies.
i s a DF s p a c e ,
k’ Exp
(E’)
is a
oq$T i s a n a l g e b r a i c isomorphism, i n o r d e r t o
F
F r 6 c h e t s p a c e and
p r o v e ( a ) i t i s enough t o show t h a t
F-’
t h e Open Mapping Theorem i m p l i e s t h a t
Now w e prove t h a t A E
Let
(O,+=],
By 3.2
j
E
N.
E > 0
p
In fact,
i s continuous.
i s continuous f o r
b e a bounded
there i s
Hence, f o r e v e r y
for all
bd
F-’
F
i s continuous.
s u b s e t of
E (0,A)
such t h a t
there i s
c(c) 2 0
k
E
(l,+m)
k Exp (E). N ,A
such t h a t
and
166
i4ARIO C . MATOS
W e have:
Since
that
W e know
Hence there is
d(e)
z 0
such that
Theref ore
N o w we choose 1
X ( k ) ( p t$ ) ( 1+e 7
e > 0 1 7
such that
--
the other values of
and k
for every
that
p
< A.
is continuous.
The proofs for all
to,+-).
and
A E
be a continuous seminorm in
Exp
p < A,
Hence
for case (a) f o l l o w the same pattern.
N o w we prove case (b) for
Let
F-l
(p+e)(l+e)
we have
k E (1,+-)
1
k'
6,
(E').
Hence,
X(k)A
>
and there is
c(p)
such
ON THE FOURIER-BOREL TRANSFORMATION
k T E [ExpN,O,A(E)]‘.
f o r every
We c o n s i d e r
m
If
03 =
for all
and
8
u j=O
$3
j E IN.
w e have
Hence
i s a b o u n d e d subset o f Now w e w r i t e
Since
k
by P r o p o s i t i o n 3 . 4 . E X P ~ ,( E~) , ~
167
MARIO C .
168
c(c) 2 0
there i s
j E N.
or a l l
MATOS
such t h a t
Hence
where
o > 0
f o r all
such t h a t
p ( l + e ) < A. F.
T h i s proves t h e c o n t i n u i t y of
I n o r d e r t o p r o v e t h e c o n t i n u i t y of bounded i n
by P r o p o s i t i o n
3.4.
Now, for e v e r y
such t h a t
for a l l
F-’
we c o n s i d e r
63
E X kP ~ , ~ , ~ ( EHence )
j E N.
We h a v e
p
>
A,
there i s
c(p) 2 0
ON T m FOURIER-BOREL TRANSFORMATION
for a l l
e
for a l l
r <
7 0
and
0
>
A.
Hence
-j--$jA-.
T h i s proves t h a t
F-l
i s continuous.
The p r o o f s f o r t h e o t h e r v a l u e s of
k
f o l l o w t h e same
0
pattern.
REFERENCES
1.
C.P.
GUPTA,
Malgrange theorem f o r n u c l e a r l y e n t i r e f u n c t i o n s
of bounded t y p e neiro,
2.
C.P.
-
Notas de Matemitica
3 7 , IMPA, R i o d e Ja-
1968.
GUPTA,
C o n v o l u t i o n o p e r a t o r s a n d h o l o m o r p h i c m a p p i n g s on
a Banach s p a c e ,
S e m i n a i r e d c A n a l y s e Moderne, n2 2 , U n i v e r -
s i t 6 d e S h e r b r o o k e , S h e r b r o o k e , 1969. 1.
L. N A C H B I N ,
T o p o l o g y on s p a c e s o f h o l o m o r p h i c m a p p i n g s ,
E r g e b n i s s e d e r M a t h e m a t i k , 47
1.
A.
MARTINEAU, Bull.
1.
M.C.
Equations d i f f b r e n t i e l l e s d f o r d r e i n f i n i ,
S O C . Math. F r a n c e ,
MATOS,
(1969)~ Springer-Verlag.
95 ( 1 9 6 7 ) , p . 109-154.
O n M a l g r a n g e Theorem for n u c l e a r h o l o m o r p h i c
F u n c t i o n s i n open b a l l s of a Banach s p a c e ,
Math.
Z.
162
( 1 9 7 8 ) , 113-123. 2.
M.C.
MATOS,
C o r r e c t i o n t o "On M a l g r a n g e Theorem N u c l e a r h o l o -
m o r p h i c F u n c t i o n s i n open b a l l s of a Banach S p a c e " ,
Z.,
1 7 1 ( 1 9 8 0 ) , 289-290.
D e p a r t a m e n t o d e MatemAtica IMECC
- UNICAMP
Caixa P o s t a l 13100
-
6155
Campinas, S . P . ,
Brasil
Math.
This Page Intentionally Left Blank
Functional Analysis, Holomorpliy and Approximation Theory 14 G.I.Zapata (ed.) @ Ekeuirr Science Publishers B. V. (North-Holland), 1984
ON REPRESENTATIONS OF DISTANCE FUNCTIONS I N THE PLANE
John McGowan
and
Horacio Porta
INTRODUCTION
ly symmetric curve assume that
r
8,sin 8 )
P(8) = r(e)(cos
Suppose that C
describes a centralr(8+rr) = r ( 8 ) ) .
in the plane ( s o that
is a continuous function and that
r(8) > 0
We on
c 0,2lTI The distance function (sometimes called the Minkowski functional
of
is the real valued non-negative function
X = IXl(cos a,sin a )
for
is convex,
Lc
C
Lc
in
by
iR2
Lc(X)
is the norm associated to
consisting of the union of
C
Lc
= IXl/r(a).
C,
defined
When
C
the unit ball for
and the region enclosed by
C.
The objective of this paper is to find integral and differ-
Lc. More specifically, for a class of
ential representations for
curves (essentially those having finite angular variation and satisfying an interior cone condition) w e obtain in $4 the formula LC(X) = where
x
dM C
Ip(e)xxl
dM(8)
denotes cross-product of vectors and where
appropriate Borei measure. tion
6'
of plane norms (where is unique, and
dM Z 0
is convex in the section
dM
is an
This generalizes the Levy representadM Z
0;
see 7 . 8 below).
on an interval
(a,p)
The measure
if and only if
a < 8 < p.
As usual, this representation for
Lc
gives when
convex the classical result that the normed space
(W2,Lc)
C
is can be
172
JOHN MCGOWAN
embedded isometrically in
and
HORACIO PORTA
L1[O,l]
(as shown in $ 2 below using
ultraproducts). I n view of the uniqueness result, it is reasonable to expect a simple expression for
dM
P(f3).
in terms of
We prove in $ 5
that dM =
where
R(8)
= l/r(e)
1 2
+
R(R
d2R T )d8 df3-
and where the derivative
interpreted in the sense of distributions. turn be used for curves of class ternative expressions of
dM
C2
d2/de2
is to be
This formula can in
(convex o r not) to find al-
in terms of geometric notions.
Among
other results, we prove in 96 that
where
n
is the curvature of
the ray through
0
and
P(0)
C
a
and
is the angle formed by
and the tangent to
C
at
P(0).
Finally, in $7 we consider several notions related to including its moments.
dM,
Among several inequalities and identities
we prove, for example, that [r2 dM
is invariant under inversion
through the unit circle. We want to thank R.P. Kaufman, H.P. Lotz and T. Morley for their valuable comments on parts of this work.
6 1.
POLYOGONAL EQUATIONS Let us consider a polygon in the
i = l,Z,...,n.
..,PZn
plane, symmetric
Pn+i = -Pi Here and in the following we do not distinguish
about the origin with vertices for
x, y
P1,P2,.
where
between points in the plane, and their position vectors with origin at
0 = (0,O).
Suppose that the
...,Pzn
P1,P2,
are all non zero
and that they form a radial sequence, i.e., they are totally ordered
ON REPRESENTATIONS OF DISTANCE FUNCTIONS
by their central angle (varying counterclockwise).
173
We assume
further that this order is strict s o that no two distinct vertices lie on the same ray from the origin.
For convenience we set P o = - P n
a .
denote the component of the cross product iJ Pi X P . in the positive z-direction, f o r 1 (= i, j 5 n (here J and in the following we consider the x,y-plane as the set in Let now
x,y,z-space characterized by 1.1 PROPOSITION.
The matrix with entries
with inverse having entries
for
1 5 i 5 n-1,
z = 0).
and
Bij
pij
= 0
laijl
is invertible
given by
for all other
i,j.
The proof of this proposition along with various observations on the matrices
([uijl) and
(Bij)
will be found in the
.
174
JOHN MCGOWAN
Appendix, $ 8 . of
(Pij)
and
HORACIO PORTA
We remark that by definition, the diagonal entries
are non-positive while the off diagonal entries are po-
sitive. Consider now the expression
n
c
L(P) =
mjlPjxPl
j=1
where
P = (x,y)
is an arbitrary point in the plane and the
m Is
are arbitrary real numbers not all zero.
It is clear that for
between the rays from the origin through
Pi
tion
L(P) = 1
P
Pi+l, the equa-
and
is equivalent to P .
-
m.P.xk (j$i
where
i
J
J
C j> i
m.P.Xk) J J
= 1
k = (0,0,1) is the unit vector in the z-direction; whence,
it is the equation of a straight line. Therefore,
L(P) = 1
represents a polygonal line with
vertices on the rays from the origin through the
P.ls,
J
ly symmetric about the origin
I /”
\ ‘
and clear-
175
ON REPRESENTATIONS OF DIST4NCE FUNCTIONS
1.2 PROPOSITION.
Given
P1,P2,...,Pn,Pn+l = -P1,...,P2n
radial sequence, there exist unique
L(P) = C m.lPxPil = 1 that order) mi
P1,P2,
ml,m2, ...,m
n
= -Pn
in
such that
is the equation of the polygon joining (in
...,Pn, -P1,...,-P n,P1.
Further, for each
is non-negative if and only if the quadrilateral
i,
OPi-lPiPi+l
is convex (hence the polygon is convex if and only if all
"'j
are
non-negative).
mn
But since the area of the triangle
-
-1 2 &i-l,i
'
1
OPi-lPi
is
1 T
IPi-lXPil =
and similarly for all the other values of
i,
from the last equation
and therefore,
mi t 0
if and only if
area(OPi-lPi+l) z
area(^^^,^^^)
+ area(OPiPi+l),
we get
J O H N MCGOWAN
and
that is, if the quadrilateral
H O R A C I O PORTA
i s convex.
OPi-lPiPi+l
$ 2 . ISOMETRIC EMBEDDINGS C o n s i d e r a s y m m e t r i c onvex p o l y g o n
r
n m.IP.XPI. J J
it is clear that
= lalJ+(P),
t o t h e convex body bounded by ed space
= 1 on
L
Since
r
with equation
r
Lr(aP) =
and
Lr
i s t h e n o r m on
r.
We s h a l l d e n o t e by
R2
associated
B,
t h e norm-
2
(IR , L r ) .
n
m =
Let
C
j=1
m. J ’
= m./m
p
J
d i s j o i n t union of i n t e r v a l s
E(3) =
[ul
+
u2,
p1 + b 2 +
and decompose
E ( 1 ) = [O,F,),
w3),
etc.
as the
[O,l)
+
E ( 2 ) = [bl,l-ll
with lengths
p2),
k(E(j)) = pjY
1 s j s n . Suppose now t h a t t h e c o o r d i n a t e s of t h e and d e f i n e f u n c t i o n s
E( j ) ,
TP = m(xf Then
j = 1,2,.
for
+
yg)
f,
g
. ., n .
where
on
r0,l)
by:
Finally define
P = (x,y)
P . J
f = b j ,
T: R 2
-t
are
P . = ( a .,b.) J J J
g = -a
j
L1(O,l)
i n an a r b i t r a r y v e c t o r i n
on by 2
IR
.
ON REPRESENTATIONS OF DISTANCE FUNCTIONS
n = m
-
C (mj/m)lxbj j=1
177
ya.1 = J
n
Thus
T
B~
is an isometry from
into
L'(O,I).
This is a particular case of the classical result that all 2-dimensional normed spaces can be isometrically embedded in 1 L1 = L (0,l)
(see for example
[4]). The following argument shows
that the general case follows from the case of polygons very simply. Suppose that convex curve polygons
)I 11
IIPI/ = 1.
r(n)
is a norm on
We can approximate
inscribed in
= lim L ~ ( ~ ) ( P ) . Let now
n
(T,)
C
of countably many copies of
we have
T:
a separable subspace of
L1/b,
so
that
LL/b,
C
(see
T
C
be the
by a sequence of IlPIl =
be isometries.
R2 + L1/k
([&], Proposition 11
IITpllL1/ll = IIPII
sublattice of
L1
and let
in the sense that
+ L1 T ~ B : r(n)
defines a linear operator
self an Ll-space
R2
The family
into the ultraproduct
r4], p.
121ff),
2.10).
Since
is an isometry.
which is it-
But
T(R2) is
and thus contained in a separable
which, by the classical Kakutani representa-
tion theorem can be identified with
L'(0,l).
It follows that all
2-dimensional normed spaces can be isometrically embedded in
L1( 0,l).
178
$3.
and
J O H N MCGOWAN
HORACIO PORTA
SOME ESTIMATES
Consider again the vertices = -p,,
,...,Pn,Pn+l
P1,P2
= -P1
,...,P2n
=
(ordered counterclockwise in radial sequence) of a symmetric
r.
polygon chosen
Denote by 0<
that
so
yi
Yi <
the angle from
Pi+l
-
Pi
to
Piml -pi,
2 ~ .
I n and let
C
milPixPl = 1
be the equation of
r.
i= 1 We know that ( s e e formula (1.2.i)
in the proof of Proposition
(1.2) ) : m
i -
area(0P i-1P.)-area(OP 1 i-1P i+l )+area(OPiPi+l) 4 area(OPi-l~i)area(OPiPi+l)
and the sign is
+
or
is not convex "at Pi since
-
depending on whether the polygon is or
(I,
area(^^,^^^^^+^) 1
=
1
Ipi,l
-
pil
- pil lsin yil
where a denotes the distance area(0P P ) = p IP.-Pkla j k J j,k j,k f r o m the origin to the straight line through Pj and P k , we get and
ON l?EPRESENTATIONS OF DISTANCE FUNCTIONS
(3.i)
m
-
-
sin y i 2ai-l,iai,i+1
179
*
Notice that the signs match automatically. Assume now that
ak,ktl Z a > 0
f o r all
that the distance from the line through
k.
Pk,Pktl
This means
(which surely
misses the origin) to the origin is bounded below by
n
c
Imil
s 2l
i=l Denote now by
-
Pi
Pi,l
Since
and
6i =
vi
2a
-
= min( 16i1 ,n
-
ISil),
lsin y i l .
6i
i s the (signed) angle formed
P i , so that
lsin y i I = Isin 6 .
[sin y i l
Then
i=l
- yi9
ft
Pi+l
We remark that, since
C2
a.
I
-n < b i <
5 16.1
= lsin 6 i
n.
we have
I
is also majorated by
we can improve (3.ii) somewhat:
(3.iii) If the polygon is convex, then
bi
% 0
for all
i,
and
180
n C
and
J O H N MCGOWAN
bi
= i ~ ,
whence (recall that
HORACIO PORTA
mi 2 0
by Proposition (1.2)):
i=l
n
C
m i s lly . i=l 2a
(3.i~)
$4.
INTEGRAL REPRESENTATION
Suppose now that
is a symmetric curve described in polar
C
P ( 8 ) = (x(e),y(8)),
coordinates by
0
S
8
g
2n
8
where
is the
central angle measured as usual counterclockwise from the positive x-axis. that
Symmetry of
2rr,
and that
n
also denote by
I P(8 ) I } ) .
C
C
= -P(e). 8 ,
periodic with We will
[ (r cos 8 , r sin 8 ) ;
For each partition
J
jl
C(n =
the interior of
Pn+j = ~ ( n + e . ) = 16
We also assume
does not pass through the origin.
r(n)
we can consider the polygon and n
P(B+n)
P ( e ) is actually defined for all values of
period
r <
means that
C
- ~ ( Je. )
= -P
P j = P ( 8 j)
whose vertices are for
j '
,...,n.
j = 1 , ~
is the total angular variation of
r(n)
on
The s u m
0
g
8
S
n
j=1
denotes the angle described in $3); the (possibly infinite) (6j quantity
is the total angular variation of
is that we want to assign to the whole curve
[O,n].
C,
T(C)
C.
The reason for the factor 2
the value that corresponds to
and this is double the angular variation on
Observe that adding a new partition point to
a larger value of
C16jl,
n
results in
and therefore, we can calculate
T(C)
as
the supremum taken only over all partitions finer than a given partition
no. Recall [ S ]
that the kernel
ker
n
of a set
n
is the set
181
ON REPRESENTATIONS OF DISTANCE FUNCTIONS
of a l l
POP
s u c h t h a t t h e segment
Po
ker R = R
R
i s contained i n
in
0.
F o r example,
= {O)
if
R
i s t h e i n t e r i o r of t h e a s t r o i d
Since
ker
n
i s a l w a y s c o n v e x ( s e e r8] f o r t h e g e n e r a l c a s e ; [ 5 ]
P
R
f o r each
if
i s c o n v e x and
ker R
P r o b l e m 111.111 f o r t h e p l a n e ) a n d c l e a r l y
a
metric i f
+
lxI2/’
i s , i t follows i n t h i s case t h a t
ker 0 =
IyI2/’
= 1.
i s c e n t r a l l y sym-
ker R
h a s non-empty
i n t e r i o r i f and o n l y i f i t c o n t a i n s a d i s c around t h e o r i g i n .
4 . 1 PROPOSITION. a
around t h e o r i g i n .
n
such t h a t f o r a l l gon
r(n)
The p r o o f
ker R
Suppose t h a t
Then f o r a n y
> 0
G
ne,
f i n e r than
c o n t a i n s t h e d i s c of r a d i u s
t h e c o e f f i c i e n t s of
f o l l o w s e a s i l y from
d e f i n e d by
C mi b g i
t h e p a r t i t i o n p o i n t s and
mi,
t h e e q u a t i o n of t h e polygon usual,
II
e
t h e poly-
satisfy
(3.ii).
n,
C o n s i d e r now, f o r e a c h p a r t i t i o n [O,rr)
there is a partition
0 = 9
where
< 9,
i = l , . .. , n ,
r(n).
<...<
The m e a s u r e
dMn
dMn
8n < rr
on are
t h e c o e f f i c i e n t s of
6g
denotes, a s
9.
t h e u n i t point m a s s concentrated a t C l e a r l y t h e t o t a l mass of
t h e measure
is
a n d a c c o r d i n g t o t h e p r o p o s i t i o n a b o v e , t h e s e numbers a r e a l l bounde d by
bounded:
n
of measures i s But t h e n t h e n e t [dM )nen, -, hence a s u b s e t ( d M 1 l ’ ] n , c o n v e r g e s i n t h e weak* t o p o l o g y
T(C)/h(a-e)*.
of measures
( = vague topology f o r continuous i n t e g r a n d s ) :
dM = l i m dMn’
the l i m i t ,
which c l e a r l y s a t i s f i e s
d e n o t e by
JOHN MCGOWAN
182
For
0
#
X E IR2,
by the property that Then, for each
X,
and
HORACIO PORTA
denote by
X/X
Lc(X)
is on the curve C ,
is the limit of the numbers L
Lc(X)
n',
defined
and set also Lc(0)= 0.
r(ni
since they actually agree as soon as the argument of titi'on point of
X
the number
X
is a par-
(n']
and this will happen eventually as
is
cofinal in all the partitions, On the other hand,
and
taking limits along
so,
LC (x) =
{n'}
we conclude that
6'
I P ( 9 ) XX I dM( 9 ) .
These remarks prove the first half of the following result:
4.2 THEOFEM.
Let
0
c R2
with boundary the curve and
Lc(X)
be a centrally symmetric bounded domain
C.
Let
P ( 9 ) = (x(Q),y(g)),
0
9 S 2r,
S
denote the equation and the distance function of
C.
Suppose that (4.2.i)
ker R
(4.2.ii)
C
has non-empty interior;
has bounded angular variation.
Then there exists a unique finite measure
xE
for all
(4.2.iii )
IR~:
LC (x) =
6
IP(e)
x
IdMl 5 T(C)/ka
2
dM
on
[O,n]
such that
xldM(e)
and it satisfies
( 4.2. iv) where
a
is the radius of the largest circle contained in
ker R .
ON REPRESENTATIONS OF DISTANCE FUNCTIONS
Further, if
dM Z 0
R
then
183
R
is convex and if
then (i) and (ii) are satisfied automatically and
dM
is convex,
2 0.
I n this
case
It only remains to establish the uniqueness of
PROOF.
1x1
denote the euclidean length of the vector
r(e)
= IP(e)l.
X,
dM.
Let
and abbreviate
Then Jp(e)xxJ = r(e)lxl Isin(8-t)l
where
X = (IXlcos t,lXlsint).
(4.3)
LC(X)
=
Suppose now that difference
dN = dM1
(4.4) t.
for all
and
r
1x1
i'
dM1
and
dM2
satisfy (4.3).
dM2
Then the
satisfies
I sin(@-t)
G = [O,n)
= 0,
Ir(e)dN(e)
dN = 0 .
Then
In fact, if we
mod n
(with addition
as invariant measure).
(4.5) where
I sin(e-t) I r ( e )dM(e).
This, however, implies that
consider the group 1 -do n
-
X E R2:
Thus, for all
as operation
(4.4) reads
Isin1 *rdN = 0
*
denotes convolution and
The characters of
are
G
lsin e
2inB
(e)
el, e
= Isin
- - -c 2
+a
4
rdN we conclude from
-
(4.5) that
1 2niB z e
4n -1
m
C
(rdN)A(n)e
G.
n = 0,*1,&2,... and
taking the Fourier transforms: Isinl(e)
E
2ni8 9
9
184
J O H N MCGOWAN
and
H O R A C I O PORTA
- -~~( r d N )A (n) = TT
and t h e r e f o r e ,
0
4n2-1
A
( r d N ) (n) = 0
for a l l
n.
Thus
rdN = 0
and
uniqueness follo w s .
05 .
DIFFERENTIAL REPRESENTATION
Let u s apply formula Since
X
( 4 . 3 ) of t h e l a s t s e c t i o n t o C,
i s on t h e c u r v e
we h a v e
Lc(X)
X = P(t).
= 1 and t h e r e f o r e ,
(4.3) reads (5.1)
1 = r ( t ) f'n I '0
I s i n ( 9 - t ) 1 r ( 9)dM(9)
or l / r = Isin1 D e f i n e now a d i s t r i b u t i o n
T
*
rdM.
on t h e compact g r o u p
G = [O,n)
by i t s F o u r i e r t r a n s f o r m m
T - - -TI
c
(4n2-l)eZnie.
-m
Clearly
T
*
[sin1 = 6 , T
whence we g e t for
*
dM
t h e u n i t p o i n t mass a t
(l/r) = T x
lsinl
*
0 E G.
But t h e n
rdM = rdM
the value dM = ( l / r ) ( T
*
l/r).
We c a n g i v e a more e x p l i c i t f o r m t o t h i s e x p r e s s i o n a s f o l l o w s . F i r s t , observe t h a t from
we g e t
-C
m
6"
-m
and t h e r e f o r e ,
-
4n2 e 2 n i 9
ON REPRESENTATIONS OF DISTANCE FUNCTIONS
n
T = T (6
185
+ 6”).
But then Under the hypothesis of 4.2, if R(8) = l/r(e) =
5.2 PROPOSITION.
= 1/IP(e)l,
where
then
d2R/de2
ticular, if
C
in taken in the sense of distributions.
In par-
is a smooth curve (of class
dM
the continuous density
+
(n/2)R(R
R“)
C‘),
then
with respect to
has
d8/w.
We can illustrate the formula (5.2.i) in the case of the 2 z p c: + m ,
p-norms f o r
= 1.
We need
5.3.
Let
p Z 2
and
i.e., when
C
is the curve
= (Cp(e) + SP(e))l/P
E1/’(e)
c ( e ) = lcos
el,
s(e)
= lsin
1x1’
+
lylp=
where
el
then
Observe now that f o r the p-norm,
P(e) = E-1/P(8)(cos
@,sin 9 )
so
that
C R(e)
can be described by
= E1lP(e)
and there-
fore, according to 5.2 and 5.3:
5.4 COROLLARY.
For the p-norm, dM = 1 (p-l)Cp-2Sp-2E-2+2/p do.
In particular the case where
1 dM = 2 d8
C
for the euclidean norm corresponding to
is the unit circle and
p = 2.
Another application of 5.2 is the following.
R2
with the norm
// 1)
Suppose that
is isometric to a two dimensional subspace
186
of
x
JOHN MCGOWAN
and
HORACIO PORTA
t 1, That is, there are vectors
Yn E
El2
such that for all
E IRc: m
c
IIXII =
Yn f 0
We may assume that
n=l
for each
yn = wnp(en
where
wn > 0 and where
where
dN = C wn 6
with
dM.
en
.
IX'YnI.
n.
Write
t i7
IIX((= 1.
is the equation of
P(e)
Then
By uniqueness, this measure must coincide
Thus:
5 . 5 COROLLARY.
The normed space
t1
space of
if and only if
(R2,1/
11)
is isometric to a sub-
In particular,
dM
is purely atomic.
+m
is not isometric to a subspace
n
IRL
with the p-norm,
of
t?
2
s p<
This result appears also in [ 3 ] , first paragraph on p.
494 and last
two paragraphs on p. 498. Observe that if
el
and
e2,
then
and therefore, gon,
dM
R
t
C
coincides with a straight line between
r = a sec(8-€I0) RN= 0
there.
has support in the set
for which
P(0)
dM = C m .b J e
j
is a vertex.
used in
$4.
8,,
for appropriate
a
and
In particular, if
C
is a poly-
{el,e2,
This gives
...,en]
of values of
back the expression
8
ON REPRESENTATIONS OF DISTANCE FUNCTIONS
$ 6 . GEOMETRIC INTERPRETATION Suppose that by
a
C
is smooth (of class
the angle formed by
6.1 THEOREM.
Let
C
1 dM = -
PROOF.
n
is the curvature of
Since
say) and denote
and the tangent vector
r = r(8)
C
and since
at
P(e),
is the equation of
R = l/r):
cotan a = r‘/r
we get also
n sinm’a so that, from (5.2.i),
Then
n dB r sin3 a
C
in polar coordi-
nates, we have (primes denote derivatives with respect to
But then (use
dP(B)/de.
be a smooth centrally symmetric curve.
(6.1.i)
where
P(0)
C2 ,
= R
+ R“
8):
188
JOHN MCGOWAN
and
HORACIO PORTA
1 1 dM = 2 R(R + R“)dB = -
r sin3a
dB
as claimed. Minor modifications allow us to obtain an expression similar to (6.1.i) for arbitrary smooth parametrizations of Let then half of 2
is
C
of
t.
C
,
P(t) = (x(t),y(t)),
(corresponding to
0
a 5 t S b
s 0
P’ = dP/dt f 0, and that
2 n)
e
C.
describe the upper
and assume that
is an increasing function
F o r simplicity, let us use the rotation
to express cross products as dot products by means of = X
P(t)
x Y
where
k = (O,O,l).
Using primes to indicate
d/dt
we have
and dB/dt = (pP’ *P)/(P.P). Use now
= (PP‘ .P”)(P.P)/(pP‘ and finally using
.P)3,
dB/dt = (pP’*P)/(P*P),
we get
(pX*Y)k =
ON REPRESENTATIONS OF DISTANCE FUNCTIONS
dll
(6.1.ii)
PP' .P"
1
= 2
189
dt.
(PP' ' P )
Of special interest is, of course, the parametrization of C ength. 1
Using again
p P ' .P = /PI)/
~ P ' . P = r sin
we get
( 6 . 1 . iii)
dM =
a
-
1
IPIsin a
Also
N
= pP' .P"
and since so that
ds r2sin2a 1 ds = de/sin a).
(which also follows from (6.1.i) using Observe now that letting that
h(8)
r sin
a
= h,
it is easy to see
is the distance from the origin to the tangent at P ( 0 ) .
Clearly we can write the last formula as
I n dM = --ds. h2
(6.1.iv)
Each of (6.1.i) through (6.1.iv) provides a formula for the distance function of
C,
according to (4.2).
For example, using
(6.1.iv) :
47. 7.1 p(e
MISCELLANEOUS REMARKS AND PROPERTIES
Suppose that
)
valid.
is a centrally symmetric curve with equation
for which a finite Bore1 measure Then, using (5.1 r(u)
so
C
-
r(t)
=
we get
dM
exists making (4.2.iii)
JOHN MCGOWAN
190
I
Ir(u)-r(t) and therefore,
and
HORACIO PORTA
5 Er(u)r(t)
in a Lipschitz function with constant not
r(e)
larger than
i,
'IT
K = (maxose zTT t
3
~
IW.
)
We can show by elementary calculations that whenever ker R
is Lipschitz then nK 2 -l/' m(l+(l + 1
z)
r(e)
contains a disc of radius
where
m = inf
r(e),
0 S 0 S R.
Here are the
details. Assume that Ir(u)-r(v)l holds f o r all
and by
a
Oo
KIU-VI
Then
u, V.
for each fixed
5
and all
8.
Denote by
m
the infimum of r ( e )
the angle indicated in the figure:
where the dotted curve has the equation
p = r(eo)
-
K18-eol
the upper half is shown in the figure, corresponding to Clearly
m = r(eo)
the symmetry of
-
C,
Ka.
8 2
(only
eo).
Also, using the Lipschitz condition and
we get
ON REPRESENTATIONS OF DISTANCE FUNCTIONS
a =
which implies that Denote by
S
We will show below that . a
m sin a . r(oo)-m cos a
+
-m/(m
ker R
al
i.e.,
. This implies that the
from the origin to the line through
with slope
But then
+ SK).
S % m/(m
larger than the distance Q
= (m cos a , m sin a ) ,
Q‘
and
s =
di stance
0 S a Z n/2.
satisfies
m)/K
the absolute value of the slope of the line through
Q = (r(eO),O)
the points
-
(r(eo)
191
and
Q
Q‘
is
from the origin to the line through
F K ) ,
that is
K 2-1/2 s) )
contains the disc of radius (1 + (1 +
as
claimed.
In order to show the inequality
o(t)
= v(t)/v(t)
m/(m
+
$K)
we define
Z m/(m
+
5K)
f o r all
S 2
where p(t) = m sin t V(t)
Thus,
S = o(a).
0 z t 2 n/2,
Kt
- uv’. =
= p ~ v
m cos t
o(t)
+ m K t cos t
> a ( n / 2 ) = m/(m
+
4 K)
-
m K sin t
< P(0) = 0
p(t) (J
S.
where
0 < t 2 ~ / 2 , and so
t < 0 for
do/dt = $ / v 2 < 0 and
U(a)
m cos t.
dU/dt = $(t)/v2(t) 2
creasing, which shows that Hence,
-
and this will include the estimate for
d$/dt = -m(m+kt)sin
S =
+
We will show that
Observe that
p(t)
= m
- m2 6
now is de-
for those values of
is also decreasing.
t.
But then
and the proof is complete.
This implies that the fact that
ker fl
has non-empty interior
is necessary for the existence of a finite measure
dM.
JOHN MCGOWAN
192
Observe that the
K
K
and
HORACIO PORTA
obtained above is too large: ker R
in terns of the interior of
the best
and the extrema of
r(8)
was
determined by Toranzos [ 8 ] . 7.2.
Concerning the hypotheses of (4.2) we observe that assuming
that
ker 0
is finite:
r(e)
= 2
ker R
T(C) =
7.3. class
T(C)
it suffices to consider the curve defined by
+ 18'
other half. that
has non-empty interior does not guarantee that
sin(n/8)l Since
r(8)
-
for
n 5 8 5 n 2 and by symmetry on the
is Lipschitz, it follows from 7.1 above
has interior, but the usual argument shows that
a.
If fl C1
is given and
y(t),
a 2 t
we can define the length of
Z
b
y
is a curve in
R2 of
according to the distance
where (t) X P(8)ldt
In particular the length of
C
=
with respect to its own
(which makes sense regardless of any smoothness assumptions); by
ON REPRESENTATIONS OF DISTANCE FUNCTIONS
for
C1
curves
C;
for
c2
curves
C.
o r finally by
It is an old result of GoZ'gb (see [ 6 ] ) that when
n
193
6
4,
n
(C)
?
8
is convex, but we don't see now how to obtain these es-
timates from the formulas above, or what happens when convexity is not assumed.
7.4.
The moments of
from (5.2.i),
are related by equations easily derived
dM
which can be written as
+ R" )do.
2rdM = (R
F o r any
Since
n, multiply the above times
r
and
R = l/r
n
and integrate to get
are periodic, integration by parts gives {r"R"de
= -n
=
I
rn-'r'
= n {rn-'
thus, for all
r
(
(rn)'R'd6
(-r'/ r 2 )d8 (1"'
=
=
) "6,
n:
[
r
(7.4.1)
rn+'dM
= (or
rn"de
+ n
Some special case are of interest:
rn-3(r')'d8.
and
JOHN MCGOWAN
194
HORACIO PORTA
ll
(7.4.ii)
2
(7.4.iii)
2
(7.4.iv)
2
de r
rdM =
n = 0
(these formulas correspond to
and
n = 1
in (7.4.i);
observe that (7.4.iii) and (7.4.i~) are actually the same formula with the last term written in two alternative ways). Using now the formula TT
A =
for the area of 2
(7.4.v)
as
i,"
R
il,'
(r') do =
Q
r'de
n = 3
and (7.4.i) with
+ 3
r4dM = A
- i," r r " d e
Td'
we get
2 (r') de = A
-
3
rr" de
by integrating by parts.
Observe that (7.4.i~) implies that r2dM
%
n y
with equality only for the circle. Since by (6.1.iii) 2 2 the last inequality reads: = ( n / r sin a)ds, ds
%
2dM =
2n
sin a (with equality only for the circle), an easy fact to establish independently of
7.5.
dM.
Consider two curves
C1
and
C2
(with corresponding
r19
r2# dM1, d M p etc.) inverse of each other with respect to the unit
ON REPRESENTATIONS OF DISTANCE FUNCTIONS
rl(e)r2(8) = 1 f o r e a c h
circle, i.e.,
R 2 = r1
and
R1 = r 2
.
8.
195
T h i s means t h a t
Hence,
+
2dM1 = R1(R1
= r 2 ( r 2+ r’i)de
R;)de
and t h e r e f o r e ,
[
Ti
dM1 = A 2
(7.5.i)
Now ( 7 . 4 . v )
+
r2r’;d8.
gives
and combining t h i s w i t h
(7.5.1)
we g e t
Combining a g a i n t h e s e two f o r m u l a s i n a d i f f e r e n t way g i v e s also
and t h e r e f o r e , r2dM2 4 2
fOn
dM1
with e q u a l i t y only f o r c i r c l e s . Finally,
and so
7.6.
according t o
[ r2 dM says t h a t
we have
i s i n v a r i a n t under i n v e r s i o n . y(8) = R ( e )
Denote now by
(5.2.1)
(7.4.iii)
y
and
dM(8) = f(8)dB.
i s a s o l u t i o n of Y(Y”
+ Y) =
f,
Then
JOHN MCGOWAN
196
and
HORACIO PORTA
subject to:
It would be interesting to investigate further this boundary value In particular, we do not know any necessary conditions on
problem.
the periodic function Obviously, if solution.
f = c,
f
for the existence of solutions
a positive constant, then
f
the period is
n
In fact, if
near a constant.
y = 1
the linearized operator near
(ffi),Df
D(y) = y(y”
D’(h) = h”+ 2h
is
is surjective.
the
+
y),
and since
By the implicit function
theorem (as in, for example, [ 7 ] ) it follows that
D
covers a neigk
D(l) = 1, as claimed.
borhood of
We can rewrite (4.2.iii) in the form LC(X) =
where
E ( 8 ) = (cos 8 ,
and where
dP(8) =
sin 8 )
r(e +
IE(e).xl
dp(e)
is a variable point in the unit circle
3)dM(e).
associated to a convex curve of
’/‘ is
Robert Kaufman has observed that existence also can be
obtained for
7.8.
y = c
y.
C
Such formula for the norm
LC is called the Levy representation
L c , and the existence of such
dP
is obtained from the fact
n
that in
RC. all norms are negative definite functions.
Fergusonfs approach in [ 2 ] where he uses for convex
C
This is a technique
similar to the one used in 9 4 above. We also remark that D o r f s treatment of higher dimensional cases in [l] uses a potential theoretic approach related to the harmonic analysis approach of our
$4. Finally, it should be mentioned that Hamel in 1903 had
already used expressions similar to our formulas for
k,(C)
in $ 7
in his study of the geometries for which the straight line segments
197
ON REPRESENTATIONS OF DISTANCE FUNCTIONS
are the shortest curves (see volume
57,
p. 251, of Mathematische
Annalen).
$ 8 . APPENDIX
P j = (aj,bj),
Let that
P1,P2,
j = 1,2,...,n
...,Pn,-P1,...,-Pn
R2
such
is a radial sequence ordered counter-
F o r convenience we set
clockwise.
be points in
-- -P1
'n+l
-
and
Po = -P
n
.
a.b and let 5 = ( 0 , 0 , 1 ) denote the unit J i vector in the z-direction. Clearly the cross product Pi X P . saDefine
aij = a.b. 1 J
tisfies
Pi X P . = a. .k
a.. > 0
if
J
1J
Denote by
1J -
i < j the
A
.
J
Further, by the assumed radial order,
and, of course,
nxn
matrix
aij
=
-aji for all
A = ( laij]) ,
i, j .
and consider the
system
where
Clearly ( 1 ) is equivalent to
-
C x.P.xPi + C x.P.xPi = yjk, isj i>j J
1
I;
j
4
n.
J
Define now
T. = J
C xiPi i<j
-
C xipi, ir j
s o that the last equations
also reads
Since each pair
pj,
pj+l
is linearly independent we can write
198
JOHN MCGOWAN
and
HORACIO PORTA
1 L j S n
Tj = 5 jPj + q jPj+l, (recall that
Pn+l
denotes
-P1).
and from
Tj+l = TJ. + 2x j+lP j+l we get also
taking cross products with (2xj
-5
j
Pj,l
+
Using ( 2 ) we get
and ( 2 ) again applied to P j+lXT .
j+l
we get
7 j-11aj,j-1
--
q jaj+l,j-1
9
whence 2 x j = -qj-1
Replacing now
q j-l
and
+
qj(aj-l,j+Jaj-l, j)
sj
+
5
j
a
by their values obtained from ( 3 ) and
(4) we conclude that
(where we define
yo = yn).
Thus the system (1) can be solved by this expression, which in fact also gives the coefficients
Bij
of the inverse
A-l = (B,j>$
ON REPRESENTATIONS OF DISTANCE FUNCTIONS
Bj,j+l
-
199
5j+l,j = m u j ,j+l
5 j,j = -(Uj-l, j+l/Zaj-l, ja j,j+l1 = 0
for all other pairs
A second method for finding
(i,j).
involves matrices with
A-'
n
coefficients in
RL.
Suppose that Uij
,
Wj,k
, etc.,
We sketch the main steps.
) , etc., where j,k We can multiply A and C
A = (V..), X = (W =J
belong to
R
2
.
the usual rule using the dot product
U.. * W I J
by
instead of number
jk
multiplication: AC = (tik)
tik = CjVij * Wjk. The result is an ordinary real matrix. We can also multiply AR or RA where R is a real matrix. But:
where
this product is not associative when three vector matrices are involved:
(Ac)r
A(CT)
and
need not coincide.
However, asso-
ciativity holds for all combination involving not more than two vector matrices, as in
(AC)R = A (CR) ,
(AR)C =
A(RC),
(AR)S = A(Rs),
etc. Using the notation introduced above, define and the matrix
0 = (U. .)
A = (PiXP.) J
with coefficients
1J
Denote also by
E = %In
and by
E
the diagonal matrix with coef-
JOHN MCGOWAN
200
ficients
2ai,l,i~i ,i+l
and
HORACIO PORTA
down the diagonal.
It is easy to see that
J A = AJ = A
and that
@A = E,
and
theref ore ,
E = @A = @ ( E A ) = (@Z A . So if we denote E'lF
= A-'.
F = @Z
(a real matrix), then
I = E -1F A ,
whence
A direct calculation gives the values obtained above
f o r the coefficients of
Am'.
We close this appendix with the remark that a direct calculation gives n n-2 det A = (-1) 2
5 uj,j+l
j=1
and in particular the fact that
A
,
is invertible follows easily.
ON REPRESENTATIONS OF DISTANCE FUNCTIONS
201
REFEmNCES
1.
DOR, L., Potentials and isometric embeddings in Israel J. Math., 24 (1976), 260-268.
2.
FERGUSON, T.,
L1
A representation of the symmetric bivariate
Cauchy distribution,
Ann.
Math. Statist., 33 (1962),
1256-1266.
3.
LINDENSTRAUSS, J.,
On the extension of operators with a
finite-dimensional range,
Ill. J. Math., 8 (1964), 488-499.
4. LINDENSTRASS, J. and TZAFRIRI, L., rtClassicalBanach Spaces", Lecture Notes # 331, Springer, 1973. 5.
PbLYA, G. and SZEGd, G., Analysis,"
"Aufgabe und Lehrsltze aus der
Berlin, 1925.
6. S C H b F E R , J.J.,
"Geometry of norms in normed spaces," Lecture
Notes in Pure Appl. Math., Vol. 20, Marcel Dekker, 1976.
7. SCHWARTZ , J .T.,
"Non-linear Functional Analysis, I '
Gordon and
Breach, 1969.
8.
TORANZOS, R., bodies,
Radial functions of convex and star-shaped
Monthly 74 (1967), 278-280.
University of Illinois, Urbana, IL, USA. Instituto Argentino de Matematica, CONICET, Argentina.
This Page Intentionally Left Blank
Functional Analysis, Holomorphy and Approximation Theory 11, G.I.Zapata ( e d . ) 0 Ekevier Science Publishers B . K (North-Holland), 1984
SPECTRAL THEORY FOR CERTAIN OPERATOR POLYNOMIALS
Reinhard Mennicken
Holomorphic operator functions T E H(C , L ( E , F ) )
and especial-
ly operator polynomials have been studied intensively by many authors.
Keldyg [ 291
, [ 301
introduced the concept of eigenvalues,
eigenvectors and associated vectors for holomorphic operator functions
T E H(C,L(H)),
operator bundles
H
T(X) = I
pact operator for each
For holomorphic
being a Hilbert space.
X E
+ K(X) CC
(1 E C)
where
K(X)
is a com-
Keldyi stated [ 2 9 ) and proved [ S O ]
the existence of a biorthogonal system of eigenvectors and associated vectors belonging to the eigenvalues of bundle
T".
T
and the adjoint (yLo): k E
H e arranged this denummerable system
eigenvectors and associated vectors of
T
(N]
of
i n a canonical way and
defined recursively
if
yi0)
is an eigenvector belonging to the eigenvalue
if
yio)
is an associated vector belonging to
Xo.
and
X,,
Keldyg called
a canonical system of eigenvectors and associated vectors n-fold complete i n
H
if the subspace : k € LN]
is dense i n the n-fold product
Hn.
For
n = 1
this concept re-
duces to the concept of completeness in the normal sense.
In [ S O ]
Keldyg proved a completeness theorem for holomorphic operator
204
REINHARD MENNICKEN
bundles assuming a certain smallness condition for the resolvent of 1 (cf.[30],p.27-31).
This assumption is, as far as we know, never
satisfied when we wish to apply this theorem to boundary value problems for differential equations.
Other results of Keldyg in
[29], [SO] concern the completeness of a canonical system of eigenvectors and associated vectors of polynomial operator bundles of order
n.
The essential assumptions which are imposed on the coef-
ficient operators are certain conditions of compactness, selfadjointness, normality and completeness (cf. [30], p.32).
The proof is
based on a linearization method, which transforms the original equation to an equivalent problem in the product space a Phragmen-Lindelbf argument.
Hn, and on
Applications of these results to
boundary value problems for linear differential operators are stated in 1291. Allahverdiev [ 11, [ 21, [3],
[4], Vizitei
&
Markus [47] and
Markus [37], [38] strengthened and generalized the results of Keldyg concerning operator polynomials.
An elaborated proof of
Allahverdievls completeness theorem in [l]
can be found in the book
of Gohberg & Krein [19], cf. Chap. V, sec. 9.
Further completeness
statements, some of them of more special character, e.g. concerning quadratic operator bundles only, are contained in the papers of Pallant [ 431 , Gorjuk [ 211 Orazov [42],
In
,
Isaev [ 261 , [ 271 , Yakubov
KostjuEenko 80 Orazov [32]
[47] Vizitei
&
&
Mamedov
1501 ,
and Kbnig [31].
Markus stated various conditions under
which the n-fold expansions (with parantheses) in eigenvectors and associated vectors of n-th order operator polynomials were proved to be conditionally or unconditionally convergent.
Their proofs
are based on the above mentioned linearization method and on perturbations statements for selfadjoint or normal operators with discrete spectrum consisting of eigenvalues only and with n o finite
SPECTRAL THEORY FOR OPERATOR POLYNOMIALS
limit point i n
C.
205
The results are applied to ordinary differential
operators with selfadjoint boundary conditions, o r with boundary conditions which are regular i n the sense of Birkhoff, and to elliptic partial differential operators which are regular in the sense of Agmon. Monien
[40] stated remarkable improvements of Markus' results
but unfortunately almost all of his proofs are incorrect (cf. Bauer
[5]
sec. 6.2).
Generalizations of Markus' expansion theorems,
different from those stated by Monien, were announced by Yakubov & Mamedov i n [ 501. Apart from the foregoing cited results i n the Russian literature (often without any proofs), completeness statements and expansion theorems similar to those of Keldyg, Allahverdiev and Markus were proved independently by Friedman and Shinbrot i n [18].
This
article also contains applications to linear partial differential operators. Further results on "nonlinear" eigenvalue problems are due to Mtiller & Kummer
[41] , Kummer [ 331 , Mittenthal [ 391 , Turner [ 461 ,
H. Langer [34] and Roach
&
Sleeman
[kk]
, [45].
F o r the present
paper the cited articles of Mittenthal and Turner are of special interest:
projections ~ ( p ) onto the principal space
2
(p),
span-
ned by the eigenvectors and the associated vectors belonging to the eigenvalue
p,
are defined in terms of the resolvent of the opera-
tor polynomial under consideration (cf. [ 3 9 ] , p.122 and [46], p.304). I t is worth mentioning that these operators i n general are not pairwise biorthogonal, i.e. eigenvalues
pl, p 2
the equation
P(b1)P(p2)
= 0
for different
may not be satisfied.
The assumptions of the cited completeness statements and expansion theorems are rather restrictive with respect to applications to linear differential operators. simple boundary value problem:
Let us consider the following
206
REINHARD M E N N I C K E N
where (for simplicity)
a,B E Cm([0,2n])
and
v E C\Z.
According
to Keldys [30], cf. the first theorem in chap. 11, or Vizitei & Markus
[47], theorem 3 . 3 almost all eigenvalues are simple, i.e.
only a finite number of associated vectors exist, and the system of eigenvectors and associated vectors is 2-fold complete in L2[0,2n].
The theorem
4.6 of Vizitei
&
Markus
[47], according to
which the 2-fold expansion in eigenvectors and associated vectors of all functions
fl,f2E H2[0,2n]
satisfying the boundary condi-
tions in (*) is unconditionally convergent with respect to the L2-norm, requires by its assumptions that
a(x)
e 0
in
[O,~TT].
The assumptions in the expansion theorems of Vizitei
&
Markus (and others) can be weakened to some extent by refining the perturbation conclusions in the proofs, cf. Bauer [6].
Then, in
the foregoing example it is sufficient to require that the function a
is small instead of zero.
The obstacle to improving the results
more essentially is to be found in the more or less rough linearization transforms.
With respect to boundary value problems more
concrete transformations, cf. Wilder seem to be better adapted.
[49] and Wagenftihrer 1481,
By the transformation
the boundary value problem (*) becomes equivalent to the linearized problem
where
SPECTRAL THEORY FOR OPERATOR POLYNOMIALS
207
and
Under a p p r o p r i a t e a s s u m p t i o n s n - t h
order d i f f e r e n t i a l equations
w i t h c o e f f i c i e n t s and boundary v a l u e c o n d i t i o n s depending polyn o m i a l l y on t h e e i g e n v a l u e p a r a m e t e r
1
can be transformed t o
b o u n d a r y v a l u e p r o b l e m s f o r f i r s t o r d e r d i f f e r e n t i a l s y s t e m s where t h e d i f f e r e n t i a l system i s l i n e a r i n t i o n s depend p o l y n o m i a l l y on h a v e b e e n s t u d i e d by R . E .
1.
1
and t h e b o u n d a r y c o n d i -
Eigenvalue problems of t h i s t y p e
L a n g e r [ 351
and C o l e [ 8 ] , [ 91
i n a more
c l a s s i c a l way w i t h o u t u s i n g f u n c t i o n a l a n a l y t i c t o o l s .
I n t h e p r e s e n t p a p e r we a r e c o n c e r n e d w i t h o p e r a t o r p o l y T E H(C,I(E,F))
nomials
F2.
F = F XF2
where
1
Accordingly t h e o p e r a t o r bundle
( T D ( X ) , T R ( X ) ) a n d we assume t h a t
T(1)
TD(X)
w i t h Banach s p a c e s F
1’
splits into
i s linear in
1.
Under
a p p r o p r i a t e a d d i t i o n a l a s s u m p t i o n s we a r e a b l e t o d e f i n e p r o j e c t i o n s P(b)
of
T
g(p)
onto the principal spaces
i n t e r m s of t h e r e s o l v e n t
which t u r n o u t t o b e p a i r w i s e b i o r t h o g o n a l . The d e f i n i t i o n of t h e s e p r o j e c t i o n s and t h e p r o o f
of t h e i r
projection p r o p e r t y and t h e i r b i o r t h o g o n a l i t y a r e t h e c o n t e n t s of section
3.
I n o r d e r t o p r o v e t h e s e p r o p e r t i e s we make u s e o f a
s l i g h t g e n e r a l i z a t i o n of Keldyg
1
t h e o r e m a s s u r i n g t h e e x i s t e n c e of
a b i o r t h o g o n a l system o f e i g e n v e c t o r s and a s s o c i a t e d v e c t o r s . t h e o r e m i s s t a t e d i n section 2.
This
S e c t i o n 1 c o n t a i n s some n o t a t i o n s
and p r e l i m i n a r y r e m a r k s .
I n S e c t i o n 4 w e a r e concerned with b i o r t h o g o n a l expansions
208
REINHARD M E N N I C K E N
c
f =
P(p . ) f
j
J
with respect t o the projections P(P)
d e f i n e d i n s e c t i o n 3.
To
p r o v e a n e x p a n s i o n t h e o r e m we assume t h a t t h e o p e r a t o r b u n d l e i s r e g u l a r which means t h a t on c e r t a i n c u r v e s i n
i n f i n i t y , the resolvent p E N.
R(),)
The f l f u n c t i o n " f
of
T(),)
5
tending t o
behaves l i k e
),'
where
h a s t o s a t i s f y some " s m o o t h n e s s p r o p e r -
.
t i e s " and c e r t a i n " b o u n d a r y c o n d i t i o n s l l I n section
(c,
T
we g i v e t h e a p p l i c a t i o n s t o b o u n d a r y v a l u e
p r o b l e m s of L a n g e r ' s a n d C o l e l s t y p e .
We s t a t e t h e e x i s t e n c e of
b i o r t h o g o n a l s y s t e m s of e i g e n v e c t o r s and a s s o c i a t e d v e c t o r s and prove a g e n e r a l b i o r t h o g o n a l expansion theorem f o r s u f f i c i e n t l y smooth f u n c t i o n s
f
s a t i s f y i n g c e r t a i n b o u n d a r y c o n d i t i o n s which
a r e d e t e r m i n e d by t h e ),-dependent
operator
TR(X).
C o l e ' s s u f f i c i e n t c o n d i t i o n s f o r t h e r e g u l a r i t y o f boundary v a l u e p r o b l e m s c a n b e weakened t o some e x t e n t .
I n t h e l i g h t of
t h e s e weaker a s s u m p t i o n s , t h e r e s u l t s o f E b e r h a r d [ 1 2 1 , [ 133, Eberhard & F r e i l i n g
[14],
[15],
1161,
Freiling
[17] and H e i s e c k e
[ 2 5 ] on b o u n d a r y v a l u e p r o b l e m s f o r n - t h o r d e r d i f f e r e n t i a l e q u a t i o n s t u r n out t o be very c l o s e t o
L a n g e r t s and C o l e l s t h e o r y .
F o r d e t a i l s we r e f e r t o Wagenfilhrer [ 4 8 ]
and s u b s e q u e n t p a p e r s of
the author.
I w i s h t o e x p r e s s my t h a n k s t o G . F i e d l e r who worked o u t a
.
m a j o r p a r t o f t h e p r o o f s of Keldyg [ 301
1. DEFINITIONS, NOTATIONS Let
H(U,G) i.e.
G
b e a Banach s p a c e and
U
d e n o t e s t h e s e t of a l l h o l o m o r p h i c mappings on a
E
H(U,G)
i f and o n l y i f
a: U
-t
CC.
be a region i n
G
U
to
G,
i s d i f f e r e n t i a b l e on U.
209
SPECTRAL THEORY FOR OPERATOR POLYNOMIALS
The function
a
there is a region
X
E U\V
a E H(V,G)
such that
there exists a neighbourhood g E H(W,C)
b E H(W,G), ga = b
and
V C U
If
on
and if for all and functions
of
W\{h)
with the properties
C
V,
g
#
F
are Banach spaces,
F.
Im(S)
If
S
E L(E,F),
y E E,
If
its image o r range.
T E H(@,L(E,F))
E
to
y @ v
:=
E C
C.
+ L(F,E)
T.
T, Up(T)
spectrum of
T,
each
y E N(T(p))\(O)
X E C.
bijective]
:= T(X) -1
R(X)
for
:= C\p(T)
a(T)
:= { X E C : N ( T ( X ) )
p E ap(T)
p(T)
#
[O))
X E p(T).
T.
is called the the point
is an eigenvalue of
T
is
T,
and any
belonging to the eigen-
u. Y = (yo,~l,...,yh)
associated vectors of the bundle if
for
It is well-known that
is an eigenvector of
A sequence
p
L(F,E).
is called the resolvent (operator bundle) of
spectrum of
value
We set
R E H(p(T),L(F,E)).
We know that
belongs to
T*(X) := T(X)*
: T(X)
is called the resolvent set of an open subset of
we set
The "adjoint" operator bundle
F.
is defined by
P(T)
v E F',
is called a holomorphic operator bundle
( o r operator pencil) from
T* E H ( C , L ( F ' , E ' ) )
E
is its null-space o r kernel,
N(S)
and state that the "tensor product"
R:p(T)
0
denotes the
L(E,F)
vector space of all continuous (bounded) linear operators on to
if
W\{X].
and
E
W
a E M(U,G),
U,
is said to be meromorphic on
yo
#
0
and
T
in
E
is called a chain o f
belonging to the eigenvalue
REINHARD MENNICKEN
210
where
T ( '
Obviously to be
#Y
(p)
denotes the 4-th derivative of
T
is an eigenvector of
yo
h+l
T
at the point
belonging to
1. Y.
and call it the length of the chain
p.
We define Let K(T,p)
denote the set of all chains of associated vectors belonging to the eigenvalue
b.
v(Y) := s~P(#Y : Y E K(T,b),
(1.2)
yo = Y]
( Y E N(T(w))\{O))
is called the multiplicity of the eigenvector p E Q:
y.
is said to be an eigenvalue of finite algebraic mul-
tiplicity i f
v(y) <
and
y E N(T(p))\(O],
for all
m
L~ := {y E N(T(p))\{O) obviously is a subspace of
Ln = ( 0 1
for some
(1.3)
multiplicity. ,b
1
: u(y) 2
n E N, :
n)
u (03
(n E N)
we conclude from
L~ 3 L ~ + n ~ , L~ = Eo3 n
m,
sudv(Y) Assume that
Since
N(T(p)),
dim N(T(CI)) < that
<
:= dim N(T(p))
nul(T(p))
i.e.
<
Y E N(T(M))\EO)]
is an eigenvalue of
T
m.
of finite algebraic
A system
. .,mk(l),
: v = O , 1,2,.
k=l,2,.
..,nul(T(b)
)]
is called a,canonical system of eigenvectors and associated vectors of
T
belonging to the eigenvalue
IJ.
if the following relation-
ships hold:
( 1.4a) (1.4b)
(Yo,w#Ypj, (k)
,Y(k) "k(M)
) E K(T,w)
,...,nu1 T(p)]
( y p ) : k=1,2
tw
9 1
,...,nu1 T ( p ) ) ,
(k=1,2
is a basis of
N(T(w)),
SPECTRAL THEORY FOR OPERATOR POLYNOMIALS
(1.4~)
mk(p)+l
= max[w(y)
: yEN(T(p))\span{yii)
:
9I-I
211
i < k]]
,...,nu1 T(p)).
(k = 1,2
F r o m the foregoing remarks concerning the spaces
Ln
we
immediately conclude that each eigenvalue of finite algebraic multiplicity has a canonical system of eigenvectors and associated vectors.
The numbers
(k = 1,2,.
mk(p)
. .,nu1 T(p-))
are indepen-
dent of the choice of the canonical system since
mk(p) + 1 = maxrn E N : dim Ln
L k].
For the following sections we state a reformulation of the relationships (1.1): be a sequence in
p E b(T)
Assume with
E
yo f 0.
and let
...,yh)
Y = (yO,yl,
We set
(1.5) and, for later convenience, Obviously
Gh(Y, * )
(1.6) PROPOSITION.
is holomorphic in
Y E K(T,p),
(1.1) if and only if PROOF.
= 0
T(')(p)
TSh(Y,-)
i.e.
if
& = -1,-2,...
.
C\[p].
Y
fulfills the relationships
is holomorphic in
A simple calculation leads to
from which the statement in (1.6) is evident.
p.
212
REINHARD MENNICKEN
THE EXISTENCE OF BIORTHOGONAL SYSTEMS, THE STRUCTURE OF THE RESOLVENT
2.
S E L(E,F)
is called a Fredholm operator if
:= dim N(S) <
nul(S)
:= codim Im(S) <
def(S)
a,
'P(E,F) denotes the set of all Fredholm operators on S E @(E,F),
Im(S)
is closed (cf. e.g. Goldberg [20],
E
m.
to
F.
If
Cor.IV.1.13).
In the following we always assume:
T E H(@,@(E,F))
(2.1)
and
#
p(T)
Q,
i.e.
T
is a holomorphic
operator bundle with values in the set of Fredholm operators and has a non-empty resolvent set. We state without proof: (2.2) THEOREM. (cf. Gramsch [22], Th. 11, p.102) (i)
U(T)
(ii)
U(T)
= Op(T),
has no finite limit point,
R E M((C,L(F ,E)),
(iii)
(2.3) THEOREM. (cf. Keldyg [30], chap. I, p.19-26) Assume Then
p
c~ c' U(T).
is an eigenvalue of
T
of finite algebraic multiplicity.
Let EY",CI (k) :
(2.4)
v = 0,1,...,m k ( b ) ,
k = 1,2
,...,nu1 T(IJ.))
be a canonical system of eigenvectors and associated vectors of belonging to
1 T"
T
p.
also is an eigenvalue of finite algebraic multiplicity of
and there exists a canonical system
SPECTFUL THEORY FOR OPERATOR POLYNOMIALS
,...,nu1 T ( p ) )
,...,mk(p),
(k) : v = 0.1
213
k = 1,2
of eigenvectors and associated vectors of
belonging to
T*
such that T(p)
(2.6) HT(R,~) =
where
HT(R,p)
mk(p) 1
C
c
k=l
j=O
(x -P) R
denotes the singular part of
Let us assume in addition that
p.
at the pole
p 2 E (S(T). Then the canon-
ical systems ( 2 . 4 ) and ( 2 . 5 ) , belonging to
T, p
and
T*, p 2
respectively, fulfill the biorthogonal relationships
(2.7)
I
0 2 h S m (p) ,
where
at
0 Z j 2 mi(p2),
(According to Prop. 1.6 A n eigenvalue
l?ignulT(D2)
denotes the n-th derivative of the function
q ( " ) ( p ,YLk))(b,)
b2,
1 2 k 5 nu1 T(p),
T
of
p
q(p,Yh(k))
is called normal if the canonical
system of eigenvectors and associated vectors belonging to consists of eigenvectors.
The operator bundle
normal if each eigenvalue of ( 2 . 9 ) PROPOSITION.
Let
p
c.)
is analytic in
T
T
p
is said to be
is normal.
be an eigenvalue of
T
and
r = nulT(U).
The following statements are equivalent: (i) (ii) (iii)
p
is a normal eigenvalue of
The resolvent For every
such that
y
R
of
T
E N(T(l))\{O]
#
0.
only
T.
has a pole of order 1 in there exists a
p.
v E N(T*(p))
214
REINHARD MENNICKEN
(iv) basis
There exists a basis ( v1,v2,.
. .,vr]
of
N(T*(w) = bij
(T(l)(!.I)yi,vj) The implication (1)
PROOF.
inverse implication (ii) eigenvectors
where
k
3
,!.I
vanish which implies
normal, i.e.
(i,j = 1,2,
...,r ) .
is obvious from (2.6).
= 0.
ml(k)
Z
1
= ml(u))
{k E M : %(p)
(iv)
(i) by contradiction:
ml(p)
and a
such that
(i) also follows from (2.6):
3
varies over the set
5
(ii)
)
N(T(p))
The
since the
are linear independent the sum
yik)
We prove (iii)
of
(y1,y2,...,yr)
,
does not
(iii) is immediate. we assume that
and there is a chain
yo (l)
p
, y1(l).
is n o t Accord-
ing to (1.1) we have
T(p)y$')
+ T(')(p)yAl)
= 0.
This equation leads to
for all
(i)
a
v E N(T*(w))
which contradicts (iii).
(iv) remains to be proved:
The implication
from (2.8) we conclude
and thus
By assumption
mk(p) = 0
according to (2.7).
for
k = 1,2,,..,r
and therefore
SPECTRAL THEORY FOR OPERATOR POLYNOMIALS
215
3. BIORTHOGONAL PROJECTIONS
I n t h i s s e c t i o n we make a d d i t i o n a l a s s u m p t i o n s w i t h r e s p e c t t o t h e o p e r a t o r bundle p o l y n o m i a l of d e g r e e
with
Tq
f 0.
F = F XF2 1
q,
T E H(C,I(E,F))
W e assume t h a t
T.
is a
i.e.
Furthermore,
let
F1
and
F2
b e Banach s p a c e s ,
and T(k) = (TD0),TR(X))
where
TD(X)
i s a p o l y n o m i a l of o r d e r
D D T (X) = To
1,
i.e.
D XTIJ
+
and
R T (k) =
.
X i~ Ti
9
Z i=O
We assume t h a t t h e c o e f f i c i e n t s
of t h e s e p o l y n o m i a l s f u l f i l l t h e
following properties: D T o , J E L(E,F1),
D T1 E L(F1,F1)
J
is injective,
and i s b i j e c t i v e ,
R T~ E L ( E , F ~ ) f o r
i = 0,1,2
We d e f i n e t h e l i n e a r o p e r a t o r
D(M) (3.2)
My :=
and t h e o p e r a t o r s
If
M j
D ( M ) C F1 + F1
by
:= J ( E ) D -1 T o J 3’
-(T!)-’
( j
E
N)
y E J’l(D(MP+l))
M ~ J YE D ( M )
M:
,...,q .
= J(E)
(Y E D(M))
i t e r a t i v e l y i n t h e u s u a l way.
then
Jy E D(MP+l) ( j = 0,1,2
and t h u s
,...,p ) ;
2 16
RE INHARD ME N N I CKE N
therefore
:= J
Y[ j l
(3.3) i s well-defined
-1M j J y
(EE)
j = 0,1,2,. ..,p.
for a l l
We set
E o := J - l ( D ( M q ) )
?, E C
For fixed to
F.
Finally,
Res ( R V )
where
P
E)
( w h i c h i s a s u b s p a c e of
if
y E Eo,
o b v i o u s l y i s a l i n e a r o p e r a t o r from
V(X)
p E a(T),
if
and,
Eo
we d e f i n e
d e n o t e s t h e r e s i d u u m of t h e o p e r a t o r f u n c t i o n
RV(X) = R(?,)V(?,)
p.
at
The r e l a t i o n s h i p
P ( K ) = qC- 1 R e s ( R T r G ' ) J -.e M 1J
(3.7)
iJ.
L=O
i s a n i m m e d i a t e c o n s e q u e n c e f r o m ( 3 . 3 ) and ( 3 . 5 ) . If q = 1
and
T
i s a n o p e r a t o r p o l y n o m i a l of t h e f i r s t o r d e r , i . e .
T ( h ) = To + XT1
(1 E
C),
then
Eo = E ,
V(x)
= T1 and
P ( p ) = Res (RT1).
v
I n t h i s " l i n e a r " c a s e i t i s well-known p. I'
( c f . e.g.
5 1 o r [ 2 4 ] p . 357-358) t h a t t h e o p e r a t o r s
Grigorieff
P(p)
[13],
f u l f i l l the
b i or t h o gona 1'I r e l a t i o n s h i p s
p(p2) for
p1,p2 E a ( T ) .
p r i n c i p a l space
The o p e r a t o r s
5 (p)
P(p)
a r e p r o j e c t i o n s onto t h e
belonging t o t h e eigenvalue
p.
SPECTRAL THEORY FOR OPERATOR POLYNOMIALS
In this section we are going to prove relationships for the operators for operator polynomials.
P(M)
217
similar biorthogonal
which were defined in (3.6)
The introduction of these biorthogonal
projections enables us to define and investigate the problem of biorthogonal expansion with respect to a system of eigenvectors and associated vectors.
(3.8) PROPOSITION. vectors of if
j < 0.
T
Let
y0,y1, ...,yh
belonging to
be a chain of associated y j := 0
For convenience we set
p.
We assert:
(3.9) (3.10) (3.11) for all
p E N
and
k
E {1,2
,....,h].
We prove ( 3 . 9 ) and (3.10) by induction with respect to Both statements are true f o r
p = 0. For
k 2 1
p.
we have
(3.12) because of (1.1) and the linear structure of also holds for
k 2 0
since
yj = 0
for
TD(?,).
j < 0.
This equality
Obviously
JYk E D(M). From (3.2) and (3.12) we conclude (3.13) so
that
Jyk E D(MP+l)
if
Jyk
and
Jyk,l
belong to
D(Mp).
Thus (3.9) is clear. According to ( 3 . 9 )
yp'
is well-defined for all
Let us assume that (3.10) has already been proved for
p.
p E N. Then,
218
REINHARD MENNICKEN
using (3.13), we obtain
P+1
=
c d=O
P+l
p+l-4
( & ) Y
'k-d
which proves (3.10). The proof of (3.11) is a bit more complicated.
We insert
the following
(3.14) LEMMA.
Let
j,p E
IN,
d E Z. Then
(3.15) (3.16)
(jiL) =
P C (-1) m= 0
),
if
h < 0
and
j + d 2 0,
The relationships (3.15), (3.16), (3.17) follow from the identity
and the equality
r!
(E),
if
s
which obviously is true f o r arbitrary
o
r E hi
and
Now we are ready for the proof of (3.11) :
Using (3.10) we obtain
s ~f
E L. jE
(N
SPECTRAL THEORY FOR OPERATOR POLYNOMIALS
of the s m s on t h e r i g h t s i d e and t h e a p p l i -
A simple rearrangement
c a t i o n of t h e f o r m u l a
where, then
if
L < 0,
L (r-m)
= 0
(3.15) l e a d s t o t h e r e l a t i o n s h i p
T4, = 0. for
4,
If
E (r-p,
m > p
then
.. . , r - m - l ]
where we a p p l i e d ( 3 . 1 8 ) f o r t h e proof
(z)
= 0
m < p
and i f
whence
the l a s t equality.
Of
Another
r e a r r a n g e m e n t of t h e sums g i v e s 4
g=o
If
m > p
yk,m-r
[P+&I =
TLYk
= 0
T(r)(p) = 0
then
P+9 P+9 C c m=O r = m
P (m ) = 0.
for a l l for a l l
k-m+l
Furthermore, i f
z r
2 p+q-m
p+q < r 5 k .
p+q
> k
and, i f
Thus,
then
p+q < k
then
from t h e f o r e g o i n g equa-
t i o n , we o b t a i n
t h e r i g h t s i d e of which v a n i s h e s b e c a u s e o f t h e D e f i n i t i o n (1.1)
s o t h a t t h e proof of
( 3 . 1 1 ) i s complete.
(3.19) PROPOSITION.
Let
c i a t e d v e c t o r s of
T
Y = (yo,yl, ...,yh)
belonging t o
b.
be a c h a i n of a s s o -
We a s s e r t
REINHARD MENNICKEN
220
where
gh(Y,A)
d e n o t e s t h e v e c t o r f u n c t i o n which h a s b e e n d e f i n e d
i n (1.5). The D e f i n i t i o n s (3.3),
PROOF.
(3.4),(3.5) and t h e r e l a t i o n s h i p ( 3 . 1 0 )
whence
j
j
(&) = ( .
because
J -4
).
A r e a r r a n g e m e n t o f t h e sums l e a d s t o
q-1 q-1-r
=
V(X)yh
(3.21)
C r=O
= q c- 1
C
'
q-1-r
L=O
j=L
q-1-r
c
q-1-r-4,
L=O
j=O
r=O
r L
j
T r + j + l'h-j+.&
j + ~
c
r L
(
)I
O n t h e o t h e r hand t h e T a y l o r s e r i e s o f D e f i n i t i o n (1.5)
Since
of
Sh(Y,X)
T r + j + L + l 'h-j T(X)
at
1
y i e l d t h e expansion
i s h o l o m o r p h i c ( i n 1) by P r o p o s i t i o n (1.5), we
Gh(Y,*)
obtain
If h
h < q-1
> q-1
then
and t h e
then T
yh,j
= 0
for
(m+ j + l )( p ) = 0
j
for
...,q-11 [ q , ...,h]
E {h+l, j
E
and, i f whence
SPECTRAL THEORY F O R OPERATOR POLYNOMIALS
The e x p a n s i o n
and t h e r e l a t i o n s h i p
(3.18)
lead t o
Because q-1-j
rn
c
c
m=O
r=O
=
q-1-j
q-1-j
r=O
m=r
c
c
we o b t a i n
q-1 q-1-j
= j =cO
c r=O
q-1
q-1-j
j=O
r=O
= c
c
q-1-j-r
c
m=O
q-1-j-r
c
m=O
9
.L=m+r+j+l
e ) X r p L - r - j-1 m+r+ j+l TLyh- j
q-1-j-r
c
t -m
.L =m
where we u s e d t h e i d e n t i t y
Since, according t o
and because
(3.16),
'
the foregoing equation (3.22)
implies that
221
222
REINHARD MENNICKEN
s o t h a t , i n c o n s i d e r a t i o n of ( 3 . 2 1 ) ,
t h e proof of t h e r e l a t i o n s h i p
(3.20) f i n a l l y i s complete.
E u(T)
( 3 . 2 3 ) THEOREM.
i) If
i s equal t o
and t h u s i n d e p e n d e n t of
Eo
@
then
I m ( P ( 1 ) ) c D(P((1))
which
p.
The b i o r t h o g o n a l r e l a t i o n s h i p
ii)
(3.24 holds f o r a l l
u1,b2 E u ( T ) .
PROOF
E u(T).
Let
(1
Obviously
Res (RV) = Res (HT(R,p)V)
1
cc
s o t h a t , by ( 2 . 6 ) ,
Hence, if
where
y E D(P(p))
V(j)(w)
then
d e n o t e s t h e j - t h d e r i v a t i v e of
V
a t the point
1.
A change of t h e o r d e r of summation l e a d s t o
whence, t a k i n g i n t o c o n s i d e r a t i o n t h e r e l a t i o n s h i p ( 3 . 9 ) ,
the
a s s e r t i o n i) i s a l r e a d y p r o v e d .
Now we a r e g o i n g t o p r o v e t h e s t a t e m e n t i i ) : A c c o r d i n g t o (1.5),
for a l l
( 2 . 8 ) and ( 3 . 2 0 ) we h a v e
h E [0,1,2,.
and b e c a u s e of ( 2 . 7 ) ,
.,
,%(p)}
and a r b i t r a r y
(3.25) we o b t a i n
?, E C .
Therefore
SPECTRAL THEORY FOR OPERATOR POLYNOMIALS
for
s mk(bl),
0P h
0 5 k S nu1 T(pl),
Thus, again in view of
(3.25), the proof of ii) is complete.
4. BIORTHOGONAL EXPANSIONS In the sequel we adopt the assumptions made in the preceding sections. spectrum
We denumerate the elements
up(T)
p E
Let p
...
of the (point)
in such a way that
We call the operator bundle
[N.
T
regular of order
if and only if the following properties are fulfilled: i)
there exist a sequence
sequence
all ii)
("j j E N
of natural numbers and a
of simply closed Jordan curves in
(rj)jcN
so that exactly the eigenvalues
9
pl,p2,
pl,p2,.. .,p
nj
a:
around 0
are inside
. for J
j E R;
dj := max{
1x1
:
E
rj}
tends to infinity and there is a
> 0 so that 6 d . S dist(0,r .) J J iii)
there exists a real number
r,
IR(X)l
( j E N); c
> 0 such that
Idxi s c dp
Obviously the property iii) is positive constants
(j
E
N).
j
cl, c2
so that
fulfilled if there exist
REINHARD M E N N I C K E N
224
l e n g t h ( r .) J
z c
1
d
j
and
j E IN.
for a l l
( 4 . 1 ) THEOREM. Let
f E
Let t h e o p e r a t o r bundle
J-l(D(MP+¶))
and assume t h a t
T f
b e r e g u l a r of o r d e r f u l f i l l s t h e "boundary
conditions
(4.2)
j=O
J
Then we a s s e r t t h a t
(4.3) PROOF.
Accorrting t o (22) and (3.3), the d e f i n i t i o n of t h e TD,ff&-1I+.;JP'
= 0
whence t h e b o u n d a r y c o n d i t i o n s ( 4 . 2 )
,
f' j1
we have
(& = 1 , 2 , . . . , p ) a r e e q u i v a l e n t 60 t h e condi-
tions
Let
for
X
E p(T)
& E [O,l,
Using
...,PI
and u n e q u a l t o
and show t h a t
(3.1) and ( 4 . 4 ) w e i n f e r
0,
We s e t
S4+l(X) = S,(X):
Obviously
p.
SPECTRAL THEORY FOR OPERATOR POLYNOMIALS
1
[ fC q"]
p+C+1
-
R(X)
-
1 -.-pi R(X)
'il
3
hkTk f[q+']
=xL+~R(X) 1
225
Tq fCq+"
k= 0
- -
q+d-l m=&
q-1 T, Tm m= 0
1 h~+1 R(X)
C,
m+tI
Tm-t.
C,
ml
which completes the proof. According to (3.4), (3.5) and (4.5) we obtain R(X)V(X)f
= R(X)
q-1 1 C m+l (T(X) m=O ?,
1 = ' c -'
,C~I
-
lm+l
m=O
q-1 m=O
-
m C
kkTk)frm'
k= 0
m
'
1 R( ?,)Tkfr m1 m+l-k
k=O
= S0(X)f whence
which, by (4.5), immediately leads to
I f we integrate both sides of the preceding equation along
rj
we
obtain
f r o m the Definition ( 3 . 6 ) .
that all
d . 2 1. J
We assume without loss o f generality
We estimate
c 1 p+l-k d
5 - -
-
for all
0 2 k 2 m-p,
p z m
j
s q+p-1 whence the assertion of the
226
REINHARD MENNICKEN
expansion theorem is proved. We would like to add the remark that according to (3.11) the eigenvectors and the associated vectors fulfill the conditions and thus the boundary conditions (4.2). conditions
(4.2) are also necessary:
frjl
(j = 0 , 1 , 2 ,. . . , q )
then
f
(4.4)
Therefore, in a sense, the
if we know that all functions
are expandable into a series of type
has to fulfill the conditions (4.2).
(4.3)
This fact is well-
known for functions with (pointwise convergent) Fourier expansions: they have to satisfy the same periodicity properties as the sine and cosine functions.
It is not difficult and therefore left to the reader to state
(4.3) is absolutely convergent
conditions under which the series
(in parantheses) which means that the sums
are bounded if
j
tends to infinity.
We would like to point out that, on the contrary to almost all other authors, we do not require that all eigenvalues with the exception of only finitely many are normal,
5 . APPLICATIONS TO DIFFERENTIAL EQUATIONS If
s E Z
and
u
is an open subset of
denotes the Sobolev space of order viated by
Hs.
If Ns
s E IN
s
over
:= ( U E Hs
: supp u
c
Hs[a,b]
:= Hs/Ns
IR\(a,b))
;
then
Hs(R)
we set
and define
(5.1)
w;
IR
Hs(w)
is abbre-
SPECTRAL THEORY FOR OPERATOR POLYNOMIALS
the corresponding quotient mapping is denoted by to show that
s E N
:
E IIs
UJ
U=W
we define := (v E I I m S : supp v
IIc [a,b]
(5.2)
It is easy
can be identified with the vector space
lIs[a,b]
(u E L2[a,b] Again for
‘p,.
227
-S
Obvious ly Lm[a,b]
(5.7)
c L2[a,b]
= Hi[a,b]
C
F o r later use we state
(Hs/Ns)’ = HC [a,b].
(5.4)
-S
It is well-known that the dual space o f the quotient space I (Hs’H:s) , i.e. the orthogonal complement Hs/Ns is isomorphic to N S of
Ns
with respect to the dual pair
(Hs,H-s). Therefore we only
have to prove that
The inclusion ” C ” immediately follows from the definition o f the (R\[ a,b]
support of a distribution because
C :
the proof o f the inverse inclusion
”1”:
Let
for
i
v E H : s [
E (1,2,
a,b]
...,s ]
.
Since
v E H-s
)
C
Ns.
We sketch
we can choose
vi E L2
such that v =
c
,(i)*
i=O
It is not difficult to show that the functions even f r o m
L2[a,b].
Thus, if
i=O
vi
can be chosen
u E Ns,
i=O
where the derivatives are taken in the sense of distribution theory.
REINHARD MENNICKEN
228
We would like to point out that in this paper
is
H:s[a,b]
understood to be the Banach space dual, i.e. the space of continuous linear functionals, and not to be the Hilbert space dual, i.e. the space of continuous conjugate linear functionals.
Consequently we
have e.g.
for
u,v E L2. If
is a vector space,
G
nxn-matrices with elements in
Mn(G)
G.
n E w\[O},
In the following
m E N,
a = a < a2 <..,< are real numbers.
denotes the space of
m 2 2
and
am = b
Furthermore f o r the present we only assume:
(5.5)
We consider the differential operator
(5.6)
TD(h)y
:= y'- A(*,X)y
and the boundary operator
(5.7) both defined for product of
y E Hy[a,b]
where
Hy[a,b]
denotes the n-fold
Hl[a,b].
For fixed operators on
X E C,
Hy[a,b]
to
TD(X)
and
L:[a,b]
TR(X) are continuous linear or to
Cn
respectively.
We
combine both operators obtaining the "boundary value operator"
(5.8)
T(X)y
:= (TD(X)y,TR(X)y)
which is a continuous linear operator on
(Y E Hy[a,bl) HyLa,b]
to
L;[a,b]
x
en.
SPECTRAL THEORY FOR OPERATOR POLYNOMIALS
(5.9) PROPOSITION. i) ii) iii) iv) PROOF.
with
229
We assert:
TD E H(C,L(H:[a,bl
,L:[a,bl)),
TR E H(C,L(H;[a,b]
,an)),
T E H(C,P(H:[a,b]
,L;[a,b]xCn)),
ind T ( X ) := dim N(T(1))
-
codim Im(T(X))
= 0
( A E C).
i) We have
A j E Mn(L,[a,b])
and
where
If y E Hy[a,b]
then
m
C
m
IXIJIAjY L;ta,bl
j=O
which already proves the first assertion. ii)
where
We set
ba
W(.,X)
denotes the delta distribution with support
is defined to be zero outside the interval
[a}
[a,b].
and By ( 5 . 3 )
V ( h ) E Mn(HCl[a,bl)
and thus
V E H(G,Mn(HCl[a,b]))
tions with respect to
If
W
because of the smoothness assump-
and the
B E Mn(HC1[a,b])
and
W(')fs.
b E H;[a,b]
to have the components n
(B,b)i
:=
C (Bij,bj) j=1
we define
(B,b) E Cn
REINHARD MENNICKEN
230
(
where
,)
dual pair
is the canonical bilinear functional belonging to the
(HC1[ a,b] ,H1[ a,b] )
(5.10)
.
We infer that
= ( V ( A ) ,Y>
TR(X)y
(Y E H?Ca,bI)
from which the assertion ii) is immediate. iii)
Let
X E
be fixed,
Q:
The relationship
T(X) E P(H?[a,bl,L:Ca,b1xCn)
(5.11)
remains to be proved.
We have codim Im(TR(X))
<
a.
We will show that
We set
If u E L:[a,b]
then the function
[
X
Y(X) belongs to that
FD(x)
Hy[a,b]
Hq[a,b]
A(*,X):
so
that
C-L:[a,b]
L:[a,b]
iv)
TD(l) +
Moreover
.?.D(X)
N(TD(X))
y'
= Cn
= u
which means
and thus fi-
is a Fredholm operator.
The embed-
is compact and the mapping
+ L:[a,b]
a compact operator. TD(X) =
(x E Ca,bl)
u(t)dt
and satisfies the equation
is surjective.
nite-dimensional ding
:=
is continuous which shows that
Therefore, cf. e.g. Kato [28],
sD(I)is a Fredholm operator.
The operator
? ( I ) := (TD(X),0) is a Fredholm operator with index dim N(F(1))
0
because
= codim Im(T(X))
= n-
p.238,
GD(I) is
SPECTRAL THEORY FOR OPERATOR POLYNOMIALS
Since
;(I)
:=
,TR(X))
(!?jD(),)
i s a compact o p e r a t o r we i n f e r , a g a i n by K a t o [ Z S ] , T(X) = ? ( A )
+ ?(X)
has index
0
p.238,
that
which c o m p l e t e s t h e p r o o f o f
Proposition (5.9). The e x p l i c i t form o f t h e a d j o i n t o p e r a t o r b u n d l e
is
T*
stated in
( 5 . 1 2 ) PROPOSITION.
For
( v , c ) E L:[a,b]
T * ( ) , ) ( v , c ) = -v'where
PROOF.
A(.,),)t
A(-,),)tv
x
Gn,
+
V(),)tc
X
d e n o t e s t h e t r a n s p o s e d m a t r i x of
We d e f i n e
A(x,),)
E G
w e have
A(.,),)
and
t o be z e r o o u t s i d e t h e i n t e r v a l
[a,b]
and t h e n T o ( ~ ) Y:=
(Y'
-
A('
,h )Y,(V(X) , Y l )
T ~ ( x ) i s a c o n t i n u o u s l i n e a r mapping on
HY
( Y E H:). to
L :
x
(Cn
and i t
i s e a s y t o show t h a t t h e d i a g r a m
i s commutative.
I t i s well-known t h a t t h i s i m p l i e s t h a t t h e
"adj o i n t " diagram
REINHARD MENNICKEN
also is commutative,
Since the adjoint mappings
cp;
are embeddings
we infer
(5.13) so
that it will be sufficient to derive the explicit form of Let
y E H;,
(v,c) E L i d .
TZ(X):
We have
which is the desired explicit form.
(5.14) REMARK. L;[a,b]
The astriction
is the restriction of
TE(1)
of
T*(X)
to the space
D(T,”(X)) := {(v,~) E L;[a,b]xCn
..
(j=1,. , m - 2 )
A
v(a
.+o)
J
-
v(aj-o)
T*(X)
: v I(aj,aj+l
to the space
n
) E Hl(aj,a.J + 1 )
= w(j)(~) tc
.., m ) j .
(j=1,.
In the foregoing defined set v(a-0) = v(al-0) = v(am+O) = v(b+O) because
L2[a,b]
= H:[a,bl.
= 0
233
SPECTRAL THEORY FOR OPERATOR POLYNOMIALS
(5.14) let (v,c) E L;[a,b]
F o r the proof of Remark
x Cn.
Then T*(X)(v,c)
E Lz[a,bl
if and only if
(5.15) because Let
Ha
A(.,X)tv
and
W(*,X)tc
belong to
Lz[a,b]
anyway.
denote the Heavisj.de function
H a w =
i
W e conclude that the relationship
0
x < a,
1
x S a .
(5.14) holds if and only if
which obviously is true if and only if
(v,c) E D(TZ(1)).
The equation
(5.16)
T(X)y
= f
(Y E H;[a,bl,
f
defines the boundary eigenvalue problem [BEVP] studied by Cole f81, [9] for Cz[a,b]
x Cn.
y E Cy[a,b]
which has been
and
f = (fl,O) E
Cole referred to a famous paper of Langer [35] which
considered BEVP's in the complex domain, i.e. for subsets of
x Cn)
E Lz[a,bl
x
varying in
C.
W e call
the "adjoint" BEVP of (5.16).
Since, for fixed
a continuous linear operator on the whole space
X E C,
T*(X)
L:[a,b]
x Cn
is the
equation (5.17) is more convenient than the adjoint BEVP
which has been considered by Cole (and Langer) under more restrict-
234
REINHARD MENNICKEN
ive smoothness assumptions on If operator
f
p(T)
T
v
and
g.
then, by Proposition ( 5 . 9 ) ,
(b
fulfills the assumption (2.1)
so
the boundary value
that all results
stated in section 2 are applicable to the B V E P l s ( 5 . 1 6 ) and (5.17). theorem (2.3) the existence of biorthogonal systems o f
By Keldyz I s
eigenvectors and associated vectors is guaranteed. boundary value operator bundles
T
F o r "normal"
this statement has been proved
by Langer, cf. also Cole, without using any functional analytic argument.
Indeed, for Langer and Cole, already the definition of'
associated vectors would have been impossible as they considered the astriction operator bundle
T;(X)
the domain of which is
1.
dependent on
In the sequel we need additional assumptions in order that the boundary value operator bundle
T
will satisfy the assumptions
of the sections 2 and 3 and the results stated there becomes applicable, We assume that
I
q
V(1)
C
=
. hJVj
(5.18)
+ lA1 with
A ( * , k ) = A.
W0 set
where without l o s s of generality
q L 1,
j=O
n
E = H1[a,b],
F1 = L:[a,b],
AO,A1 E Mn(Lm[a,b]). F2 = Cn
and
(5.19)
J
denotes the canonical embedding of
Hy[a,b]
into
L:[a,b].
: f(j)
E Lm[a,b])
define H
for
jE N
j ,m
[a,b]
and infer
:= [f E H.[a,b] J
We
SPECTRAL THEORY FOR OPERATOR POLYNOMIALS
Mn(H. [a,b] ).Hn[a,b] J tm J from the Leibniz-formula since f(i)E f 5 Hj[a,b].
r E N.
Let
235
c Hn[a,b] J
Co[a,b]
for
i < j
if
In addition to (5.18) we assume that
AO'A1 E Mn(H
[a,bl), rtrn
det A , ( x )
(5.20)
#
0
[ a,b]
almost everywhere in
,
AY1 E Mn(Lm[a,bl). If
r > 0
A1 E Mn(Co[a,b]).
then
Therefore, in this case, the
second and the third condition are fulfilled if and only if det A,(x)
#
0
everywhere in
[a,b].
We infer that the boundary value operator bundle
T
fulfills
all the assumptions made in section 3 so that the results stated there and in the subsequent section 4 become available for this operator bundle.
Here the operator
M,
defined in ( 3 . 2 ) , has the
following concrete form:
(5.21)
where we could omit subspace of
J-l
because we understand
Under the assumption (5.20) D(M
PROOF. Let
to be a
Li[a,b].
(5.22) PROPOSITION.
for all
Hy[a,b]
c) =
Hn[a,b] 1,
4, 5 r + l . The assertion is true f o r
y E Hi+l[a,b].
& = 0
According to (5.20),
thus, by the induction hypothesis, Conversely, let
y E D(M'+l).
From
because (5.21)
L
My E D ( M ) ,
D(M
0
) = Li[a,b].
My E HF[a,b]
i.e.
and
y E D(ML+').
REINHARD M E N N I C K E N
236
y'= -AIMy and
My E D(ML) = Hy[a,b]
y E H;+,,[a,b]
-
AOY
it follows that
y'E
H:[a,b]
and thus
which already completes the proof.
According to (3.3)
[ O l := y
is recursively defined if
y E Hyfa,b].
A n explicit form of the resolvent
operator bundle matrix. h E a!,
T
R
of the boundary value
can be derived in the usual way using Green's
For this purpose it is useful to know that, for fixed V(X)
does not only belong to
Mn(H"[a,b])
also given by a Lebesgue-Stieltjes-measure:
[
but that it is
if we define
X
F(x,X)
(5.23)
:=
C
W(j)(h)
+
a .<x J then
V(X) = dF'(*,X).
Let
W(t,X)dt
Y(.,X) E Mn(Hl[a,b])
be a fundamental
matrix of the equation Y'+
for each
(.,I)
X E a!.
We may suppose that
E [a,b] x a!
x E [a,b].
Let
is well-defined.
A(*,X)y = 0 Y
is continuous in
and is holomorphic with respect to
X E p(T).
Then
We conclude that
for each
SPECTRAL THEORY FOR OPERATOR POLYNOMIALS
237
The boundary value problem (5.16) is called regular of order p
if and only if the corresponding boundary value operator
this property which has been defined in section
T
has
4. Langer [35] and
Cole [8], [9] stated sufficient criteria for the regularity of (5.16), cf. especially Cole [9], p. 541, Theorem 8 which he proved by a profound analysis of the corresponding Green's matrix.
Langer and Cole
made rather incisive assumptions with respect to the coefficients A and
A 1 . These assumptions can be weakened to some extent as we are
going to show in a subsequent paper.
We would like to point out
that Langerls and Cole's regularity criteria only concern the resolvent
R(X)
restricted to
L:[a,b]
x
(0)
s o that they did not
have to estimate the boundary part
However, their criteria remain true for
R(1)
itself since the
additional boundary term can be estimated in a similar manner. We summarise our result in (5.25) THEOREM. operator and ( 5 . 2 0 )
T
Assume that the coefficients of the boundary value
T, defined in (5.8), satisfy the conditions (5.5),(5.18) (with a fixed
is regular of order
p.
r E N).
Let
Suppose that
#
and assume that n p+q 5 r, f E Hp+q[a,b]
p(T)
(b
and that it fulfills the boundary conditions (5.26)
C Tjf R fj+.e-11= j=O
(.e=1,2,.
..,P)
Under these assumptions we have the following biorthogonal
0
238
REINHARD MENNICKEN
expansion in parentheses: n j c P(iri)fl i=l H :[:
If
-
where the natural numbers and the projectors
P(1)
n
j
-I
o
(j+
m )
a ,bl
T
are given by the regularity of
have been defined in (3.6), see also (3.7)
or (3.25). The proof is immediate from the Theorem
(4.1).
The expansion theorems proved by Langer and Cole (cf. e.g.
[9], p.547, Theorem 9) only apply to normal boundary value operator T. Furtheremore, Cole's expansion of the function
bundles
f
in
general is not biorthogonal in the sense defined in the present f
satisfies boundary condi-
paper.
And Cole did not require that
tions.
Indeed he seemed to be a bit astonished that, according to
his convergence theorem, in most cases not function
f
-
. a
f
but a "modified"
is expandable with respect to the eigenvectors of
T, cf. [9], p.549, section 14.
SPECTRAL THEORY FOR OPERATOR POLYNOMIALS
239
mFERENCE S 1.
ALLAHVERDIEV, DI. E.,
O n the completeness of the systems of
eigenvectors and associated vectors of nonselfadjoint operators close to normal ones, 207-210 (Russian). 2.
ALLAHVERDIEV, Dg. E.,
Dokl. Akad. Nauk SSSR 115 (1957),
On the completeness of the system of
eigenelements and adjoint elements of nonselfadjoint operators,
Dokl. Akad. Nauk SSSR 160 (1965), 503-506 (Russian),
Engl. transl.: Soviet Math. Dokl.6 (1965), 102-105.
3.
ALLAHVERDIEV, D%. E.,
On the completeness of the system of
eigenelements and adjoint elements of a class of nonselfadjoint operators depending on a parameter X ,
Dokl. Akad.
Nauk SSSR 160 (1965), 1231-1234 (Russian), Engl. transl.: Soviet Math. Dokl. 6 (1965), 271-275.
4. ALLAHVERDIEV, DI. E., Multiply complete systems and nonselfadjoint operators depending on a parameter 1 ,
Dokl. Akad.
Nauk SSSR 166 (1966), 11-14 (Russian), Engl. transl.: Soviet Math. Dokl. 7 (1966), 4-8.
5. BAUER, G.,
fiber Entwicklungsfragen bei Operatorenbiischeln,
Dipolomarbeit Regensburg 1979.
6. BAUER, G.,
StBrungstheorie fiir diskrete Spektraloperatoren
und Anwendungen in der nichtlinearen Spektraltheorie, to be published.
7. COLE, R.H.,
Reduction of an n-th order linear differential
equation and m-point boundary conditions to an equivalent matrix system,
8. COLE, R.H.,
Amer. J. Math. 68 (1946), 179-183.
The expansion problem with boundary conditions
at a finite set of points,
9. COLE, R.H.,
Can. J. Math. 13 (1961), 462-479.
General boundary condtions for an ordinary linear
differential system,
Trans. Amer. Math. SOC. 111 (1964),
521-550. 10.
COLE, R.H.,
The two-point boundary problem, Amer. Math.
Monthly 72 (1965), 701-711.
240
11.
REINHARD MENNICKEN
D i PRIMA, R.C. and HABETLER, G.J.,
A completeness theorem
for nonselfadjoint eigenvalue problems in hydrodynamic stability, 12.
Arch. Rat. Mech. Anal. 34 (1969), 218-227.
EBERHARD, W.,
tfber das asymptotische Verhalten von L6sungen
der linearen Differentialgleichung Mfy] = XN[yl Werte von 1, EBERHARD, W.,
filr grosse
Math. Z. ll9 (1971), 160-170. h e r Eigenwerte und Eigenfunktionen einer
Klasse von nichtselbstadjungierten Randwertproblemen, Math. Z. 119 (1971), 171-178.
14. EBERHARD, W. und FREILING, G., Nicht-S-hermitesche RandEigenwertprobleme, Math. 2 . 133 (1973), 187-202. 15.
EBERHARD, W. und FREILING, G.,
und
Das Verhalten der Greenschen
Matrix und der Entwicklungen mch Eigenfunktionen N-regullrer Eigenwertprobleme,
Math. Z. 136 (15174)~ 13-30.
16. EBERHARD, W. und FREILING, G., Stone-regullre Eigenwertprobleme, Math. 2 . 160 (1978), 139-161.
17
FREILING, G.,
Regullre Eigenwertprobleme mit Mehrpunkt-
Integral-Randbedingugen,
Math. Z . 171 (1980), 113-131.
18. FRIEDMAN, A. and SHINBROT, M., Nonlinear eigenvalue problems, Acta Math. 121 (1968), 77-125.
19
GOHBERG, I.C. and KREIN, M.G.,
Introduction to the theory of
linear nonselfadjoint operators, SOC.
20.
Providence, Amer. Math,
1969.
GOLDBERG, S., tions,
Unbounded linear operators:
theory and applica-
New York-Toronto-London-Sydney, McGraw-Hill 1966.
21.
GORJUK, I.V., A certain theorem on the completeness of the system of eigen- and associated vectors of the operator 2 bundle L ( h ) = 1 C+XB+E, Vestnik Moscov Univ. Ser. I Mat. Meh. 25 (1970), 55-60 (Russian, Engl. summary).
22.
GRAMSCH, B.,
Meromorphie in der Theorie der Fredholmoperatoren
mit Anwendungen auf elliptische Differentialoperatoren, Math. Ann. 188 (1970), 97-112. 23
GRIGORIEFF, R.D.,
Approximation von Eigenwertproblemen und
Gleichungen zweiter
183 (1969), 45-77.
Art im Hilbertschen Raume,
Math. Ann.
SPECTRAL THEORY FOR OPERATOR POLYNOMIALS
24.
GRIGORIEFF, R.D.,
241
Diskrete Approximation von Eigenwertproble-
men, I. Qualitative Konvergenz, Num. Math. 24 (1975), 355-374. 25.
HEISECKE, G.,
26.
ISAEV, G.A.,
= XM(y) bei 61 (1981), 242-244.
Rand-Eigenwertprobleme N(y)
X-abhlngigen Randbedingungen,
ZAMM
The completeness of a certain part of the eigen-
and associated vectors of polynomial operator pencils, Uspehi Mat. Nauk 28 (1973), 241-242 (Russian). 27.
ISAEV, G.A.,
The numerical range of operator pencils and
multiple completeness in the sense of M.V. Keldyg, Funkcional Anal. i Prilogen 9 (1975), 31-34 (Russian), Engl. transl.: Functional Anal. Appl. 9 (1975), 27-30. 28.
KATO, T.,
Perturbation theory for linear operators,
Berlin-
Heidelberg-New York, Springer Verlag 1976. 29.
KELDYS, M.V.,
On the eigenvalues and eigenfunctions of certain
classes of nonselfadjoint equations,
Dokl. Akad. Nauk SSSR
77 (1951), 11-14 (Russian). V
30.
KELDYS, M.V.,
On the completeness of eigenfunctions of some
classes of nonselfadjoint linear operators,
Uspehi Mat.
Nauk 26 (1971), 15-41 (Russian), Engl. transl.: Russian Math. surveys 26 (1971), 15-44.
31. K6NIG, H., A trace theorem and a linearization method f o r operator polynomials, 32.
preprint 1981.
KOSTSLICENKO, A.G. and ORAZOV, M.B.,
The completeness of the
root vectors of certain selfadjoint quadratic pencils, Funkcional Anal. i Prilogen 11 (1977), 85-87 (Russian), Engl. transl.: Functional Anal. Appl. 11 (1978), 317-319.
33.
KUMMER, H.,
Zur praktischen Behandlung nichtlinearer
Eigenwertaufgaben abgeschlossener linerer Operatoren, Mitteilung Math. Sem. Giessen 62 (1964), 1-56.
34. LANGER, H.,
Zur Spektraltheorie polynomialer Scharen selbstadjungierter Operatoren, Math. Nachr. 65 (1975), 301-319.
35.
LANGER, R.E.,
The boundary problem of an ordinary linear dif-
ferential system in the complex domain, SOC. 46 (1939), 151-190.
Trans. Amer. Math.
242
REINHARD MENNICKEN
The convergence of multiple expansions in the system of eigen- and associated vector-valued functions of a
36. MAMEDOV, K.S.,
polynomial differential-operator pencil, Izv. Akad. Nauk Azerbaidgan. SSR Ser. Fiz.-Tehn. Mat. Nauk 1 (1977), 36-40 (Russian, Engl. summary).
37. MARKUS, A.S.,
Some criteria for completeness of the system of
root vectors of a linear operator in a Banach space, Mat. Sb. 70 (1966), 526-561 (Russian), Engl. transl.: Amer. Math. SOC. Transl. 85 (1970), 51-91.
38. MARKUS, A.S.,
The completeness of a part of the eigen- and associated vectors for certain nonlinear spectral problems, Funkcional Anal. i Prilogen 5 (1971), 78-79 (Russian),
Engl. transl.: Functional Anal. Appl.
5 (1971), 334-335.
39. MITTENTHAL, L., Operator valued analytic functions and generalizations of spectral theory, Pacific J. Math. 24 (1968), 119-132.
40. MONIEN, B.,
Entwicklungssltze bei Operatorbiischel, Dissertation Hamburg 1968.
41. MtfLLER, P.H. und KUMMER, H.,
Zur praktischen Bestimmung nicht-
linear auftretender Eigenwerte, Anwendung des Verfahrens auf eine Stabilitltsuntersuchung (Kipperscheinung), ZAMM 40 (1960),
136-143
42. ORAZOV, M.B.,
The completeness of the eigen- and associated vectors of a selfadjoint quadratic pencil, Funkcional Anal. y Prilogen 10 (1976), no.2, 82-83 (Russian), Engl. transl.:
Functional Anal. Appl. 10 (1976), 153-155.
43. PALLANT, Ju. A.,
A test for the completeness of a system of eigenvectors and associated vectors of a polynomial bundle of operators, Dokl. Akad. Nauk SSSR 141 (1961), 558-560 (Rpssian), Engl. transl.: Soviet Math. Dokl. 2 (1961),
1507-1509.
44.
ROACH, G . F .
and SLEEMAN, B.D.,
operator bundles,
On the spectral theory of
Applicable Anal.
7 (1977), 1-14.
45. ROACH, G.F. and SLEEMAN, B.D., operator bundles 11,
46.
TURNER, R.E.L.,
On the spectral theory of Applicable Anal. 9 (1979), 29-36.
A class of nonlinear eigenvalue problems, J. Functional Anal. 2 (1968), 297-322.
SPECTRAL THEORY FOR OPERATOR POLYNOMIALS
47. VIZITEI, V.N. and MARKUS, A.S.,
243
On convergence of multiple
expansions in eigenvectors and associated vectors of an operator bundle, Engl. transl.:
48. WAGENFmRER, E.,
Mat. Sb. 66 (1965), 287-320 (Russian),
Amer. Math. SOC. Transl. 87 (1970), 187-227. Transformations of n-th order linear dif-
ferential equations whose coefficients are nonlinear in
p
to systems of n first order equations which are linear in the parameter, to be published.
49. WILDER, C.E.,
Reduction of the ordinary linear differential
equation of the n-th order whose coefficients are certain polynomials in a parameter to a system of
n
equations which are linear in the parameter,
first order Trans. Amer.
Math. SOC. 29 (1927), 497-506.
50.
V
YAKUBOV, S. Ya. and Mamedov, K.S.,
Multiple
completeness of
the system of a eigen- and associated elements of a polynomial operator bundle and multiple expansions with respect to this system, Funkc onal Anal. i Prilozen 9 (1975), 91-93 (Russian), Engl. transl.: Functional Anal. Appl. 9 (1975)9 90-92
This Page Intentionally Left Blank
Functionu/.4nulysis, Holoniorpliy mid A p p r o x i m a t i o n Theory 11, G I . Zupato ( e d , ) 0 Elsevier Science Publishers B. K (h'orth-Holland), I 9 8 4
245
INTEGRO-DIFFERENTIAL OPERATORS AND THEORY
OF SUMMATION
M. Mikolds
I. HISTORICAL REMARKS
The theory of differential and integral operators of arbitrary complex order which originates in the 19th century with some results of Abel, Liouville, Riemann and Heaviside, has been property established since the turn of the 20th century by certain fundamental works of Hadamard, Hardy and Littlewood, M. Riesz, H. Weyl.
At
the end of the fourties, Marcel Riesz summarized the results of the field due to him and his school, discussing in the work [13] of more than 200 pages also some applications of the theory in modern mathematical physics, connected with the so-called Riesz potentials. The kernel of the Riesz method is the analytic continuation of the Riemann-Liouville "fractional" integral:
(1)
1 xo :
=T(s)
&"
f(t)(x-t)
f (Re
" ' d t
1
bounded, integrable s
7
0
0
which is nothing else but a natural extension of Cauchyls solution of the initial value problem
y(m) (x) = f (x) ;
(2)
y(xo)
... = y(m-l)(xo) m
=
= Y' (x,)
...
= 0,
corresponding to the complex parameter
s
in (1).
in case of a Lebesgue integrable function
f
and any fixed
with
Re
Note that s
s > 0, the existence of the integral (1) is assured for
M. MIKOLAS
246
almost all
x
I :
and the operator
satisfies the so-called
0
index law (or semigroup property):
(3) r
e
s1 > 0 ,
x
Re
s2
> )0
< t 5 x
The analytic continuation of
: 1
is meant of course with
0
respect to the order of integration
s
and depends on the fact that
the integral (1) is a holomorphic function of existence.
s
in its domain of
On the same basis, supplemented by the observation that
the widening of the semigroup of fractional integral operators of the type (1) to an Abelian group is equivalent to the introduction of fractional differential operators in the Hardy-Littlewood sense [2]
-
this has been recently the starting-point of a newly develop-
ed branch of semigroup theory, namely the Itoperational calculus of fractional powerst1. The results are also interlinked with differential equations in Banach spaces and a few modern topics in functional analysis, e.g. about certain operators in abstract Hilbert spaces.
(Cf. e.g. [l]
and [15].)
In the pertinent works [3]-[lO]
of the author which have been
published since 1958 i n several periodicals in English, French o r German and partly also in the Hungarian-written book 1111, some complex analytical procedures, the idea of Riesz about analytic continuation, furthermore certain deeper tools from summation theory and operator theory are linked, in order to get the most general, unified treatment of integro-differential operators for arbitrary Lebesgue integrable functions, based mainly on Weylls concept [16] of fractional integration.
Let us mention that the words "integro-
differential operatorsll are utilized in what follows f o r the sake of brevity in a much wider sense than itls usual.
B y this term,
INTEGRO-DIFFERENTIAL OPERATORS AND THEORY OF SUMMATION
247
integral or differential operators of arbitrary complex order will be meant, referring s o to the fact that it is about a common generalization
of integration and differentiation of any positive in-
tegral order.
(.A
similar meaning has the word "diffintegration"
in a recent book of Oldham and Spanier [12].)
There exists also a
longer work in English which yields a survey on all main topics on integro-differential operators published during the last decades. This is author's summarizing report [lo], which was given at the first international conference on the field held in New Haven (Conn., U S A ) in 1 9 7 4 , and was published in the Proceedings of that congress edited by Springer-Verlag as Vol. 457 of the series "Lecture Notes in Mathematics".
(Cf. [ l b ] .)
It may be pointed out that the theory of integro-differential operators is becoming now a new branch of analysis, between the classical and functional one, whose applications reach from the theory of functions, integral transformations, theory of approximation and from a large scale of differential and integral equations to the modern operator theory and the theory of generalized func-
t ions.
11. EXPOSITION OF THE METHODS
In the sequel, according to the character of this seminar, we will elucidate the close connection between integro-differential operators of arbitrary complex order and some strong summation methods. The one side of the inherence in question is that the analytic continuation of the Weyl fractional integral and herewith the introduction of integro-differential operators can be realized by means of powerful processes of summation, due to Abel-Poisson, Borel, Le Roy and Lindelbf etc.
As an illustration, lekus formulate
248
M. M I K O L A S
only the central results of this theory, given in [ 3 ] and [lo]. Consider Weyl's fractional integral of order tion
where
f E L(0,l)
Re
(5)
s > 1
with the period
s
of a func-
in the form
1
and
an(.)
= (an-ao)cos 2nrrx
bn(x)
=
(an-a )sin Pnnx
+ p,,sin
-
Znnx,
pncos 2nnx,
@ n denoting the corresponding Fourier coefficients of f. If we apply the most effective of the above-mentioned summation methods, namely a suitable extension of the so-called Mittag-Leffler summation* (due to M. Riesz) to the series in
-mined
(4), then it can be deter-
completely the characterizing Mittag-Leffler star for both of
these series, so that we get an explicit expression of
fC
s ] (XI 9
holding everywhere in the common part of the mentioned domains. This means that the method yields about the maximal infornation on the holomorphy domain and the singularities of as a function of
s
f[s] (x)
which can be hoped in full generality,
By the
way, we have the integral representation:
(6)
fcs,(X)
=
f(x-6) [s,(t)
-
,S,(x)ldt
(Re
5
where
*This
process is defined for an arbitrary series m
(ML)
C
m= 0
m
um = lim 6++0
C m=O
r(1+6m)''
urn.
Cum by
> 1)
INTEGRO-DIFFEmNTIAL OPERATORS AND THEORY OF SUMMATION
249
Thus the result depends also upon the properties of the generalized (Hurwitz) zeta-function
C(s,u),
defined by analytic continuation
m
C (u+m)-'. m= 0 tion ( 7 ) the formula:
of the series
It holds namely for the kernel func-
The other side of the exposed connexion will be discussed more in detail, i.e. that certain function series can be handled very effectively by means of integro-differential operators.
It is
about a qew summation method which was introduced in its simplest form in the works [4]-[6]
of the author and has been developed
further since the sixties in several directions, e . g . in [9]-[11]. Let
'pn(x)
(n=1,2,...)
a sequence o f functions bounded and
Lebesgue integrable in an interval
c
series
: I (pn
(xo,xl)
converges at a point
x E
and suppose that the (xo,xl)
for any v > 0.
0
Then the limit
(9) is called the (W)-= existence of ( 9 ) ,
of the series
C cpn
C (pn;
and in case of the
is said to be (W)-summable at
x.
This (W)-method leads to sharp results e.g. for trigonometric series and ordinary Dirichlet series. x
=
In particular, putting
and using certain properties of the Hurwitz zeta-function,
-m
a simple
-
necessary and sufficient
-
summability condition for
trigonometric Fourier series can be deduced.
Moreover we find that
the local "strength" of the (W)-method is beyond that of any classical summation process. By formal grounds, it is reasonable for these applications to define two variants of the method: We say that the series
M.
250
. a
is
+
MIKOLAS
C (an cos nx + bn sin nx) n=1
(W-)-summable at a point
(W+)-or
x,
if there exists,a > 0
being chosen sufficiently small, the limit for
8
-t
+O
of the sum
of the series
or
co
. a respectively.
+
n-9[an cos(nx
n=1
T8 - Tr) +
bn sin(nx
- Trrs) ]
,
These limits are called the (W+)- resp. (W-)-sum of
(10)
111. MAIN RESULTS
On the above mentioned lines, the following theorems can be obtained.
--1. f
The trigonometric Fourier series of a bounded function
at a point
x
is (W+)-summable if and only if the limit 5 f(x+O)
(11)
exists, where
6
=
is an arbitrary positive number (but a fixed one).
does not depend on
f(x+O)
f(x+t)t'-ldt]
6
and in case of its existence, it
yields also the (W+)-sum o f the Fourier series.
-2 .
We have especially
f(x+O)
= f(x+O)
the function has a limit from the right.
at every point where
Furthermore, the (W+)-sum-
mability holds uniformly in each closed interval where tinuous.
--3 .
f
is con-
(Two-side continuity at the end-points being assumed.) Analogous statements hold also for the (W-)-method.
have only to put
x-0
instead o f
x+O.
We
INTEGRO-DIFFERENTIAL OPERATORS AND THEORY OF SUMMATION
4. -
There exist such trigonometrical series which are not
summable neither by any ( C , r ) -
-
( W i ) summable 5. -
251
nor by the (A)-method, yet are
.
In case of a related summation method, defined for (10)
m
. a
+
all our theorems are valid with of
f(xit)
cos nx + bn sin nx),
C n-'(an W + O n=l
lim
rpx(t)
= 1 [f(x+t)+f(x-t)]
instead
and with
[
b
(13)
f((x))
instead of tion
f
f(xf0).
= '++o lim
['
rpx(t)tO-ldtl
In addition, we obtain for any bounded func-
the result, that the set of points at which
f((x))
exists
is wider than that of the points where the so-called Lebesgue condition holds i.e.
the limit /
lim
exists.
x+e
-
f(t)dt
As it is well-known, this last one is the most general
summability condition of practical use. We remark that these results are based essentially on the connection between the closed form of the series occuring in the definition of the (W,)-method
and the
Hurwitz zeta-function.
By
certain propositions of Tauberian type it can be shown still:
the
effectiveness of the (W*)-methods in case of bounded functions exceeds not only the effectiveness of the Abel-Poisson method but also that of a more general class of processes, namely the so-called Abel-Cartwright methods. The latters are defined for an arbitrary series
c
Un
by
252
M. MIKOLAS
where
is a fixed positive number.
q
Recent investigations indicate wide application possibili-
ties to boundary
asymptotics of power series in Hadamardls sense,
namely concerning improvement and localization of the results in question.
IV. PROOF OF A TYPICAL SUMMATION THEOREM For illustration, we will consider the above-mentioned theorem
2. -
which can be formulated in detail as follows:
The trigonometric Fourier series of a bounded function is summable in the sense (12) at a point
x
if and only if the
limit (13) (with an arbitrarily small but fixed
In particular we have
f( (x))
f
6 > 0)
= [ f (x+O)+f (x-0)] /2
exists.
whenever f (x*
exist, and the summability is uniform in each closed continuity interval of the function.
The domain of effectiveness of our sum-
mation process is greater than that of any CesAro method o r of the Abel-Poisson summation. PROOF,
We start with some elementary lemmas which can be deduced
easily from the classical theory of the Hurewitz zeta-function Ccf.
(811 *
LEMMA 1.
<(s,u)
satisfies the formula
( O < S < l ,
and the inequality
LEMMA 2.
We have the representation
O < x < l )
INTEGRO-DIFFERENTIAL
O P E R A T O R S AND T H E O R Y O F S U M M A T I O N
253
n=1
and for the partial sums of the left-hand series the estimation
These propositions imply that the sum o f the series
a.
with
= 2-
n
2rr
8, = L
r=
f(t)sin
(0
< 9 < 1,
f(t)dt,
nt dt
rl
0
5;
x < 2lT),
;[
27
an
=
(n=1,2,. . . )
f(t)cos
nt dt,
and a bounded function
f,
can be written in the form:
(21) where
Furthermore the kernel function
v = 0
and obeys the inequality
Z9(v)
has the only singular point
254
M.
MIKOLAS
s o that the difference on the left-hand side of ( 2 3 ) tends in
0 < v s 2n
uniformly to
0
as
9 + +O.
Let now split (21) into three parts: 2ll 1
(24)
12=
=
vx(v)Zg(v)dv
2i7
cpx(v)[Zs(v)
= ;n
-
2n(cos n9 7) -1 r(9)-' v'-l]dv
+
6 vx(v)v
LZn
9-1
dv +
JO
+
-1
(COS
"9) 2
= J1+ J
with a fixed
2
+
r(9)"
(~,(v)v'-~
dv =
J3
6 E (0,l).
A s regards the first term, with any given
9; < 1
a number
can be associated such that
I Jll
(25) where
E 7 0
K =
sup t€C 0 , 2 n l
On the other hand, using simple properties of the gammafunction we get
provided that
9 < 9%
.
Finally, there exists a number the inequality
8; > 0
such that for 9
7
8;
I N T E G R O - D T F F E R E N T I A L OPERATORS AND THEORY O F SUMMATION
holds. By
(24)-(27),
1% whenever
9
r
we see that
6
-
vx(v)Zg(v)dv
9 [
is sufficiently small.
cpX(v)v9-ldvl
< 7KC,
This is equivalent to the
statement that the limits
[a
lim W+O
Jo
cpx ( v ) v6
-dv] '
can exist only simultaneously and in case of existence, they are equal.
If b o t h of the limits 0 < 77 <
6 < 1: 6
[o
19
cpx(v)v9-1dv
+ T1 (f(x+O)+f(x-O)I
1
-
exist, then we have f o r
f(x*O)
2 1 [f(x+O)+f(x-O)]
v9-'dv
19
+ p If(x+O)+f(x-O)1(1-6
9
);
-
11
I
4
I;
255
M. MIKOLAS
256
and the last bound becomes as small as we please, if first thereafter
(q
being fixed)
9
q,
is chosen in a suitable manner.
Since the bounds figuring in ( 2 5 ) - ( 2 7 ) are independent of
x,
it
follows also the assertion on uniform summability.
In order to show that the method (12) is more effective than any Cesaro o r the Abel-Poisson summation process, we refer to the W C n (l+iT ) (Tho), by a well-known fact that the series n=1 Tauberian theorem of Hardy and Littlewood, is not summable by any
-
o f these methods.
Nevertheless it is plainly summable in (12)
sense, because the continuity of
implies for
9
-+
c(6,u)
as a function of
+0:
= c(l+iT, 1) = c(l+iT). Thus the verification of the theorem is completed.
6
INTEGRO-DIFFERENTIAL OPERATORS AND THEORY OF SUMMATION
257
REFERENCES 1.
BUTZER, P.L., TREBELS, W . ,
Hilberttransformation,
gebrochene Integration und Differentiation.
Kbln-Opladen
1968, 81 pp. 2.
HARDY, C . H . ,
LITTLEWOOD, J.E.,
integrals1.-11.
34 (1932) 3.
MIKOLAS, M.,
Some properties of fractional
Math. Zeitschrift 27 (1928), 565-606;
403-439. Differentiation and integration of complex
order of functions represented by trigonometric series and generalized zeta-functions. 10 (1959)
4. MIKOLAS, M.,
Acta Math. Acad. Sci. Hung.
77-124. Sur la sommation des series de Fourier au moyen
de lfintegration dlordre fractionnaire.
Comptes Rendus
Acad. Sci. Paris 251 (1960), 837-839.
5.
MIKOLAS,
M.,
Application dlune nouvelle methode de sommation
aux series trigonometriques et de Dirichlet.
Acta Math.
Acad. Sci. Hung. 11 (1960), 317-334.
6.
MIKOLAS, M.,
a e r die Dirichlet-Summation Fourierscher Reihen.
Annales Univ. Sci. Budapest, Sectio Math. 3-4
(1960-61),
189-1515,
7.
MIKOLAS, M.
Generalized Euler sums and the semigroup property
of integro-differential operators.
Annales Univ. Sci.
Budapest, Section Math. 6 (1963), 89-101.
8. MIKOLAS, M.,
S u r la proprie't6 principale des operateurs
diffgrentiels g8n6ralises.
Comptes Rendus Acad. Sci. Paris
258 (1964) 5315-5317 9. MIKOLAS, M., Fourier. 10.
MIKOLAS, M.,
Procedes de sommation (A,hn) dans l'analyse de Communications CIM Nice, 1970, 132 p. O n the recent trends in the development, theory
and applications of fractional calculus.
Lecture Notes in
Math., vol. 457 (1975), Springer-Verlag. 11.
MIKOLAS, M.,
Real Function Theory and Orthogonal Series.
(In Hungarian.) Budapest 1978, 494 pp.
258
12.
M. MIKOLAS
OLDHAM, K.B., SPANIER, J , , The Fractional Calculus.
Academic
Press, New York-London 1974, 234 pp. 13.
RIESZ, M.,
Ltint6grale de Riemann-Liouville et le problhme
de Cauchy. 14.
Acta Mathematica, 81 (1949), 1-223.
ROSS, B. (ed.), Fractional Calculus and Its Applications. Berlin-Heidelberg-New York 1975, Springer-Verlag, 381 pp.
15.
WESTPHAL, U . , Ein KalkCll fllr gebrochene Potenzen infinitesimaler Erzeuger von Halbgruppen und Gruppen von Operatoren I.-II. Compositio Math. 22 (1970), 67-103;
104-136, 16. WEYL, H.,
Bemerkungen zum Begriff des Differential-
koeffizienten gebrochener Ordnung.
Vierteljahrschr.
Naturforsch. Ges. ZClrich, 62 (1917), 296-302.
Technical University of Budapest, I. Dept. of Math. and Mathematical Research Institute of the Hungarian Academy of Sciences Budapest, Hungary
Funrtiona124nolysis, Holoniorphy und Approwirnotiun 7lreory 11, G.1. Zaputa (ed.) 0 Elsevier Science Publisliers B. V. (Nortli-Holland), 1984
259
APPROXIMATION-SOLVABILITY OF SOME NONCOERCIVE NONLINEAR EQUATIONS AND SEMILINEAR PROBLEMS AT RESONANCE
WITH APPLICATIONS ( * )
P.
s. MILOJEVI~
1. INTRODUCTION
In the first section of the paper, we study the (approximation) solvability of operator equations o f the form
T
A-proper or strongly A-closed and such that
as
IIx((4
m
for
u E Tx,
v E Kx,
where
IIuII
f E Tx
with
-
+ (U,V)/II~II -t
K: X -+ PY*.
Applications
to operator equations involving monotone like and other classes o f mappings and to elliptic
BV
propblems are also given.
The results
o f this section complement the earlier ones of the author t20-221.
In Section 2, we first prove several new abstract results on the (approximation) solvability of semilinear equations o f the f o r m Ax
+
Nx = f ,
where
A
is a Fredholm mapping o f index zero and
a given quasibounded nonlinear mapping such that o r strongly A-closed.
is A-proper
The results are then applied to studying
semilinear elliptic equations o f the form
= f
A + N
exhibiting (double) resonance with
F
..
Au + F(x,u,Du,. ,D2m~)= having a linear growth.
The obtained results extend earlier ones of Dancer r l l ]
and Berestycki
and Figueiredo rl] involving nonlinearities that depend on U.
(*)
N
The work was partially supported by a CNPq Grant, 1980/81.
x
and
260
MILOJEVI~
P.S.
2. SOLVABILITY OF CERTAIN NONLINEAR NONCOERCIVE OPERATOR EQUATIONS
Let
X
and
Y
b e normed linear spaces,
(En)
Fn
continuous linear mappings of
Wn
respectively,
(x,VnEn)
admissible scheme for C
Y
x E X.
Let
K(Y)
and
x,
-+
C
X
T: D
BK(Y)
that
be a subspace,
n
and
V -+ '2
Tn
y
-+
and
Tn
onto
dist is a n
Qn: Y
for each
YnC
-+
x,y,
is a projectionally complete scheme for
(X,Y).
be the families of all nonempty closed convex
and bounded closed and convex subsets of
V
and
= (En,Vn;Fn,Wn)
Qny
Y
and of
m
Pn: X -+ X n C X
If
are linear projections, Pnx
To = (Xn,Pn;Vn,Qn)
r
Then
(X,Y).
X
into
= sup IIVn\/<
6
injective,
for each
0
-+
Vn
En
(F,)
dim En = dim Fn,Vn
two sequences of finite dimensional spaces with and
and
I-
D C X, 6
WnTVn:
Y
respectively.
a scheme for Dn
is upper demicontinuous.
Dn = Vil(D),
We shall always assume
K(Fn).
-t
(V,Y),
Let
The classes of mappings to be
studied are introduced next. DEFIAJTION 2.1.
T: n
(a) We say that
n
V -+ '2
closed (A-closed) with respect to a scheme ever
D
n
V 3 V
and
f E Y,
(b)
T: D
"k
E
V
3
n
V
flJ -+ nk f E Tx.
IIynk-w and
for
o
f o r some
is strongly A-closed w.r.t.
'2
(V,Y)
r
V
-+
{Vnkunk E D
n
V)
T: D
if, when-
y E T u nk nk nk
(V,Y) if,
for
for some
V
n
(c)
and
r
-W fll -+ o u E D n V is bounded and IIynk nk nk nk T u and f E Y , then f E T x for some x E D nk "k
whenever y
n
u -+ x nk nk then x E D
is approximation-
Y 2
A-proper w.r.t.
E T u and f E Y, nk nk nk T h e upper semi-(demi-) continuity o f
r
for
IIynk-w fll -+ o nk then some subsequence
is bounded and
y
trictions on
To
and/or
(X,Y)
T
(V,Y)
n
V.
if, whenever
for some V
U
-+
x.
nk(i) nk(i) together with some res-
imply its A-closedness (cf. [ 2 8 ] ) .
APPROXIMATION-SOLVABILITY
261
Monotone like mappings are examples of strongly A-closed ones and, as will be seen later on, many classes of nonlinear mappings are of the A-proper type,
Theory of A-proper mappings, their uniform
limits and of strongly A-closed (i.e., pseudo A-proper) mappings unifies and extends theories of compact and ball-condensing vector fields, of monotone and accretive like and other classes of mappings an is, more importantly, useful in studying various classes of mappings to wich the other existing theories are not applicable.
The
theory is constructive in case of A-proper mappings and was initiated by Pertyshyn and we refer to [SO] for a survey of the theory until 1975.
Our terminology is somewhat different from the usualone.
In this section we shall study the (approximation) solvability of operator equations
x,
(x E
f E Tx,
(2.1)
f E Y)
r
where
T
(X,Y)
and satisfies a rather weaker than coercivity condition
is either A-proper or strongly A-closed w.r.t.
assume that whenever that
T
is K-quasi-bounded for some
Exn] c X
for each
K: X
y, E Tx,,
is bounded and
(yn,fn) 5; cl1xnll
bounded.
In most of the cases we shall
o r (2.8) below).
(see (2.5)-(2.6),
for
Y"
2
-t
,
fn E Kxn
and some c > 0 then
n
i.e., are such {yn]
is
For o u r study of Eq. (2.1) we need introduce a multivalued
bounded mapping
G: X -+ K(Y)
such that
0 $f G x
for
)IxII
large
and which satisfies the following conditions: (2.2)
F o r each large
large
n,
where
r > 0, deg(WnGVn,Bn(O,r),O) Bn = Vil(B(0,r))
#
0
for each
and the degree is that
.
defined in [ 17,181
(2.4)
There exists
Kn: Vn(En)
-I
K:
2
for each
(WnY,u) = (Y,v) f o r each u E Knx,
n
v E Kx,
such that
x E VnEn, y E Y.
262
MILOJEVIC
P.S.
Y = X
When
Y = Y"
or
there are natural choices f o r
t h a t s a t i s f y (2.2)-(2.4).
Y = X
If
i s a lll-Banach
G = I,
Kn = K I X n
p i n g , and
x E Xn,
each
K = J: X
we
n z 1.
(2.2)-(2.4)
X
-t
possible t o find a
mapping. if
X
K:
K = JG,
J: Y
r
Then (2.4) h o l d s f o r
Q ~ K X= K X
x E Xn.
for
"'2
Let
J:
a semi-inner (=min ( @ ( x )
X
G:
-t
i t i s always
Y
= {Xn,Vn,Yn,Qn)
with K n = K'Xn I n applications i t i s often possible
K, K n l G
r
and a scheme
2.2)-(2.4). X
"'2
b e t h e n o r m a l i z e d d u a l i t y mapping a n d d e f i n e
product
I
Kn = I l x n we s e e
and
i s t h e normalized d u a l i t y
-I
t o c o n s t r u c t o h e r t y p e s of mappings which s a t i s f y
and
s u c h t h a t (2.3) h o l d s ; f o r e x a m p l e ,
"'2
-t
where
K = I
, " ' 2
for
JX
c o m p l e t e scheme f o r
F o r a given
t h a t t h e y s a t i s f y (2.2)-(2.4).
IIPn(I = 1,
P ~ J Xc
since
Y = X*
i s reflexive,
G = J: X
X,
t h e n o r m a l i z e d d u a l i t y map-
i s a projectionally
then taking
we c a n t a k e
, * ' 2
have
If
T o = {XnlPn;R(P:),P:] (X,X*),
-t
Kn
s p a c e and
T o = {XnlPn) i s a p r o j e c t i o n a l l y c o m p l e t e scheme f o r t h e n choosing
G, K,
@
( a , * ) - :
XXX
-t
R
(x,y)- = inf{@(x)l@EJy}
by
We a r e i n a p r o p o s i t i o n t o g i v e o u r b a s i c
E Jy)).
approximation-solvability
r e s u l t f o r Eq.
(2.1) w i t h
T
being
A - p r o p e r which was announced f i r s t i n [ 281.
T H E O R E M 2.1.
T: X
Let
+
-$
'2
G,T
Suppose t h a t
(2.2)-(2.4) h o l d a n d t h a t f o r e a c h
e x i s t s an
Then E q .
rf > 0
A - p r o p e r and A - c l o s e d
w.r.t.
bounded a n d
pG
K = JG
be K - q u a s i b o u n d e d ,
f
r in
and
G
1
2
for
Y
0.
there
such t h a t
(2.1) i s f e e b l y a p p r o x i m a t i o n - s o l v a b l e
(i.e,,
there exists a solution
large
n
and a s u b s e q u e n c e
V
un E E n "k
u
"k
-t
x
of with
f o r each
Wnf
E WnTVnu
f E Tx).
f
in
Y
f o r each
263
A P P R O X I MAT1ON- S O LVABI LITY
PROOF.
Let
e r n e s s of
Y
in
f
and ( 2 . 5 )
T
I t i s e a s y t o s e e t h a t t h e A-prop-
be f i x e d .
imply t h a t t h e r e e x i s t s a n
n
1
2
such
that
Therefore,
deg(lJnTVn-\Jnf,
Bn,
0 ) = deg(WnTVn, B n ,
0)
To show t h a t t h i s d e g r e e i s n o n z e r o , we d e f i n e on H n ( t , u ) = tldnTVnu
homotopy
n znl
each
w i t h some
n1
+ (1-t)WnGVnu.
E $Bn
u
and
a
tk E r O , l ] ,
f o r each
tk + t0
k.
)
Suppose
(0,l)
and l e t
+ (1-tk)W
tkWnkvk
vk w
E
u and wk E GV u nk nk "k "k = 0, k 2 1. Then f o r e a c h TV
nk
k
w
(l-t0)tilw
+
v nk
( 'nkUnk )
= rf.
u E Tx
3
+ (l-to)t;l w
w
-
1
( l - t k ) t ; ]wnwk
+
and A - c l o s e d n e s s
of
{Vmum)+ x
with
0 €
o
k +
as
T
+
toTx
+
pG,
v E Gx
1
(l-to)ti,
then
a r e such t h a t
+
tou
T h e r e f o r e , i t remains t o c o n s i d e r t h e c a s e
Then wnkVk = e r n e s s of T,
- ( lmtk)
ti
'nkWk
+ 0
some s u b s e q u e n c e
2
U
and if
and
X(v,v)- = 0 .
to = 0 , l .
k + m
as
-
some s u b -
since,
(1-to)v = 0
))uI/IIvII + ( u , v ) - = XIIvll
1
=
a,
(1-to)Gx
This leads t o a contradiction with (2.6)
and
w nk
nk
a n d , by t h e A - p r o p e r n e s s sequence
w
= -(l-tk)t;l
w nk
C ( l - t o ) t-1 o
=
=
)
0 C Hnk(tk,u
such t h a t
to E
be s u c h t h a t
X
gn
nk
first that
IIxll
X
[O,l]
no.
2
fixed
nk
large
n
W e n e e d show t h a t for
( 2 . 7 ) d i d n o t h o l d , t h e n t h e r e would e x i s t
If
for
t o = 1.
Let
a n d , by t h e A-prop+ x
with
0
E
Tx
vnk(i) "k(i) and
IIxII = r f ,
i n contradiction with ( 2 . 5 ) . and
Then f o r
xk E KV
and ( 2 . 4 )
and n o t h i n g t h a t
yk
E
K
nkUnk
tk f 0
V u nk nk "k
Finally,
let
to = 0.
we o b t a i n u s i n g ( 2 . 3 )
f o r i n f i n i t e l y many
k
264
P.
which i m p l i e s t h a t
= ( 1-tk)
T h i s completes t h e proof
of
ri 2 n1
Wnf E
tha
osed,
A-C
(b)
WnTVnun
(a)
tkll Wnkwk((
n
for
(2.7).
2
V
nl
.
u k n nk
(2.5)
Conditions
Analysing t h e proof
of T.
a contradiction.
-t a ,
deg(WnTVn
-
T
Since -Ix
u n E Bn
i s A-proper
and ( 2 . 6 )
of Theorem 2 . 1 ,
0
such
and
0
f € Tx.
with
#
Wnf,Bn,O)
and c o n s e q u e n t l y t h e r e e x i s t s a
some s u b s e q u e n c e
REMARK 2 . 1 .
MILOJEVI~
(2.7) t h a t
Now, i t f o l l o w s f r o m f o r each
.
i s bounded by t h e K - q u a s i b o u n d e d n e s s
{v,}
-1
(1 Wnkvk((
Hence,
s
a r e i m p l i e d by
we s e e t h a t c o n d i t i o n
( 2 . 6 ) c a n be r e p l a c e d by (2.9)
f E Y
For e a c h
x
L e t u s now e x t e n d Theorem 2 . 1 t o t h e c a s e when
T
Whenever and
{x,)
f E Y,
THEOREM 2 . 2 .
C X
for
x E aB(o,rf)
i s bounded and
then there e x i s t s an
<
0.
is just
un
-+
x E X
f o r some
f
such t h a t
un E Txn f E Tx.
Suppose t h a t a l l t h e h y p o t h e s e s of Theorem 2 . 1 h o l d
w i t h t h e A-properness condition
(*) for
f o r each
r > 0
0
$
or s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n :
s t r o n g l y A-closed
#
such t h a t and
~ ( x n) X G ( X ) =
(*)
rf > 0
t h e r e e x i s t s an
T
of
T
r e p l a c e d by e i t h e r
or t h e s t r o n g A - c l o s e d n e s s
l a r g e and e a c h
f o r each l a r g e
remains v a l i d i f
and A - c l o s e d n e s s
> 0
n. Then T(X) = Y .
(2.6)
of
T
and t h a t
s m a l l , deg(pWnGVn,Bn(Qr),O) Moreover, t h e c o n c l u s i o n
i s r e p l a c e d by ( 2 . 9 ) .
#
265
APPROXIMATION-SOLVABILITY
PROOF.
Let
po > 0
s u c h t h a t for e a c h
p E
Let
in
f
(O,yo)
-
Wnf,Bn,O) = deg(l\rnTVn
d e f i n e on
x fin
[O,l]
+
and
nl(pl)
exist
s n2(p2)
t k E [O,l],
for e a c h
Let
k.
t k l J n k v k + pWnkwk
W
v
+
z no
H
E aB
"k
u nk nk
and by t h e A - p r o p e r n e s s 0 E
toTx
+ pGx
i 1,.
-1 = p(to
of
and
T
.t
-
yG
no.
;r
+
0
-1
tk )Wn
-
Next,
FWnGVnU
-
such t h a t f o r each
If n o t ,
u
Hpnk(tk,unk )
be s u c h t h a t
nk nk
< to
then there
0 E
such t h a t
nk
Suppose f i r s t t h a t
-1 p t o WnkwX
n
( t , u ) = tWnTVnu
wk E GV
and
"k
with
Pn
for
deg(WnTVn+pWnGVn
f o r each
p 2 < p1
t k -+ t o , u
= 0.
Hence,
and A - c l o s e d ,
+ pWnGVnu
WnTVnu
nl = n l ( p ) z no
whenever
vk E TV
d
.
i-(WnGVn,Bn,O)
a hornotopy
i s A-proper
pG
tWnf
n
We claim t h a t t h e r e e x i s t s a n
and
+
T
Since
t € [ 0,1]
there exists a
(O,po)
z 1 such t h a t
n
u 5 a B n ( 0 ,rf),
E
i-(
be f i x e d .
there e x i s t s an
By condition (2.5)
be f i x e d .
Y
1.
I
wk + 0
as
Then
k
-t
m
k some s u b s e q u e n c e
I(xI/ = r f .
V
+ x nk(i)unk(i)
A s i n Theorem 2 . 1 ,
this
leads
t o a contradiction with (2.6).
t o = 0.
Suppose now t h a t many
k,
we h a v e t h a t
xk E KV
any
1J v nk
Since
-1
= -btk T
for i n f i n i t e l y
f o r such
and f o r
k,
u nk nk
-1 (vkSxk) = ( W n vkSYk) = - p t k k
Since
o
tk f
-1 = -btk Wnkwk
\Iwkl( ((xkll +
i s K-quasibounded,
-1
= -btk -a
we g e t t h a t
as
k +
{vk]
(Wk'xk)
=
-. i s bounded and con-
MILOJEVI~
P.S.
266
-1
ness of
( / W v 1) = ptk llWnkwkl1 -+ m as k -+ m by the A-propernk G. This contradicts the boundedness of {v,] and there-
fore
#
sequently
to
0. Hence, ( 2 . 1 0 )
is valid.
Now, it follows from ( 2 . 1 0 )
= deg(kWnGVn,Bn,O) f E Tx
+ pGx
)Ik -+ 0
and
for each
is solvable in xk E n(O,rf)
dition (*) we obtain an
n
e
g(O,rf).
e
( 0 , ~ ~be )
such that
fixed, condition
X
T
theref ore
deg(WnkTVn
k
un nk k
be such that as
k -+
a,
+
WnkVk
holds for each
-
W
f,B
nk
"k'
0)
#
n
2
nk
and
0.
lkWnkwk = W x E X
-
f. Since ))Wnkvk Wnkfll -+ 0 nk such that f E T x by the strong
0
T.
(2.1) with
ing ( 2 . 5 ) - ( 2 . 6 )
H
k
u
there exists
A-closedness of Eq.
'
F o r each
decreasingly.
E B such that W f E W T V u + nk nk nk nk nk nk for each k . Let vk E T V u and wk E GV u nk nk nk nk
Hence, there exists a
+ pkWnkGV
and using con-
is strongly A-closed and let
kn k W GV nk nk
+
with
f € Tx.
such that
p k -+ 0
for
(2.10)
pk E ( O , k o )
Taking
f E Txk + bkGxk
Let us now suppose that pk
deg(WnTVn+lWnGV,-W,f,BnsO)
n l , and therefore, the equation
2
with x
that
T
A-proper o r strongly A-closed and satisfy-
or (2.8) has been earlier studied by the author
under other additional conditions on
T.
So,
i n [ 2 0 ] have announced
the following results (see also Note added i n proof). THEOREM 2 . 3 .
(a)
Eq.
(2.1)
T: X
Let
H(t,x)
closed homotopy on T(X) = Y
K = JG
2',
and
G
be bounded and
is feebly approximation-solvable f o r each
if, in addition,
(b)
3
[O,l]
= tTx
+ (1-t)Gx
f
in
Y
is a n A-proper and A-
x X.
if, in addition,
e r and A-closed homotopy at
0
HW(t,x)
on
= tTx
+ (IGX is a n A-prop-
[O,l]xX\B(O,R)
f o r some large
APPROXIMATION-SOLVABILITY
R
p E
and a l l
(0,p)
6 > 0,
f o r some
T
267
e i t h e r s a t i s f i e s con-
( * ) or i s s t r o n g l y A - c l o s e d a n d f o r e a c h
dition
f
deg(pWnGVn,Bn(O,r),O)
0
r > R
and p
e
(O,p),
r.
f o r each large
The f o l l o w i n g s p e c i a l c a s e i s u s e f u l i n a p p l i c a t i o n s
(C
20-221
)
COROLLARY 2 . 1 .
y
(p > 0 ,
2 0
for
If
u E Tx,
H(t,x)
X
and
resp.) v E
Kx
(Hp(t,x),
x
[O,l]
G
a r e A-proper
yG
and e i t h e r
a n d some
i n a n A-proper
t h e c o n c l u s i o n s of Theorem 2 . 3
(a)-(b)
c
> 0,
( u , v ) 2 -cllvli then
and A - c l o s e d homotopy on
R 2 Ro,
[O,l]XX\B(O,R),
and-A-closed f o r
i s bounded or
T
IlxII 2 R o
with
resp.)
( a t 0 on
+
T
resp.).
Therefore,
a r e v a l i d provided t h e o t h e r
i t s hypotheses hold. We n o t e t h a t i n t h e above r e s u l t s we h a v e n o t assumed ( 2 . 4 ) . D e t a i l e d p r o o f s of Theorem 2 . 3
and C o r o l l a r y 2 . 1 a n d t h e i r a p p l i c a -
t i o n s t o v a r i o u s s p e c i a l c l a s s e s of n o n l i n e a r m a p p i n g s a n d BVP f o r p a r t i a l d i f f e r e n t i a l e q u a t i o n s c a n be f o u n d i n t h e a u t h o r ' s p a p e r s
[21,22]
( c f . also [28]).
C o r o l l a r y 2 . 1 we s e e t h a t
A n a l y s i n g t h e p r o o f s of Theorem 2 . 3 a n d
(2.6)
c a n be r e p l a c e d by ( 2 . 9 ) .
We h a v e
a l s o proven t h e r e t h e f o l l o w i n g PROPOSITION 2 . 1 .
for
v E Knu
Mn: E n -+ F n .
and
(2.3),
Suppose t h a t 0 f u E En
f 0
(Mnu,v) ' 0
a n d some l i n e a r i s o m o r p h i s m
Then, f o r e a c h
deg(pWnGVn,Bn(O,r),O)
( 2 . 4 ) h o l d and t h a t
r
for
and l a r g e
n
p > 0,
large.
The a b o v e r e s u l t s a r e a p p l i c a b l e t o s t u d y i n g p e r t u r b e d equations
f
(2.11) with
F
such t h a t
E
Tx
+ Fx
P
268
( u , v ) - 2 -allvll
(2.12)
.s . M I L O J E V I ~ u E F x , v E Gx, IIxIl
for
T
i s K-coercive,
next r e s u l t ,
We r e c a l l t h a t
If
if
(x,}
in
Y.
then (2.12)
([28]).
T: X
Let
( u , v ) - b c ( ~ ~ x I I ) ~ \ vfIoI r
u E Tx,
such t h a t
r
2
c(r)
as
m
-b
- c ( ~ ~ x ~ /f )o r~ ~u vE ~ F~x ,
Suppose t h a t
G,
i s A-proper and A-closed
> 0.
8,
Then, i f
+
T
c o n d i t i o n s ( + ) and
(*),
F:
-+
X
2'
x E X\B(O,R)
Let
f E Y
be g i v e n .
By c o n d i t i o n
rf
and
y > 0
such t h a t
I ( u + v - t f ) )2 y
((~11=
rf
t E [O,l].
and
0, E ( p o , l )
6 E
Therefore, f o r each
t E [O,l] T
+
BF
w e have t h a t s a t i s f i e s (2.5) u E Tx,
over, f o r each
B E (61,1)
Since
( O , , 1) ,
we obtain t h a t
and t h e r e f o r e
T + pF
x E E(O,rf)
(u+OVtW)- 2
s a t i s f i e s (2.6)
such t h a t Since
F
exists a
8,
-t
1
such t h a t and
E
v
v E Fx,
/\xll = r f .
11 x(I
Fx,
= rf
E
on
B(O,rf)
such t h a t
+
Fx.
0
More-
(fj,,l).
-
C(IIXII)llWII
and
BC(IlXli)IIWII
> 0,
f o r each
E (Bl,l) Let
and i.e.
\lxll = rf
with
f E Tx + BFx.
f E Tx
e
f o r a given
xk E f 3 ( 0 , r f )
such t h a t
(T+F)(X) = Y .
v E Fx,
i s bounded, i t f o l l o w s f r o m c o n d i t i o n
x E X
satisfies
u E Tx,
for
f o r each
w E Gx,
B E ( ~ ~ ~Hence, 1 ) . by Theorem 2 . 2 , exists
w i t h some
= Ilu+v-tf-(l-B)v// t y / 2 ,
BB(O,rf)
v € Fx,
F
pG
(+), there e x i s t s an
for
u E Tx,
IIu+Bv-tfll on
+
+
BF
i s bounded, t h e r e e x i s t s a
F
( l - ~ l ) l l v l \ 5 y/2
such t h a t
T
is surjective, i.e.
PROOF. t R
p E (po,l)
and
R > 0.
+
T
Rf
-t
(u,v)- 2
and some
i s K-quasibounded and
T + F
i.e.
b e bounded a n d
p > 0
f
and some c:R+
a r e a s i n Theorem 2 . 2 and
f o r each
BF
x E X,
and
I< = J G ,
be K - c o e r c i v e ,
v E Gx,
Kn
and
K
y n E Axn
f o r some
v EGx,
m ,
-t
(+)
s a t i s f i e s condition
f
2'
-b
c a n b e weaken a s i n t h e
2'
-t
y n -+
i s bounded whenever
THEOREM 2 . 4
X
A:
R and some a > 0 .
b
Bk
E
f E Txk
there
(p,,l) +
be
BkFxk.
(*) t h a t t h e r e
2 69
APPROXIMATION-SOLVABILITY
We c o n t i n u e our e x p o s i t i o n by d e r i v i n g s o l v a b i l i t y r e s u l t s f o r Eq
(2.1)
pings.
Recall that X + 2
K:
f o r some
i n v o l v i n g v a r i o u s s p e c i a l c l a s s e s o f n o n l i n e a r map-
z E K(x-y)
X
A:
Y"
if
and some
ro
I
allx/I
for
u
E
Kx
2
c l l x - y ~ ~f ~ or
c = 0,
If
0.
W e s h a l l a l w a y s assume t h a t and a g i v e n scheme
i s s a i d t o be c - s t r o n g l y K-monotone
(AX-AY,~)
>
c
Y
-t
i s c a l l e d K-monotone.
A
= { X ~ , P , ; Y ~ , Q ~f ~o r
>
and some
0.
When
f o r x E Xn
QEKx c Kx
i s such t h a t
K
y E X,
x,
and
Y = X"
(Y = X,
s u c h mappings a r e c a l l e d c - s t r o n g l y monotone ( c - s t r o n g l y with
K = J,
t h e normalized d u a l i t y mapping, r e s p . ) .
m e a s u r e of n o n - c o m p a c t n e s s by
= inf(r > 0
x(D)
-t
5 kX(Q)
f o r each
BK(Y)
Q
c D
EXAMPLE 2 . 1 and
F:
X
-t
(a)
of a bounded s u b s e t
D C X
Q
I
D C
c
i t i s ball-condesing i f
D;
f 0.
The b a l l i s defined
([19,25])
Let
X + Y
A:
k-ball-contractive
with
F
of p a r t
(u.s.c.)
upper semicontinuous
6 = supllQnII.
k6 < c ,
w . r . t . a p r o j e c t i o n a l l y c o m p l e t e scheme
+
x(Q)
be c - s t r o n g l y K-monotone
i s A-proper
where
x(T(Q))<
We h a v e
6 = c = 1,
continuous with
accretive
u
densing i f
A
resp.)
n
x(Q)
and
BK(Y)
(b) ([24,28])
I
B ( x i , r ) , xi E X , n E N ) . A mapping i=l i s s a i d t o be k - b a l l c o n t r a c t i v e i f X(T(Q)) <
T: D c X
whenever
llull
(x,y),
Then
( a ) i s A-closed w . r . t . and
A
or b a l l - c o n A+F: X + B K ( Y )
To =
To
(Xn,Pn;Yn,s]. if
F
i s e i t h e r c o n t i n u o u s , or demi-
reflexive.
X
I n Example 2 . 1 we c o u l d h a v e assumed more g e n e r a l l y t h a t i s a-stable w.r-.t.
To.
By Theorem 2 . 3 i n [l9] a n d t h e a b o v e
a b s t r a c t r e s u l t s we o b t a i n t h e f o l l o w i n g e x t e n s i o n of Theorem in [
(a)
2.7
191 and C o r o l l a r y 1 i n [ 2 1 ] .
THEOREM 2 . 5 . Let
Let
T = A + F
is
A
and
F
b e a s i n Example 2 . 1 ( a ) - ( b ) .
s a t i s f y condition
following conditions holds:
( + ) and e i t h e r one of t h e
A
P
270
.s . M I L O J E V I ~
(2.13)
T
i s odd on
(2.14)
A
i s bounded and ( 2 . 9 )
(2.15)
2 (up.) 2 -al\xll f o r
a
some
f o r each
in
f
(*),
then
(b)
Let
f
then
T(X) = Y .
in
If
Y.
If
Y.
T
i s feebly approximation-solvable
k6 = c
+ Fx
and
T
s a t i s f i e s also condition
and ( 2 . 9 ) .
i s f e e b l y approximation-solvable
k6 = c
and
be K-quasibounded,
(u,.)
2
T
Then for
s a t i s f i e s also condition (*),
s a t i s f y c o n d i t i o n s ( + ) and ( * ) ,
-allxl12 f o r u E T x , z E Kx, I ( x ( (2 R and some a 7 0.
A + cK
i s j u s t K-monotone, t h e n
A
a n d , u s i n g Theorem
2.5 with
F
i s c - s t r o n g l y K-monotone
THEOREM 2 . 6 .
K = J
or
compact, A
Y = X*
-IY
and
K = I, T = A
and
+
F
F: X
with e i t h e r -t
BK(Y)
be
Y = X U.S.C.
and and
s a t i s f i e s c o n d i t i o n s ( + ) and ( * )
i s e i t h e r continuous o r demicontinuous with T
we o b t a i n
[ l 9 ] f o r s u c h mappings.
be K-monotone,
A: X
Suppose t h a t
Suppose t h a t T
Let
c > 0,
compact f o r e a c h
t h e f o l l o w i n g e x t e n s i o n o f Theorem 2 . 7 i n
or
and
T(X) = Y .
Then
and
R
and
(2.16)
If
2
7 0.
f E Ax
each
kb = c
IIx11
be K-quasibounded and s a t i s f y ( 2 . 5 )
T
Let
E Kx,
z
T(X) = Y .
the equation
(c)
holds;
u E Tx,
f E Ax + Fx
Then t h e e q u a t i o n
R > 0;
f o r some
X\B(O,R)
X
reflexive.
s a t i s f i e s e i t h e r one of c o n d i t i o n s ( 2 . 1 3 ) - ( 2 . 1 5 )
i s K-quasibounded
and s a t i s f i e s ( 2 . 1 6 ) .
Theorem 2 . 5 w i t h
F = 0
a r e s u l t of F , Browder [ 61. e n t i a l e q u a t i o n s , our p r o o f
Then
( T + F ) ( X ) = Y.
y i e l d s t h e f o l l o w i n g e x t e n s i o n of
Unlike h i s approach based on d i f f e r i s much s i m p l e r .
271
APPROXIMATION -SOLVABILITY
Let
COROLLARY 2.,?
be a n l - s p a c e
X
and
A:
+
X
i.e.
a c c r e t i v e and
X
-+
X = X**.
e i t h e r c o n t i n u o u s or d e m i c o n t i n u o u s w i t h m-accretive,
X
i s s u r j e c t ve € o r e a c h
A
A
Then ),
0.
7
The f o l l o w i n g r e s u l t i s u s e f u l i n a p p l i c a t i o n s
is
of Theorem
2.6 ( c f . c91). LEMMA 2 . 1
Let
be r e f l e x i v e ,
X
s t r i c t l y c o n v e x and condition
(*).
whenever
x
then
n
+ F
A
-
be m - a c c r e t i v e .
X -+ X
A:
have normal s t r u c t u r e ,
I f , i n addition, x
Axn -+ f
and
Then
and
X
satihfies
A
i s s t r o n g l y demiclosed,
A
then
Ax =
r,
and
X*
i.e.
i s compact,
F'
(*).
satisfies
L e t u s n o w d i s c u s s more g e n e r a l t h a n c - s t r o n g l y K-monotone mappings.
Let
Cb(D,Y)
d e n o t e t h e normed l i n e a r s p a c e w i t h t h e
supremum norm of a l l c o n t i n u o u s bounded f u n c t i o n s f r o m t h e t o p o l o g i c a l space
DEFINITION 2 . 2 a-stable for
([28]).
if
A mapping
U:
there exists a
-I B K ( Y )
x
XXX
ij
(Xt.1
-+
-+
BK(Y)
2
s
= (Xn,Pn;Yn,Qn}
such t h a t
Tx = U ( x , + ,
x E
(ii) F o r each
In particular,
U:
XXX
>
c
for some
x, 0
-t
i s a-stable
U(X,*
and e a c h l a r g e
T: X -+ B K ( Y )
BK(Y)
is
and
K:
&
U
> 0
As before,
and
GKx
X -+ ZY*
we assume t h a t C
into
Kx,
x
E
w.r.t.
To,
cJIT,Y) i.e.,
n
c - s t r o n g l y K-monotone
( i ) of D e f i n i t i o n 2 . 1 h o l d s and for e a c h l y K-monotone.
6
i s compact f r o m
D c X.
f o r e a c h bounded s u b s e t
and some
i s s a i d t o be
and
( i ) t h e mapp n g
exists a
T: X
Y.
a p r o j e c t i o n a l l y c o m p l e t e scheme
w.r.t.
(X,Y)
x E X,
i n t o t h e normed l i n e a r s p a c e
U
Xn.
such t h a t x E X,
there
Tx = U ( x , x ) ,
U(x,.)
)IuII < U//xll
if
for
i s c-strong-
u E Kx
It i s c l e a r t h a t s u c h map-
P.S. MILOJEVI~
272
of semi K-monotone m a p p i n g s .
f 25,281
To.
w.r.t.
p i n g s a r e semi a - s t a b l e
c = 0,
If
we h a v e t h e c l a s s
F o r s u c h mappings we h a v e p r o v e n i n
t h e f o l l o w i n g r e s u l t whose p a r t ( a ) a n s w e r s p o s i t i v e l y t h e
q u e s t i o n r a i s e d by Browder
[ 7 ] and e x t e n d s some of h i s r e s u l t s ( c f .
c 6,71) . THEOREM 2.7
(a)
To.
w.r.t.
A-proper
If
( + ) and i s e i t h e r i s a l s o A-closed
T: X + B K ( Y )
then
I f , a l s o , QnT i s i n j e c t i v e i n X n ,
U.S.C.
or d e m i c o n t i n u o u s w i t h
and t h e e q u a t i o n
s o l v a b l e f o r each
i s semi a - s t a b l e ,
f
Y.
in
If
f
E
is
T satisfies
Y = Y**,
then i t
i s f e e b l y approximation-.
Tx
i s continuous a - s t a b l e
T
T
and
s i n g l e v a l u e d , i t i s a n A - p r o p e r and A - c l o s e d homeomorphism. (b)
If
i s a s i n ( a ) and
T
contractive with T + F
kb
i s A-proper
< c
F: X + BK(Y)
is
or b a l l - c o n d e n s i n g if
and A - c l o s e d
Now, i n v i e w of Theorem
w.r.t.
and k - b a l l
U.S.C.
b = c = 1,
then
r o o
2.7 ( b ) , o u r g e n e r a l r e s u l t s y i e l d
(r 281 ) THEOREM 2 . 8
The c o n c l u s i o n s of Theorem 2 . 5 r e m a i n v a l i d i f we
(a)
assume i n i t t h a t (b)
that
A
i s s e m i c - s t r o n g l y K-monotone w i t h
The c o n c l u s i o n of Theorem 2 . 6
r e m a i n s v a l i d i f we assume i n i t
i s s e m i K-monotone.
A
REMARK 2 . 2
I n Theorem 2 . 8
k-ball-contractive
with
( a ) one c a n assume t h a t
kb < c
(i.e.
t i s f y i n g ( i ) of D e f i n i t i o n 2 . 1 and f o r each
x),
x
+ Y
F
F ( x ) = U(x,x)
u(x,.)
or semi b a l l - c o n d e n s i n g i f
We n o t e t h a t ( i ) of D e f i n i t i o n 2 . 1 h o l d s i f
U( * ,x):
K = JG.
i s semi with
U
sa-
i s k-ball-contractive
6 = c = 1 X
( c f . [ 281).
i s r e f l e x i v e and
i s completely continuous uniformly f o r
x
in a
bounded s e t , I n [ 311
,
Pohogaev s t u d i e d a c l a s s of A - p r o p e r mappings T:X
* X"
APPROXIMATION-SOLVABILITY
such t h a t
0: X
R
-t
c: R+
(Tx-TY, x - y )
c(llx-yII)
2
-
c(0) = 0
i s continuous,
-I R +
x,y E X ,
~(x-Y),
i s weakly u p p e r s e m i c o n t i n u o u s a t
0
-t
where
@(O) = 0
and
0
r
and
273
and
whenever c ( r )
0.
-t
A s l i g h t l y more g e n e r a l c l a s s i s g i v e n by
PROPOSITION 2 . 2
q(r)
constant
Let 0
7
(2.17) Then
=
and
T
i s of t y p e I
0,
Let
PROOF.
x
there exists
7
and 0
with
x),
r,
2
-
q(r)c(llx-yll)
2
(i.e., -t
Ilx-yll
whenever
xn
Q(x-Y).
-
x
and
l i m sup
and i s t h e r e f o r e A-proper w . r . t .
l i m sup (Txn,xn-x)
such t h a t
(/xn-x/I z
r
2
0.
If
xn
f x,
then
n.
f o r i n f i n i t e l y many
n
(2.17) holds f o r such n l s l e a d i n g t o a c o n t r a d i c t i o n .
Hence,
some
To.
or
n -x r
xn
r > 0,
be s u c h t h a t for e a c h
X*
x-Y)
(S+)
then
(xn,vn;xn’v;l *
-t
x,y
(TX-TY,
(Txn,xn-x)
ra
T: X
P a r t i a l d i f f e r e n t i a l e q u a t i o n s s t u d i e d r e c e n t l y by H e t z e r
[15],
i n a much more c o m p l i c a t e d way u s i n g g e n e r a l i z e d d e g r e e t h e o r y
o f Browder [ 6 ] , s a t i s f y ( 2 . 1 7 ) modulo a compact mapping. be a bounded r e g i o n .
L e t Q c Rn
We a r e i n t e r e s t e d i n a g e n e r a l i z e d s o l u t i o n
of
u E V
(2.18)
with Let
c
f E L2(Q),
sm = # ( a
> 0
and
I
where
lalsm].
m V c W2
om
i s a clo s ed subspace with
Suppose t h a t
aaB
E L ~ ( Q ) for
W2
la1 ,
c V.
l ~ 5:l
m,
P.
274
la1
For e a c h
(2.20)
5
(2.21)
E
1 E R,
.
MILOJEVI~
A,:
m,
dory c o n d t i o n s and and some
s
QxRSm
IAa(x,y)I
-t
R
6
L(yI
s a t i s f i e s t h e CarathBo-
a(.)
+
for
E
x
Q(a.e.),
L2(Q).
p: R'\(O]
There e x i s t s a f u n c t i o n
[O,c)
4
([15])
such t h a t
s -s for
x
E
Q(a.e.),
Iz-z'
I
2
r.
y
r > 0,
E Rsm-',
z,z'E
and
D e f i n e c o n t i n u o u s and bounded mappings
(N2u,v) =
and
C
(Aa(x,u,Du
l a I-
i s a w e l l known f a c t t h a t
A2
i n Hetzer [15]
t h a t , for e a c h
(N1~-N1vy u-v)
2
T(r)
-s(r)c(llu-vll)
,...,Dmu),Dav) and
+
A1
N1
i s A-proper
and A-closed
satisfies
- o(u-v)
ro.
veloped i n [ 24,26-28] In particular,
[27]
V
-t
It
I t was shown
E
w i t h IIu-vI/ 5 r ,
V
where q ( r ) = ( c 0 + 3 r ( s ( r ) ) ) / 4 ,
Since
T
@(u-v) =
i s applicable t o (2.18)
t h e r e s u l t of H e t z e r [ l 5 ]
+
A
5
+
A = A
s o l v a b i l i t y of
f o r a more g e n e r a l c a s e when
V
u , v € V.
for
u,v
( 2 . 1 7 ) and s o
holm of i n d e x z e r o , t h e t h e o r y of
(see
and
with
L2
a r e compact.
N2
r > 0,
w.r.t.
m-l
A1,A2,N1,N2:
= r / 4 1 ~ 1 ~ / ~c , a s i n P r o p o s i t i o n 2 . 1 and
Therefore,
R
Ax
A2
N1
+
Nx = f
de-
assuming ( 2 . 2 0 ) , ( 2 . 2 1 ) .
i s d e d u c i b l e by t h i s t h e o r y A = A*).
When
N(A)
= (01,
w e have: THEOREM 2 . 9 (a)
If
Suppose t h a t T
(2.19)-(2.21)
s a t i s f i e s condition
hold.
( + ) and e i t h e r
=
A,(X,-Y)
S
= -A,(x,y)
for
x
E
Q(a.e.)
and
y
E
N2
i s Fred-
A2
+
+
R r n , la1
5
m,
or
T
sa-
APPROXIMATION -SOLVABILITY
t i s f i e s (2.9),
then Eq.
If
(b) s i o n of
N(A)
= {O}
i s feeby approximation-solvable
(2.18)
V
the v a r i a t i o n a l sense i n
f o r each
and
27 5
f E L2
.
i s s u f f i c i e n t l y small, t h e conclu-
2
(a) holds.
(c)
N(A)
If
the value
c
= (03,
and
T
PROOF.
Part
Remalk 2.1,
p
i s sufficiently small,
s a t i s f i e s condition
a generalized solution
u
E
(*),
A1
+
Now
A1+N1
+ a1
N1
Part
s a t i s f i e s (2.17)
for
a. > 0 .
r > 0
Hence,
+ a1
{Xn,Pn]
for
Theorems 2 . 1 - 2 . 4
V
is
f o r each i n [28]
a. >
(or Co-
127,281.
and s t r o n g l y A-closed mappings d i s c u s s e d
We c o n l c u d e t h e s e c t i o n w i t h a c o u p l e of a l s o C o r o l l a r y 3.1N in [ S O ] ) .
more a p p l i c a t i o n s p r o v e n i n [ 2 8 ] ( c f .
x
=
ro
x**,
= {x,,P,},
))P,J) = 1,
s: x
-+
c(x)
a g e n e r a l i z e d c o n t r a c t i o n ( i n t h e s e n s e of B e l l u c e a n d K i r k ) a n d C:
v
X
C(X)
-+
E Jx,
u.s.c.,
/ / x / / 2- R ,
0
and C o r o l l a r y 2 . 1 a r e a p p l i c a b l e t o many
o t h e r c l a s s e s of A - p r o p e r
Let
T
and
n
r o l l a r y 1 i n [24] ) .
THEOREM 2.10
Let us prove
for e a c h
V\B(O,r)
To =
w.r.t.
Theorem 2 . 1 and
1231.
a n d t h e c o n c l u s i o n now f o l l o w s from Theorem 4.5.2
in detail in
(2.18) has
( b ) f o l l o w s from t h e g e n e r a l i z e d
i s monotone on
A-proper and A-closed
t a k e s on a l s o
V.
(a) follows f r o m a r e s u l t i n [30,31],
respectively.
p
then Eq.
f i r s t F r e d h o l m t h e o r e m for A - p r o p e r mappings i n (c).
in
compact a n d c
> 0.
( u , v ) 2 -cllx))* f o r
Then t h e e q u a t i o n
f e e b l y approximation-solvable
f o r each
p i n g s we h a v e
([ 281 ) :
THEOREM 2 . 1 1
Let
X
be r e f l e x i v e ,
T: X
e i t h e r p s e u d o monotone a n d d e m i c l o s e d ,
x
-
Sx
-
Cx
is
f E X.
2.4.
The p r o o f f o l l o w s f r o m Theorem
f E
u E Cx,
F o r monotone l i k e map-
+ 2’”
quasibounded and
or g e n e r a l i z e d p s e u d o mono-
276
MILOJEVIC
P.S.
t o n e , or q u a s i - m o n o t o n e ,
and
F: X
-+
such t h a t
2"'
Ta = [ X , V n ; X n x, V z }
s t r o n g l y A-closed w . r . t .
T t F
(e.g.,
c o u l d be
F
c o m p l e t e l y c o n t i n u o u s , or, f o r t h e f i r s t two t y p e s o f
(u,x)
bounded and g e n e r a l i z e d pseudo monotone w i t h
f o r some c o n t i n u o u s Then, i f
T
+
DEFINITION 2 . 3 R > 0
v
E Ty,
and
where
c(R,r) 2 0
t
-+
0'
~
c : R + -+ R',
x
5
11 - 1 1 '
COROLLARY 2 . 2
Let
yz
~R , ~ ( u , - v~, x - y~) i s a norm on
i s continuous i n
f o r fixed
T
and s a t i s f y ( 2 . 5 ) - ( 2 . 9 ) .
( T + F ) ( X ) = X".
R
-~ c ( R~, l / x - y l l ' )
for
u
E
\l*ll
compact r e l a t i v e t o
X
r
and
and
c(R,tr)/t
Tx,
-+
0
and
as
r.
and
R
-c(IIxII)II xi1
i s of semibounded v a r i a t i o n i f f o r
T: X -+ BK(X") ~
quasi-
T,
or t h e sum of two s u c h m a p p i n g s ) .
satisfies (2.5)-(2.9),
F
3
is
be h e m i c o n t i n u o u s , Then
of semibounded v a r i a t i o n
T(X) = X".
Theorem 2 . 1 1 f o l l o w s f r o m Theorem 2 . 2 and e x t e n d s t h e known s u r -
It i n c l u d e s t h e s u r -
j e c t i v i t y r e s u l t s f o r monotone l i k e n x p p i n g s . j e c t i v i t y r e s u l t s of Brezis [ Z ]
f o r bounded c o e r c i v e pseudo mono-
t o n e mappings, of Browder and H e s s [ S ]
f o r g e n e r a l i z e d p s e u d o mono-
t o n e mappings, of H e s s [ 1 4 1 , C a l v e r t and Webb [ 101 and F i t z p a t r i c k
[ 131 f o r quasimonotone mappings, a n d of W i l l e [ 341 and Browder [ 51 f o r maximal monotone and bounded g e n e r a l i z e d pseudo monotone mappings t h a t s a t i s f y (2.8)
and
(Tx,x)
z -1IxII z R ,
l a r y 2 . 2 f o l l o w s from Theorem 2 . 1 1 s i n c e
respectively. T
i s pseudo monotone,
It extends the
d e m i c l o s e d and q u a s i b o u n d e d by a r e s u l t i n [ 2 9 ] . e a r l i e r r e s u l t s of Browder [ b ]
and D u b i n s k i i [ 121.
r e s u l t s a r e v a l i d f o r mappings between
X
Corol-
and
Y
B o t h of t h e s e under s u i t a b l e
r e s t r i c t i o n s on t h e s p a c e s . F i n a l l y w e s h a l l l o o k a t i n t e r t w i n e d p e r t u r b a t i o n s of mappings of semibounded v a r i a t i o n . Banach s p a c e s ,
X
Let
X
and
Xo
be s e p a r a b l e r e f l e x i v e
c o n t i n u o u s l y a n d d e n s i l y embedded i n
X
and l e t
-
APPROXIMATI ON SOLVABI LITY
I: X
the injection
xo
of
-t
( C 331 ) .
T: X
-t
X*
mappin5 i f
Tx = U ( x , x )
such t h a t
for
and
(i) F o r each
y E X,
U(y
(ii) F o r each
x E X,
U(.
R 7 0
(iii) F o r each
and
c : R+XR+
where
r > 0,
each
R
-t
R+
>
0.
x
+
X*
i s c o m p l e t e ~ yc o n t i n u o u s ;
x): X
-+
X"
i s hemicontinuous;
a ) :
IIXII
*
,x-Y)
(U(X,X)-U(Y,X
2
dition
(*),
COROLLARY 2 . 3
holds,
l i m c(r,tR)/t t+0+
(see
= 0
for
[ 3 3 ] ) and s a t i s f y con-
gives the following extension of t h e r e s u l t
.
C33l
T
R,
-C(R,Ilx-Yllo)
i s c o n t i n u o u s and
Theorem 2 . 2
*
IIYII
R,
S i n c e s u c h m a p p i n g s a r e p s e u d o monotone
if
the n o r m
i s c a l l e d a G:rding
X X X -+ X*
U:
t h e r e e x i s t s a mapping
Oden
1/'110
D e n o t e by
.
DEFINITION 2 . 4
x E X
he compact.
Xo
277
Let
T
be a G i r d i n g mappings and
(2.5) hold.
i s q u a s i b o u n d e d and e i t h e r o n e o f c o n d i t i o n s ( 2 . 6 )
Then,
and ( 2 . 9 )
= X*.
T(X)
We s h a l l now a p p l y C o r o l l a r y 2 . 3 t o f i n d i n g a g e n e r a l i z e d s o l u t i o n u E V
of
C
(2.22)
( - l ) l a lD a ( x , u , D u
,...,D m u )
= f,
f
E Lq(Q),
la I s m where
Q
c Rn
i s - a hounded domain w i t h t h e smooth b o u n d a r y a n d
i s c l o s e d s u b s p a c e of
with
WF(Q)
im C V P
and
p E
V
(-1,m).
Assume (2.23)
F o r each
la1
5
m,
Aa:
QXRSm
c o n d i t i o n s and t h e r e e x i s t that
-t
R
K > 0
s a t i s f i e s t h e CarathAodory and
k ( x ) E Lq(Q)
such
P.
278
IA,(x,y)l
s
. MILOJ-EVI~ +
S
k(x))
x E Q a.e.,
for
A s b e f o r e , a g e n e r a l i z e d form a s s o c i a t e d w i t h ( 2 . 2 2 ) T: V + V".
bounded and c o n t i n u o u s mappings
(wr,v) = (Qfvdx
that
s o l u t i o n s of
TU = w f
Y
D u = { (Dau)
induces a
E V"
wf
be such
Then f i n d i n g g e n e r a l i z e d
i s equivalent t o solving the operator equation
(2.22)
(2.24) Let
v E V.
f o r each
Let
y E RSm.
1
)a1 g m - 1 1 ,
,
U E V .
W e n e e d r e q u i r e t h e f o l l o w i n g con-
ditions. (2.25) c:
Let
R+XR+
Rf
+
(2.26)
v E V,
R > 0
i n e q u a l i t y i n (2.25)
a n d for e a c h
.$
0
if
c,(r)
v E V,
R > 0
I I w ~ ~ S ~ R, ~ t h e
s R,
and
c1
(2.27)
and ( 2 . 2 6 ) ,
lyil
+
lzil
each S
R,
(2.25)
The f o l l o w i n g a l g e b r a i c c o n d i t i o n s i m p l y
respectively.
There a r e c o n s t a n t
y E Rsm-l,
integral
0.
3
Some a l g e b r a i c c o n d i t i o n s t h a t i m p l y a s t r o n g e r v e r s i o n o f
(2.25)
and
b e a s i n D e f i n i t i o n 2 . 4 and s u p p o s e t h a t
holds with
c a n be f o u n d i n [ 1 2 1 .
+ 0,
k S m-1
we h a v e f o r some
c : R + x R + + R+
Let
f o r each
R
I
R
1"
i s weakly u p p e r - s e m i c o n t i n u o u s
c(R,*)
= 0 f o r each
Ilwllm,p
R,
b e c o n t i n u o u s and
be s u c h t h a t
a t 0 and c ( R , O ) IIullmrp
R+ + R+
cl:
R
>
0
and
i = 1,2,
c1
>
0,
(yi,zi) and
c
5:
0
s u c h t h a t for e a c h
E Rsm-l
x E Q
a.e.
x R
s -s
we h a v e
m-l
with
APPROXIMATION-SOLVABI LITY
(2.28)
y E R Sm-l ,
F o r each
z1 z2
279
Rsm-sm-l
x E Q
and
a.e.
we have
c
THEOREM 2.11
-
[Aa(x,y,zl)
la I =m
Aa(x,y,z2)](z1-z2) a a
Let (2.23) hold and
T
0.
2
satisfy condition(2.5).
Sup-
pose that either one of the following conditions holds:
T
(2.29)
is odd on
= -Aa(x,I)
la1
V\B(O,r) for
for some
E Q
x
r
> 0, i.e.
I
15
a.e., and all
Aa(x,-5)
=
large and
m.
(2.30)
h E V"
For each Tu
where
J: V
(a)
-t
V"
#
XJu
there exists an for
rh > 0
X <
u E aB(O,rh),
such that 0,
is the normalized duality mapping.
Then
If (2.25) holds, the generalized boundary value problem
(2.22) is feebly approximation-solvable in
V
for each
f E Lq.
If (2.26) holds, (2.22) has a generalized solution in
(b)
for each PROOF.
f E Lq
.
Define the mappings
(u(u,v),w)
V
=
c
U: VxV + V
and
C: V -+ V"
Aa(x,DYv,Dm~)Da~dx,
for
by
u,v,w E V,
la(=m (cu,w) =
Then
C
Aa(x,D Y u,Dmu)Daw
dx,
for
is known to be completely continuous and
is A-proper w.r.t.
an injective scheme
u,w E V.
Tu = U(u,u) + Cu
*
TI = (Xn,Vn;Xn,Vz] for (V,Vy)
P.
280
if
(2.25)
since
h o l d s by Example 1 . 4 . 5
i n [28]
i s of t y p e
(2.26) h o l d s , then
T
+
= U(u,v)
If
(S+).
i s a G z r d i n g mapping
Cv
rl
A-closed w . r . t .
=
U1(u,v)
and t h e r e f o r e
T
(*).
in [27])
1.4
Example
i s strongly
Consequently,
f r o m Theorem 2 . 1 and Remark 2 . 1 a n d Theorem
r e s p e c t i v e l y when ( 2 . 3 0 )
results in
(i.e.,
and s a t i s f i e s c o n d i t i o n
the conclusions follow 2.2,
.
s MILOJEVIC:
[31,30] when
h o l d s , and from t h e c o r r e s p o n d i n g
0
i s odd.
T
Theorem 2 . 1 1 e x t e n d s t h e e a r l i e r r e s u l t s of Pohozaev " j l ] , Browder
[4],
Dubinskii [ 1 2 ] ,
(S+)
a n d of s e m i -
Bondary v a l u e p r o b l e m s s a t i s f y i n g ( 2 . 2 7 )
bounded v a r i a t i o n . c = 0
i n v o l v i n g mappings of t y p e
with
or ( 2 . 2 8 ) h a v e b e e n a l s o s t u d i e d e a r l i e r by t h e a u t h o r i n (cf. also [29]).
[21,26,28]
3. SOLVABILITY O F SEMILINEAR EQUATIONS AT RESONANCE
T h r o u g h o u t t h e s e c t i o n we s h a l l a l w a y s assume t h a t H i l b e r t s p a c e which c o n t a i n s t h e Banach s p a c e space.
We s h a l l s t u d y t h e ( a p p r o x i m a t i o n - )
is a
H
a s a v e c t o r sub-
X
s o l v a b i l i t y of e q u a t i o n s
of t h e f o r m
(3.1) where ping
AX
A:
D(A)
+
NX = f ,
V c X -+ H
3
i s closed with
R(A)
Xo = N(A) R(A)
n o n l i n e a r mapping of c e r t a i n t y p e . complement i n and s i n c e
H
Ail:
graph theorem.
x E R(A).
a
Then
Xo
of
R(A)
-t
V
Let
c
>
C
f
E H)
i s a l i n e a r d e n s i l y d e f i n e d c l o s e d map-
such t h a t i t s n u l l space
t h e range
(x E D(A),
H.
Let
= X
h a s f i n i t e dimension, l.
Here A1
= A
and
N:
is a
V -+ H
denotes t h e orthogonal
X i
restricted t o
V
n
R(A)
i s c l o s e d , i t i s c o n t i n u o u s by t h e c l o s e d 0
be s u c h t h a t
llA~l~lL l
( A x , x ) 2 -IIAxl(llxll 2 - ~ " l l A x ( ( ~ for
b e t h e supremum of a l l s u c h
C.
Then
a E
[O,m]
1 ; IIxII
for
x E V. and
Let
-
A P P R O X I M A T I O N S O LVAB ILITY
(3.1) w i t h
Equation R(A)
A;':
i s compact
R(A)
-+
N i r e n b e r g [ 31
,
X = H
and
a s above and s u c h t h a t
A
has
been s t u d i e d by B r k z i s and
f 11
B e r e s t y c k i and de F i g u e i r e d o
u n d e r v a r i o u s c o n d i t i o n s on t h e p e r t u r b a t i o n b i b l i o g r a p h y on t h e s e p r o b l e m s ) .
s t r o n g l y A-closed.
and many o t h e r s ( c f . [1,3] for t h e
N
Our r e s e a r c h h a s f e e n m o t i v a t e d
( 3 . 1 ) such t h a t
by [ 13 and d e a l s w i t h E q .
281
+
A
N
i s A-proper
or
U s i n g t h e d e g r e e t h e o r y f o r m u l t i v a l u e d map-
p i n g s i n s t e a d of t h e B r o u w e r f s d e g r e e , we s e e t h a t t h e r e s u l t s of t h i s section are also v a l i d
f o r m u l t i v a l u e d n o n l i n e a r i t i e s of
the
same t y p e . We b e g i n w i t h t h e f o l l o w i n g r e s u l t proven i n [ l ] w i t h and
R ( A ) -+ R ( A )
A;':
compact.
s e e s t h a t t h e compactness of
X = H
However, a n a l y s i n g i t s p r o o f one
i s n o t needed and we g i v e i t s
A;'
proof for t h e s a k e o f c o m p l e t e n e s s . LEMMA A;'
+
(Ai'x
3.1
Let
1
+ a-
x
> 0).
( A X ~ , X ~ =)
+
-a
2
+
x
E
and some
i.e.
and
(3.3) holds. x1 E R ( A ) . x1
f
setting
Then
= 0,
or
t h e con-
and t h e n ( 3 . 3 ) becomes
0
u =
Ax1
x = x o + x1
x1 = 0 ,
If
AX^,
we g e t
The s t r o n g m o n o t o n i c i t y of a-lu
R(A)
H.
in
Suppose t h a t
= 0. A;u '
for e a c h
a-lI)xIl
l l ~ ~ ~, l or, l
a-'u,u)
implies t h a t
2
+
be such t h a t
-1
and
(0,m)
be s t r o n g l y monotone ( i . e . ,
xo 6 N ( A )
clusion follows.
a E
be such t h a t
I f e q u a l i t y holds i n (3.2),
x E- V
u n i q u e l y , where
1 (A; u
(A;'
H
-+
N ( A ) CB N ( A + a I )
Let
PROOF.
cII
2
X,X)
c
E
V c X
R ( A ) -+ R ( A )
U-%:
constant
then
A:
+
a x l = 0.
A;'
+ 0
U'lI
P.
282
s. MILOJEVIC
Lemma 3.1 i n c l u d e s t h e c a s e when
REMARK 3 . 1 selfadjoint.
Assuming a d d i t i o n a l y i n Lemma
i s compact t h e n a s i n r l ]
X = H
and
A
3.1 t h a t
AY1:
R(A)
-a
one o b t a i n s t h a t
is -t
R(A)
i s a n e i g e n v a l u e of
A. Introduce i n
11 ' \ I O .
a new norm
V
Then t h r o u g h o u t t h e
s e c t i o n we s h a l l assume t h a t t h e n o n l i n e a r mapping quasibounded,
N: V
-I
H
is
i.e.
(3.4) We s a y t h a t
i t has the p r o p e r t i e s discussed
has Property I i f
A
a t t h e b e g i n i n g of t h e s e c t i o n and i f Let
ra
= (Xn,Vn;Yn,Qn)
ped w i t h t h e norm
3.1
THECREM
Let
II.IIO. A:
G
a(u,v)
Whenever
IIxnllo
i s compact.
and
N(A)
(V,H)
with
equip-
V
We a r e r e a d y now t o p r o v e v a r i o u s s u r -
(3.1).
V c X
s t r o n g l y monotone,
( i i i ) (2w,v)
onto
a n a d m i s s i b l e scheme f o r
j e c t i v i t y r e s u l t s f o r Eq.
+ a1
H
b e t h e o r t h o g o n a l p r o j e c t i o n of
Q
R ( A ) -+ R ( A )
A;':
-t
H
O u r first result is
have P r o p e r t y I ,
and
C,N:
V
-t
H
0 < a < m;
A;'
+
quasibounded and such t h a t
i n ( i )and ( i i )
a n d one of t h e i n e q u a l i t i e s
is strict.
(3.6)
uo
E
N(A),
-t
m ,
> o
A-closed
w.r.t.
xn
uo
u1
H
in
Nxn/\\xnllo -L v ,
(v,ul)
< a\\ul\12 i f
+
+
with
then u1
#
v
0
#
uul;
and
u1 = 0 .
if
Suppose t h a t
mr0++
and
u1 E N ( A + d I )
i n p a r t i c u l a r , t h i s i s s o when
(v,uo)
=
Un
A
and
ra
H ( t , x ) = Ax for
(V,H)
(1-t)Nx
tCx
and f o r each l a r g e
are A - p r o p e r and R,
-
283
APPROXIMATION SOLVABILITY
deg(QnH1,B(O,R) r l X n , O )
#
0
f o r each l a r g e
f e e b l y approximation-solvable PROOF.
f E H
Let
IIx[lo < R
that
f o r each
be f i x e d .
t E [O,l].
I f not,
such t h a t
t n -+ t o
Then t h e r e e x i s t s a n
/Ixn/(
Ex,]
R = R(f)
C V
such
and
and t n E [ O , l ]
and
m
-+
(3.1) i s
x E V
for some
t h e n t h e r e would e x i s t
,
Then E q .
f E H.
H(t,x) = (1-t)f
whenever
n.
0
(3.7)
H ( t n , x n ) = (1-tn)f
Set
un = xn/l1Xnllo.
and
uln E R ( A ) ,
u n = uon
Since
( 3 . 7 ) by
dividing
NXn + (l-tn)
AUln
f o r each
+
uln
IIxnlI
u n i q u e l y w i t h uon E N ( A ) we o b t a i n
0
wiC xn
+
tn
n o
n.
f
(l-tn)
1 3=
0,
or
(3.8)
+
(l-tn)Al
-1 f (I-Q)
= 0
n
(3.9)
Since
{Nxn/llxnll
t h e q u a s i b o u n d e d n e s s of
and
N
{Cx,/IIx,II
and
C,
v e r g e weakly t o
v
t i n u i t y of
we o b t a i n f r o m ( 3 . 8 )
u
In
-+
u1
AY1 in
H,
and
w,
u1 E R ( A ) ,
(3.10)
a r e bounded i n
H
by
we may assume t h a t t h e y c o n -
respectively,
and by t h e c o m p l e t e c o n -
passing t o the l i m i t t h a t
and
u1 + ( 1 - t o ) A y 1 ( I - Q ) v
or
o]
+
t o A ; 1( 1 - Q ) w
= 0,
P,
284
dim N ( A ) <
Since
= u o + u1
+
(1-to)Qv
o
s
Au +
(3.5)-(iii),
+ au)
to
If
#
u
E
2
(l-to)
5
to = 0,
N(A+aI) C R(A),
+ t
(l-to)V
v a l i d and l e t
3.1.
by Lemma
E
>
Now, s i n c e
2
+ to/lw/I
2
N(A+aI)
+ au) = 0 ,
-
Au = Aul = - a u l
and
(3.6).
0
be a such one.
N
and
(1-t)gfll 2 y
and a
for
Y >
( 3 . 5 ) - ( i ) and
and t h e r e f o r e and by ( 3 . 1 1 )
= -v
Therefore, our claim i s
a r e bounded on
C
no 2 1
t h a t t h e r e e x i s t an
-
u =
ato(w,u).
+ u1
u = u
Since
aB(0,R)
and
A
and
it i s easy t o see
t,
a r e A-proper and A-closed f o r each
II&,H(t,x)
-t
= 0.
W
(Au,Au
i n contraction t o
R
un
( 3 . 9 ) we o b t a i n
rIJvll 2 - a ( u , v ) I
we g e t t h a t
u1
v = uu1
H(t,x)
and s o
i t f o l l o w s f r o m ( 3 . 2 ) and (3.11) t h a t
N(A) @ N(A+aI)
Hence,
u
t h i s l e a d s t o a c o n t r a d i c t i o n i n v i e w of
0,
(ii). I f
-t
it follows t h a t
a n d by ( 3 . 1 0 )
toQw = 0 ,
(AU,AU
u on
Passing t o the l i m i t i n
(3.11) Using
. MILOJEVI~
w e may assume t h a t
m ,
H.
in
s
such t h a t f o r each n 2 no
0
x E aB(O,R)
n
Xn,
t E [O,l].
Con-
s e q u e n t l y , b y t h e homotopy t h e o r e m f o r t h e B r o u w e r l s d e g r e e we n 2 n
o b t a i n t h a t f o r each
Hence, t h e r e e x i s t s a that
&,Axn -+ x
x
+
and
xn E B ( 0 , R )
S N x n = Q,f, Ax
+
n
Xn
f o r each
and t h e r e f o r e ,
n 2 no
such
some s u b s e q u e n c e
Nx = f .
nk The f o l l o w i n g s p e c i a l c a s e i s s u i t a b l e i n many a p p l i c a t i o n s . COROLLARY
<
m,
A?
satisfy
3.1
+ a1
Let
(V,ll-I/)o
be c o m p a c t l y embedded i n
s t r o n g l y monotone a n d
( 3 . 6 ) and
N : V -+ H
H,
0
< a <
q u a s i b o u n d e d and
285
APPROXIMATI ON -SOLVABILITY
(3.12)
(V,H)
+
A
Suppose t h a t
tN
i s A-proper
t E [O,l].
for each
solvable f o r each
in
f
V c X
A:
Since
H
-D
in
(3.1)
(V
n
R(A),
a
C = - I.
A;':
AY1:
R(A)
(3.12),
Since
A
+ (1-t)N
t E [O,l]
f o r each
f r o m Theorem
+ tC
i s compact.
II
-+
Set
i t i s easy t o see t h a t (3.5) i s compact,
C
and A-closed w . r . t .
i s A-proper
and t h e c o n c l u s i o n of
3.1.
n
R(A)
A;':
c,
i s c o m p a c t l y embedded
V
R(A)
-t
f o r some
holds with t h e s t r i c t i n e q u a l i t y i n ( i i ) . Since we s e e t h a t
for
i s compact.
-+ R ( A )
R(A)
IIx/l s cIIxIIo
i s continuous.
T h e n , i n v i e w of
2
ra
€I.
i s c l o s e d and
i t follows that
H,
in
11
and
i s f e e b l y approximation-
( V , ~ ~ - ~-+~ Ho ) i s a l s o c l o s e d and t h e r e f o r e
A:
H
and A - c l o s e d w . r . t .
Then E q .
We s h a l l f i r s t show t h a t
PROOF.
-+
xn
11 xnll
Whenever
Ta
the corollary follows
9
When a Landesman-Lazer
t y p e c o n d i t i o n h o l d s i n s t e a d of
( 3 . 6 ) , we
have
THEOREM 3.2
w
f
0
and
(3.13)
Suppose t h a t a l l c o n d i t i o n s of Theorem 3 . 1 h o l d w i t h
( 3 . 6 ) r e p l a c e d by
Whenever
IIxnl/
-+
m,
un =
-1xn
-+ u o
+ u1
in
H
with
0
uo E N ( A ) , ( v , u l ) < olIu1112 some
f
in
H,
Then t h e e q u a t i o n
PROOF.
u1 E N ( A + a I )
if
u1 f 0 ,
if
u1 = 0.
Ax
A s i n Theorem
+ Nx = f
and
and,
Nxn/llxn/l0-
l i m inf
v,
then
(Nxn,uo) 7 ( f , u o )
for
i s f e e b l y approximation-solvable.
3 . 1 i t s u f f i c e s t o show t h a t t h e r e e x i s t s a n
286
R
MILOJEVI~
P.S.
>
)IxIIo < R
such t h a t
0
= (1-t)f
+
un = uon
uln
Then, a s i n Theorem
= 0.
t
tn
= (1-tn)f
-I
to
and
x
3 . 1 we s e e t h a t
n
n.
(Axn,uo) = 0.
COROLLARY 3.2
on
+ a1
(V,(I*llo)
in
H
H(tn,xn) =
,Uo)
,
R
>
0
r7
exists.
i s compactly embedded i n
s t r o n g l y monotone and
ra
A-closed
E R(A).
3.1, w e o b t a i n
A-proper and A-closed w . r . t .
I n c a s e when
In
uln -+ 0
T h e r e f o r e , such a n
Suppose t h a t
f
u
and
and
Therefore
Suppose t h a t
A;"
=-
,
-+ u
and s a t i s f i e s ( 3 . 1 2 ) - ( 3 . 1 3 ) .
Nx
un = xn/lIxnIlo
Set
uon E N ( A )
u
-+
I/xnll 0
+ t n ( c x n , u o ) = ( 1-tn)( f
A s i n t h e c a s e of C o r o l l a r y
+
with
u o , we o b t a i n for e a c h
i n contradiction t o (3.13).
Ax
E V
Taking t h e i n n e r p r o d u c t of t h e e q u a t i o n
with
a E (o,m),
+ ( 1 - t ) N x + tCx =
Ax
I
Again, a r g u i n g by c o n t r a d i c t i o n ,
uniquely with
( l - t n )( N X n , U o ) since
H(t,x)
H(tn,xn) = ( l - t n ) f f o r each
such t h a t
and
.
t E [O,l]
for some
suppose t h a t t h e r e e x i s t
write
whenever
for
V + H
N: A
(V,H).
+
tN,
H,
quasibounded
t E [O,l], i s
Then t h e e q u a t i o n
i s f e e b l y aporxirnation-solvable.
Ho = A
+
N
i s n o t A-proper b u t j u s t s t r o n g l y
i n s t e a d , we have t h e f o l l o w i n g e x t e n s i o n s of t h e above
results.
THEOREM 3.3
Let
and A-closed w . r . t .
A
and
ra
H ( t , x ) = Ax for
(V,H)
+ ( 1 - t ) N x + tCx for e a c h
be A-proper
t E (0,1] and A
+
N
APPROXIMATION -SOLVABI LITY
N
also that
and
Suppose
Then
= H.
(a)
If a l l o t h e r c o n d i t i o n s of Theorem 3 . 1 h o l d ,
(b)
If a l l o t h e r c o n d i t i o n s of Theorem 3 . 2 h o l d , t h e e q u a t i o n
PROOF.
Let
f
3 . 1 and 3 . 2 ,
E E
Let
on
E
aB(0,R)
n
B(0,R)
ck E (0,l)
-
ek(Nxk
R = R(f) > 0
Since
condition
such t h a t
i s A-proper
H(t,*)
t E (0,1], t h e r e e x i s t s an
f (l-t)Qnf
for
x
such t h a t
Ax
c
be such t h a t
Cxk) + f
E
+
(l-e)Nxe
ek + 0
n
Q(O,R)
el < e2
whenever
H(ek,xk) = ( l - a k ) f .
such t h a t
Then, a s i n t h e p r o o f s of Theorems
and A-closed
n(c) 2 1
such
n(c)
2
2 n(e2)
"(El)
E
be g i v e n .
f o r each
QnH(t,x) and
H
be f i x e d .
(0,l)
R(A+N)
i s solvable.
t h e r e e x i s t s an
t h a t for e a c h
+
(*).
and s a t i s f y c o n d i t i o n
a r e bounded.
C
Ax + Nx = f
x E
Ta
w.r.t.
be s t r o n g l y A-closed
287
.
Therefore,
+
there e x i s t s a
E
d e c r e a s i n g l y and
Then
,1]
= (1-e)f.
cCx
xk
Let
E
B(0,R)
Axk + Nxk = ( l - e k ) f
by t h e boundedness of
(*) there e x i s t s a
t E
Xn,
x E G(O,R)
N
and
+
F i n a l l y , by
C.
Ax + Nx = f .
such t h a t
0 COROLLARY 3 . 3 0 <
a <
m ,
Let
A;'
+ a1
quasibounded and (V,H)
(V,l/*llo)
f o r each
A-closed w . r . t .
A
+
s t r o n g l y monotone,
tN
t E [O,l).
ra
be compactly embedded i n N:
V
+
A
and s a t i s f i e s c o n d i t i o n
( 3 . 6 ) and ( 3 . 1 2 ) h o l d ,
R(A+N)
+
bounded and
Ta
w.r.t.
A-proper and A-closed Suppose t h a t
H
H,
for
i s strongly
N
(*).
Then
= H.
(a)
If
(b)
If ( 3 . 1 2 ) and (3.13) h o l d , t h e e q u a t i o n
Ax
+
Nx = f
is
solvable.
I n our
l a s t r e s u l t we show t h a t t h e A-properness
r e q u i r e m e n t of
P.
288
H(t,x)
s . MILOJEVIE
r
c a n b e r e l a x e d i f a scheme Pnx + x
j e c t i o n a l l y complete ( i . e . and
y E H
for
y E H
,
= Ax
%Ax
=
ro
= (Xn,Pn;Yn,Qn)
Qny + y
and x E Xn,
f o r each
i s pro-
x E V
f o r each
n z 1,
and
+
Q y:
y
Such schemes h a v e b e e n e x t e n s i v e l y u s e d i n our works
[24,26-28]
and when
has a n i c e approximation p r o p er ty such a
X
scheme c a n b e a l w a y s c o n s t r u c t e d u s i n g t h e p r o p e r t i e s of
X
example, suppose t h a t
i s such t h a t
( i ) There a r e f i n i t e dimensional subspaces that
Xo
I
For
A.
N ( A ) c X1 C X 2
...
C
of
Xn
X
such
with the i n c l u s i o n s being
proper.
(ii) There a r e p r o j e c t i o n s P x n
+
x
and
D e f i n e a l i n e a r mapping y E H
each
yo E N ( A ) , Q y n l
y1 E R ( A ) ,
To
.
H + H,
Yn = R(&,)
n z
y = yo
(b)
If
with
%yo = y o
and
Let
we have w i t h
A+N:
V + H
X = (V,ll
*llo),
b e A - p r o p e r and A - c l o s e d w . r . t . f
E H
a n d j u s t for a g i v e n
(3.13) and a l l o t h e r c o n d i t i o n s of Theorem 3 . 2 h o l d , A+N:
c o n c l u s i o n s of
V
+ H
i s s t r o n g l y A-closed w . r . t .
ro,
then the
( a ) hold e x i s t e n c i a l l y .
The p r o o f f E H,
y1
(3.1) i s f e e b l y approximation-solvable f o r each
Then E q .
f o r which
+
since
Then we h a v e t h a t
i f a l l o t h e r c o n d i t i o n s o f Theorem 3 . l h o l d , f
as follows:
.
Now, w i t h o u t r e q u i r i n g t h e A - p r o p e r n e s s
t,
f o r each
3.4 ( a )
.
x E D(A).
f o r each
n = 0,1,2,...
for
1,
such t h a t
s o c o n s t r u c t e d i s p r o j e c t i o n a l l y complete and x E Xn.
for
H(t,x)
THEOREM
C D(A)
i t suffices t o define
Set
To = (Xn,Pn;Yn,%)
of
%:
APnx + Ax
and
Pn(D(A))
Xn
onto
X
has t h e unique r e p r e s e n t a t i o n
-1 = APnAl y l .
&,Ax = A x
of
x E X
f o r each
( i i i ) popn = PnPo
Pn
of Theorem
3 . 4 c o n s i s t s i n showing, f o r a given R
t h a t t h e r e e x i s t an
>
0
and
no z 1 s u c h t h a t
( ( ~ ~ 1<1 0
whenever
QnH(t,xn)
E
%Axn
+
(1-t)Qn
Nxn
+
tQnCx,
= (1-t)Qnf
for
R
APPROXIMATION-SOLVABILITY
t E [O,l]
some
,
xn E Xn
and
with
n z n
289
.
T o t h a t end one
To
a r g u e s by c o n t r a d i c t i o n u s i n g t h e p r o p e r t i e s o f s i m i l a r t o those i n t h e previous r e s u l t s .
and arguments
W e omit t h e d e t a i l s .
Let u s now look a t some a p p l i c a t i o n s o f t h e a b s t r a c t r e s u l t s
t o t h e s o l v a b i l i t y of
(3.15) where
+
Au
Q
c Rn
F 5
,...,D 2 m
U)
i s a bounded domain, A :
l i n e a r mapping and
(3.16)
F(x,u,Du
F:
QXR S 2 m + R
= f,
f
E L2(Q)
D(A) C L2(Q)
+
+
Y(x)
x E
for
a
such t h a t
s a t i s f i e s t h e C a r a t h e o d o r y c o n d i t i o n s and Ply1
L2(Q)
y E RS2m
Q(a.e.),
IF(x,y)l and some
4
p E R,
E L2(Q);
x E
exist for
(3.18)
0
,
Q a.e.)
k*(x) = l i r n sup F ( x ' y ' z ) Y y+*m z E
uniformly i n
~ 2 m - 1and s a t i s f y
&+(x) 5 k+
L
V = D(A)
Let
,
&,(x) = l i r n i n f z y++m
(3.17)
with
Our f i r s t r e s u l t for
II.112m
,
V C W;"(Q)
uf ( x ) = max(*u(x),O).
and
(3.15) i s a consequence o f C o r o l l a r i e s 3.1 and
3.3. 3.5
THEOREM
A;'
+ aI:
hold.
Let
A
R ( A ) -+ R ( A ) c La
Suppose t h a t
(V,L2)
have P r o p e r t y I ,
for e a c h
A+tN:
T
lif a l s o
a <
dim N ( A ) < m ,
m ,
(3.16)-(3.18)
be s t r o n g l y monotone and
V
t E [O,l)
-+
L2
To
i s A-proper w . r . t .
for
and
~ ( 3 . 2 0 ) r ~ - k + ( x ) ] u + ( x )+ [ a - k - ( x ) ] u - ( x ) hen E q .
0 <
f
0
for
0
#
u E N(A+UI).
(3.15)
i s f e e b l y a p p r o x i m a t i o n - s o l v a b l e for e a c h
A + N
i s A-proper;
i t i s j u s t solvable f o r each
f E L2 f E
L2
290
P.
if also
+
A
s,
MILOJ-EVI~
s a t i s f i e s condition
N
If z e r o i s an e i g e n v a l u e of
either
L+
if
u > 0
4.-
and
u < 0
or
are
a.e.
f 0
in
(*) and i s s t r o n g l y A-closed. and i f f o r any
A
0
f u E N(A)
t h e n (3.19) h o l d s i f and only
Q,
on s u b s e t s of
w i t h p o s i t i v e measure.
Q
A second o r d e r s e l f - a d j o i n t e l l i p t i c o p e r a t o r w i t h
as i t s eigen-
0
v a l u e i s such an example.
W e now c o n s i d e r t h e c a s e when t h i s does
n o t happen, i .e. we assume
(f11 )
I n (3.21) Q
4.-, k-
one c o u l d have
C+
t h e r o l e s of
a
equal t o
on s u b s e t s of
An a n a l o g o u s problem c o u l d be s t u d i e d w i t h g-
and
k+
Hence, problem (3.15) e x i b i t s d o u b l e
w i t h p o s i t i v e measure.
resonance i n t h i s case.
or
interchanged.
Double r e s o n a n c e boundary
v a l u e problems have been e a r l i e r s t u d i e d by Dancer [ l l ] and B e r e s t y c k i and F i g u e i r e d o [l]
.
A s an a p p l i c a t i o n of C o r o l l a r y 3 . 2 ,
we
obtain THEOREM 3.6 A i l
+ aI:
Let
R(A)
-t
have P r o p e r t y I ,
A
R(A)
C
( 3 . 2 0 ) and ( 3 . 2 1 ) h o l d . change sign i n
(3.22)
dim N ( A ) <
(O,m),
be s t r o n g l y monotone and ( 3 . 1 6 ) ,
Suppose t h a t any
f u E
0
-, (3.17),
does not
N(A)
and:
There e x i s t s a
for
(3.23)
Q
L2
a E
c ( x ) E L2(Q)
x E Q(a.e.),
y 2 0
F+(x) = l i m inf F ( x , y , z ) y 3 +a ~ 2 m - l and in z E R
[ [Q
for
F u
fu <
and
such t h a t z ER
e x i s t s for
o f
F(x,y,z)
2 C(X)
S2rn-1
x E Q(a.e.)
u E N(A),
u
>
uniformly
0.
+
(3.24) with a.e.
in
Q
u
>
0,
for e a c h
t h e n t h e r e e x i s t s an n 2 no
.
no
such t h a t u n ( x ) > 0
291
APPROXIMATION-SOLVABILITY
Then,
if
A
V
tN:
ro
i s A-proper w . r . t .
L2
-t
( t E C 0 , l ) and
t E [O,l] Eq.
+
A+N
(*)
satisfies
(3.15) i s f e e b l y approximation-solvable
i n d e x z e r o , we h a v e shown i n
i s A-proper
and A - c l o s e d
A t y p i c a l e x a m p l e of
c
( -1) l a
l e 1-l
la1 9
I
(3.15)
i n Eq.
F
V = D(A)
[24]
.
L2
-t
is
t h a t A+tN:(V,jl*II)
t E [O,l]
with
k
-t
L2
pro-
s u f f i c i e n t l y small,
i s a s t r o n g l y uniformly e l l i p t i c o p e r a t o r
A
D'
and
v a r i o u s schemes f o r
i s k-ball-contractive
V -+ L2
N:
vided
w.r.t.
A A:
Since
F r e d h o l m of
and is s t r o n g l y A - c l o s e d ) ,
(solvable, resp.)
L e t u s now d i s c u s s some e x a m p l e s of t o which t h e above r e s u l t s a p p l y .
(v,L~)for
for
( a a B ( x ) D'u) N:
lowing c o n d i t i o n i m p l i e s
with
D(A)
-+
L2
2m o m = W2 fl W2
D(A)
.
The f o l -
is k-ball-contractive:
c i e n t l y small.
C aa(x)Da be s t r o n g l y u n i f o r m l y e l l a l ~ 2 m for la1 5 2 m , l i p t i c with r e a l coefficients a E Co"(G) a
More g e n e r a l l y ,
o
<
and
x
< 1,
let
A =
as E c m , v
H = L2(Q).
n B:(Q)
= W;"(Q)
= D(A)
By a r e s u l t o f S k r y p n i k [ 3 2 ]
there e x i s t a linear
C
K =
s t r o n g l y uniformly e l l i p t i c o p e r a t o r
11 *l12m-norm
with
bu(x)Du
with
.l a l. r 2 m
C m - r e a l c o e f f i c i e n t s and c o n s t a n t s
holds with
p < c,
A+tN:
t E [O,l].
V + L2
(V,L2)
for
(3.26)
There e x i s t s a f u n c t i o n
for
x E
Q(a.e.),
Now,
y E RS2m-1
c
> 0
such t h a t
ro
i s A-proper w . r . t .
(3.25)
,
> 0,
c
for
can be r e l a x e d t o
p: R+\{O]
-I
[O,c)
such t h a t
S'
r > 0,
and
z,z'E
R 2m w i t h
Iz-z'Izr.
P
292
A + t N r V + L2
Then
A + N
and, i f
A
+ N
Po
p
A
of
+
t N
can be found i n [ 2 4 , 2 7 , 2 8 ] .
3.5
we s e e t h a t Theorems
and
for
t E [O,l]
by a
t a k e s on a l s o t h e v a l u e
i s j u s t s t r o n g l y A-closed.
t h a t imply t h e A-properness of
MILO J E V I ~
i s A-proper w . r . t .
v a r i a n t of P r o p o s i t i o n 2 . 2 then
.s .
O t h e r c o n d i t i o n s on
c,
N
or t h e s t r o n g A-closedness I n view of t h i s d i s c u s s i o n ,
3.6 e x t e n d some r e s u l t s o f Dancer [ l l ]
and B e r e s t y c k i and F i g u e i r e d o [ 13 i n v o l v i n g n o n l i n e a r i t i e s t h a t depend o n l y on
x
and
u,
a r e a l s o c o n s t r u c t i v e when
and,
i n c o n t r a s t t o t h e i r s , our r e s u l t s
A+N
i s A-proper.
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P.S.
P. S . MILOJEVI6,
A g e n e r a l i z a t i o n of L e r a y - S c h a u d e r t h e o r e m
and s u r j e c t i v i t y r e s u l t s f o r m u l t i v a l u e d A-proper A-proper mappings,
N o n l i n e a r A n a l y s i s , TMA, 1
and pseudo
(3) ( 1977),
263-276. 20.
P.S. MILOJEVId,
S u r j e c t i v i t y r e s u l t s f o r A-proper,
their
u n i f o r m l i m i t s a n d p s e u d o A - p r o p e r maps w i t h a p p l i c a t i o n s , N o t i c e s Amer. Math.
21.
P.S.
MILOJEVId,
SOC. January
1977, 77T-B27.
O n t h e s o l v a b i l i t y and c o n t i n u a t i o n t y p e
r e s u l t s f o r nonlinear equations with applications, I, Third Intern.
Symp. on T o p o l o g y and i t s A p p l i c . ,
Proc.
Belgrade,
1977. 22.
P.S.
MILOJEVIE,
O n t h e s o l v a b i l i t y and c o n t i n u a t i o n t y p e
r e s u l t s f o r n o n l i n e a r e q u a t i o n s w i t h a p p l i c a t i o n s 11, n a d i a n Math. B u l l .
23.
P.S. MILOJEVI6,
Ca-
2 5 ( l ) ( 1 9 8 2 ) , 98-109.
Some g e n e r a l i z a t i o n s of t h e f i r s t F r e d h o l m
t h e o r e m t o m u l t i v a l u e d A - p r o p e r mappings w i t h a p p l i c a t i o n s t o nonlinear e l l i p t i c equations,
P.S.
Anal. A p p l i c .
65
( 1 9 7 8 ) , 468-502.
(2)
24.
J . Math.
MILOJEVI6,
A p p r o x i m a t i o n - s o l v a b i l i t y r e s u l t s f o r equa-
t i o n s i n v o l v i n g n o n l i n e a r p e r t u r b a t i o n s of F r e d h o l m m a p p i n g s with applications t o d i f f e r e n t i a l equations,
Proc. I n t e r n .
Sem. F u n c t i o n a l A n a l y s i s , Holomorphy a n d Approx. T h e o r y , Rio de J a n e i r o , August Applied Math.,
25.
P.S.
MILOJEVI6,
1979,
L e c t u r e Notes i n Pure and
M a r c e l D e k k e r , N.Y.
(Ed. G.
Z a p a t a ) , Vol.
83.
Fredholm a l t e r n a t i v e s and s u r j e c t i v i t y r e s u l t s
f o r m u l t i v a l u e d A-proper
a n d c o n d e n s i n g mappings w i t h a p p l i -
c a t i o n s t o n o n l i n e a r i n t e g r a l and d i f f e r e n t i a l e q u a t i o n s , C z e c h o s l o v a k Math, J .
26.
P.S.
MILOJEVIE,
30 ( 1 0 5 ) , ( 1 9 8 0 ) , 387-417.
C o n t i n u a t i o n t h e o r e m s a n d s o l v a b i l i t y of
e q u a t i o n s w i t h n o n l i n e a r noncompact p e r t u r b a t i o n s of F r e d holm m a p p i n g s , ITA,
27.
P.S.
Atas 12n Semin6rio B r a s i l e i r o de AnAlise,
Sgo J o s e d o s Campos, S g o P a u l o , O c t o b e r ,
MILOJEVI6,
1980,
163-189.
C o n t i n a u t i o n t h e o r y f o r A-proper and s t r o n g l y
A - c l o s e d mappings and t h e i r u n i f o r m l i m i t s and n o n l i n e a r p e r t u r b a t i o n s of F r e d h o l m m a p p i n g s ,
Proc.
Intern.
Sem.
F u n c t i o n a l A n a l y s i s , Holomorphy and Approx. T h e o r y , R i o de J a n e i r o , August 1 9 8 0 , N o r t h H o l l a n d P u b l . B a r r o s o ) , M a t h e m a t i c s S t u d i e s , No
Comp.
(Ed. J . A .
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APPROXIMATION-SOLVABILITY
28.
P.S.
MILOJEVId,
pings,
T h e o r y of A - p r o p e r a n d p s e u d o A - c l o s e d map-
H a b i t a t i o n Memoir, U n i v e r s i d a d e F e d e r a l d e Minas
G e r a i s , B e l o H o r i z o n t e , December
29.
P.S.
MILOJEVId a n d W.V.
1 9 8 0 , 1-208.
PETRYSHYN,
C o n t i n u a t i o n and s u r j e c t i -
vity t h e o r e m s f o r u n i f o r m s l i m i t s o f A - p r o p e r m a p p i n g s w i t h applications,
30.
W.V.
PETRYSHYN,
J . Math. A n a l . Appl.
6 2 ( 2 ) ( 1 9 7 8 ) , 368-400.
O n the approximation-solvability of equations
i n v o l v i n g A-proper
and p s e u d o A - p r o p e r m a p p i n g s ,
Bull.
AMS,
81 ( 1 9 7 5 ) s 223-312. 31.
S.I.
POHOZAEV,
The s o l v a b i l i t y of n o n l i n e a r e q u a t i o n s w i t h
odd o p e r a t o r s ,
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I.V.
SKRYPNIK,
Funct.
On t h e c o e r c i v i t y i n e q u a l i t i t e s f o r p a i r s o f
l i n e a r e l l i p t i c operators,
(1978) 33.
J.T.
,
ODEN,
F . WILLE,
S o v i e t Math.
Doklad,
19 (2)
324-327. E x i s t e n c e t h e o r e m s for a c l a s s o f p r o b l e m s i n n o n -
linear elasticity,
34.
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Anal.
J . Math.
Anal.
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6 9 ( 1 9 7 9 ) , 51-83.
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R a t i o n a l Mech.
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4 6 ( 1 9 7 2 ) , 269-288.
Departamento d e Matemgtica U n i v e r s i d a d e F e d e r a l d e Minas G e r a i s B e l o H o r i z o n t e , MG, B r a s i l
Arch.
This Page Intentionally Left Blank
Functional Analysis, Hu lo mo rp h y and Approximation Theory II, GI. Zapata ( e d . ) 0 Ekevier Science Publizlrers B. C< (Nurtlr-HuNmrd), 1984
297
HOLOMORPHIC FUNCTIONS ON HOLOMORPHIC INDUCTIVE LIMITS AND ON THE STRONG DUALS O F STRICT INDUCTIVE LIMITS
L u i z a Amdlia Moraes
INTRODUCTION
The c o n c e p t s of ho l o m o r p h i c a l l y b o r n o l o g i c a l ( h b o ) p h i c a l l y b a r r e l e d (hba
,
,
holomor-
h o l o m o r p h i c a l y i n f r a b a r r e l e d ( h i b ) and
h o l o m o r p h i c a l l y Mackey (hM) s p a c e s h a v e b e e n i n t r o d u c e d by B a r r o s o , Matos a n d N a c h b i n i n [ l ] .
I n t h i s note
we w i l l s u r v e y r e s u l t s con-
c e r n i n g t h i s h o l o m o r p h i c c l a s s i f i c a t i o n of t h e s p a c e s .
More e x p l i c -
i t l y , we w i l l b e c o n c e r n e d w i t h t h e f o l owing s i t u a t i o n s : be a i n d u c t i v e l i m i t o f
E
Let
l o c a l l y convex s p a c e s
Ei
i E I.
1.
2.
E'
is a
F i n d s u f f i c i e n t c o n d i t i o n s on
i E I
Ei
-
(a)
Ei
hbo
f o r every
i E
(b)
Ei
hba
f o r every
i E I
(c)
Ei
hib
f o r every
i E I
(d)
Ei
hM
f o r every
i E I - E
F i n d s u f f i c i e n t c o n d i t i o n s on hbo
I
E
hbo
=
E
hba
3
E
hib
Ei
a n d on
E
so that
hM i E
I
and
E
so that
space.
PRELIMINARIES r e v i e w o f what w i l l b e n e e d e d h e r e .
L e t u s make a b r i e f Unless s t a t e d otherwise, spaces,
U
E
and
F
d e n o t e complex l o c a l l y c o n v e x
i s a non v o i d open s u b s e t of
a l l m a p p i n g s of
U
into
F.
If
I
E
and
i s a s e t and
Fu
F
i s t h e s e t of is a seminord
298
L U I Z A AMALIA MORAES
s p a c e , we d e n o t e by mappings of
into
I
morphic mappings of of a l l mappings of
t h e seminormed s p a c e of a l l bounded
lm(I;F) F.
#(U;F) into
U
and
F;
into
U
i s t h e v e c t o r space of a l l holo-
F
H(U;F)
i s t h e v e c t o r space
which a r e h o l o m o r p h i c when c o n s i d 4
e r e d a s mappings of
i n t o a f i x e d completion
U
F
of
F.
We w i l l
s a y t h a t a mapping
f: U
-t
F
i s holomorphic i f f
#(U;F).
f: U
-t
F
i s a l g e b r a i c a l l y holomorphic ( e q u i -
A mapping
v a l e n t l y G-holomorphic) i f t h e r e s t r i c t i o n
flu
fl S
S
of
f o r every f i n i t e dimensional v e c t o r subspace where
c a r r i e s i t s n a t u r a l topology.
S
P,(%;F)
PHy(%;F),
and
k-homogeneous
o f a l l k-homogeneous p o l y n o m i a l s bounded s u b s e t s of polynomials of
aM(U;F)
E;
-+
P: E
meeting
E
F,
-t
F
-t
P(%;F),
t h e v e c t o r space
t h a t a r e bounded on
E
and t h e v e c t o r s p a c e o f a l l k-homogeneous
F
t h a t a r e c o n t i n u o u s on t h e compact s u b s e t s
(respectively:
#=(U;F))
s p a c e of a l l G-holomorphic mappings of e d on t h e compact s u b s e t s of
on t h e compact s u b s e t s o f
U
U
into
t h a t a r e bound-
F
( r e s p e c t i v e l y : t h a t a r e continuous F = 6,
When
U).
w i l l denote t h e v e c t o r
it i s not included
i n t h e n o t a t i o n f o r f u n c t i o n s p a c e s ; s o , we w i l l w r i t e P(%;C),
for
#(U)
l m ( I >f o r
#(U;C),
s e t of a l l c o n t i n u o u s seminorms on mapping every U
f: U
+ F
0 E SC(F);
$nto
F
E
p o f
the collection
p 6 CS(F).
z
A given
E
A
U
of mappings o f
=
p o x
A given
morphically b o r n o l o g i c a l space i f , f o r every
CS(E).
The
i s l o c a l l y bounded f o r
more g e n e r a l l y , a c o l l e c t i o n
i s l o c a l l y bounded f o r e v e r y
for
P(%)
and s o on.
lQ(I;6),
i s d e n o t e d by
i s amply bounded i f
i s amply bounded i f
#(U;F) = gM(U;F).
U,
t h e v e c t o r s p a c e of a l l
P: E
P: E
i s holomorphic,
We d e n o t e by
respectively,
continuous polynomials
belongs t o
f
E
and
f p o f
I
f E Z]
i s a holoF,
we h a v e
i s a holomorphically b a r r e l e d space
( r e s p e c t i v e l y : a holomorphically i n f r a b a r r e l e d space) i f f o r every U
and e v e r y
F,
we h a v e t h a t e a c h c o l l e c t i o n
bounded i f , and a l w a y s o n l y i f ,
Z
Z c H(U;F)
i s amply
i s bounded on e v e r y f i n i t e d i -
299
HOLOMORPHIC FUNCTIONS ON HOLOMORPHIC INDUCTIVE LIMITS
m e n s i o n a l compact s u b s e t of s u b s e t of
U).
f o r every
U
belongs t o
A given and e v e r y
H(U;F)
lo the
s e n t by
s e t s and by
lof
iff
E F,
(respectively:
on e v e r y compact
i s a h o l o m o r p h i c a l l y Mackey s p a c e i f f: U
we have t h a t e a c h mapping
$of E W(U)
$ E F’.
f o r every
of s c a l a r s .
A net
( ha
(Xa)a€A
la E A
of e l e m e n t s i n
-t
F
An i n d u c t i v e l i m i t o f ( E . ) 1
we have
SECTION 1:
f E #(U;F)
iff
E
is
c o n v e r g e s t o z e r o f o r any n e t by
iEI
i s s a i d t o b e a holomorphic i n d u c t i v e l i m i t i f f o r e v e r y U
We r e p r e -
t h e t o p o l o g y of u n i f o r m convergence on t h e f i n i t e
very strongly convergent i f
f:
F
-t
t o p o l o g y of u n i f o r m convergence on t h e compact sub-
d i m e n s i o n a l compact s u b s e t s .
(‘a )a€ A
U
fapi E #(Ui;F)
(Pi)iEI U, F ,
f o r every i E I.
STRONG DUALS O F STRICT INDUCTIVE LIMITS O F FmCHET-
MONTEL SPACES
B a r r o s o , Matos and Nachbin prove i n [ l ] t h a t S i l v a s p a c e s a r e holomorphically bornological spaces. of Dineen
r4]
W e i n f e r from t h e r e s u l t s
t h a t t h e s t r o n g d u a l s of Fr6chet-Monte1 s p a c e s (DFM-
s p a c e s ) a r e a l s o h o l o m o r p h i c a l l y b o r n o l o g i c a l s p a c e s and t h i s i m p r o v e s t h e r e s u l t of B a r r o s o , Matos and Nachbin a s e v e r y S i l v a s p a c e i s a DFM-space and t h e r e a r e DFM-spaces spaces.
t h a t a r e not S i l v a
Boland and Dineen g i v e i n [ 21 an example of a s t r i c t i n -
d u c t i v e l i m i t of Fr6chet-Monte1 s p a c e s (FM-spaces)
t h a t i s n o t ho-
l o m o r p h i c a l l y b o r n o l o g i c a l ( s e e P r o p o s i t i o n 1 4 ( a ) of going t o prove i n t h i s s e c t i o n t h a t i f
E
t 21 ) .
i s a s t r i c t inductive
l i m i t of FM-spaces and t h e r e e x i s t s a c o n t i n u o u s norm on E’
i s a holomorphically b o r n o l o g i c a l space.
Proposition quently Q‘
3 of [ 81 :
#=(U;F)
w e have now
= #(U;F))
U
E,
then
T h i s r e s u l t improves
HM(U;F) = # ( U ; F )
f o r every
We a r e
( a n d conse-
and for e v e r y
s a t i s f i e s o u r c o n d i t i o n s , we have i n p a r t i c u l a r
F.
As
XM(U;F) = # ( U ; F )
300
LUIZA A d L I A MORAES
U c i9’
f o r every of
f
and f o r e v e r y
F.
T h i s improves P r o p o s i t i o n 1 4
21.
The r e s u l t s c o n t a i n e d i n Lemmas 1 & 3 a r e well-known and we h a v e s t a t e d t h e n h e r e o n l y f o r t h e s a k e of c o m p l e t e n e s s . LEMMA 1. E‘
i s t h e s t r i c t i n d u c t i v e l i m i t of FM-spaces
E
If
PROOF.
T: E ‘ + F
Let
compact s u b s e t of E’
= 12m E n ,
“n f o r every
n E N,
nn: E’ -+ E L
i s the canonical surjection.
t h e r e e x i s t s a l i n e a r mapping Let
By h y p o t h e s i s ,
bounded on
Kn.
Eh
Tn
i s a DFM-space,
i s continuous.
So,
T
i s bounded on
E‘
such
such t h a t
and s o ,
K
So,
By Example
EA.
of
K
+ F
Tn: E’ n
be a compact s u b s e t of
Kn
K = nn(K).
So,
of [ 5 ] t h a t
We know f r o m P r o p o s i t i o n 2 . 8
of [ 5 ] , t h e r e e x i s t s a compact s u b s e t
2.10
E’
9
be a l i n e a r mapping t h a t i s bounded on e v e r y
E’.
where
T = Tnonn.
that
as
En
i s a bornological space.
Tn
is
i s bounded on e v e r y compact s u b s e t o f E h ; we i n f e r from C o r o l l a r y 11 of r k ]
T = T on n n
that
Tn
i s c o n t i n u o u s and t h i s p r o v e s t h a t
i s a bornological space.
DEFINITION 2. and
flX
LEMMA
n
3.
A set
U = 0
Let
E
X
imply
f: U + G
i s determining f o r F
if
X
n
U
$6
0.
I
be a Monte1 s p a c e s u c h t h a t
E’
i s bornological.
The f o l l o w i n g a r e e q u i v a l e n t : (a)
T h e r e e x i s t s a c o n t i n u o u s norm
p
on
= (cp E E : p(cp) 4 11 f o r e a c h P t h e r e e x i s t s a t l e a s t a p E SC(E) such t h a t (b)
If
f o r every of
V
f E HM(W),
where
W
There e x i s t s a s u b s e t
i n g for e v e r y
E E = (E’ ) ‘
.
p E SC(E),
then
Vo
i s determining P i s any convex b a l a n c e d open s u b s e t
E‘ ; t h i s i s t r u e f o r e v e r y c o n t i n u o u s norm (c)
E.
K
of
E’
p
such t h a t
on K
E.
i s determin-
301
HOLOMORPHIC FUNCTIONS ON HOLOMORPHIC INDUCTIVE LIMITS
PROOF.
(a)
(b):
=)
Let
p
i s t h e Minkowski f u n c t i o n a l o f E'
K = Vo P
i s Montel,
E.
be a c o n t i n u o u s norm on
V p = (rp E E
11.
: p(tp) sz
p
Since
a n d we c l a i m i t
E'
i s a compact s u b s e t of
Then
satisfies (b). f E HM(W)
We show t h a t if and
f7 W =
flK
0,
it is clear that = 0
fl
W
=
f
f Q
0.
i s convex b a l a n c e d o p e n i n E '
W
Since
a n d if
flK
n
W = 0
f
I
0
H e n c e , t o show
= E
(E')'
= V tc 1 E Q:
tc T E P M ( % ' ) ; p ( h T ) 1 1 %' X E C
3
for a l l e l e m e n t s of be s u c h t h a t
P/K
PM(%')
= 0,
form c o r r e s p o n d i n g t o
f a c t o r s ) a n d v a n i s h e s on
Lx:
E'+
d e f i n e d by
Q:
PM(n-%')
= L(z,y, ...,y)
every
y E E'
induction every
K x...x
Then
K, Ly:
and
Ly/K
= 0.
But t h e n i n p a r t i c u l a r
E'.
K
k E IN.
=)
X T E Voo = T
3
0.
i s determining
K
Let
P
E
P,(%')
K.
z)
and hence E'+
C
x E K.
Now f i x
...,
= E'xE'x
(E')n
...x E '
Then
i s a n e l e m e n t of Lx
I
0.
d e f i n e d by
Next l e t Ly(z) =
i s a l i n e a r f o r m t h a t i s c o n t i n u o u s on t h e bounded
E'
s u b s e t s of
E'
d e n o t e t h e symmetric n - l i n e a r
Lx(z) = L ( x , z ,
w h i c h v a n i s h e s on
be a r b i t r a r y .
as
From t h e p o l a r i z a t i o n f o r m u l a we s e e
(n
on
L
n = 1
p(T) = 0
3
k < n.
i s bounded on t h e bounded s u b s e t s of
L
y E E'
f o r every
and l e t
P.
E K
V x
1 p ( T ) 4 T~;TV X E C
3
for e v e r y
i s a c o n t i n u o u s norm on E :
p
n > 1 and s u p p o s e we h a v e shown t h a t
Now l e t
that
i t s u f f i c e s t o show PM(%')
T(x) = 0
so
V l K r 7 W =
then
I t i s c l e a r for
We p r o v e t h i s by i n d u c t i o n .
i s a b o r n o l o g i c a l s p a c e (Lemma 1) a n d T E
i s convex a n d b a l a n c e d ,
K
i s d e t e r m i n i n g f o r t h e e l e m e n t s of
K
n E N.
K
n E N.
f o r each
that
then
where
.
T h i s shows t h a t
By t h e i n d u c t i o n h y p o t h e s i s Ly(y) = L( y , . . . , y )
P
P
0
on
E'
(b)
3
(c):
obvious.
(c)
3
(a):
L e t K be a compact s u b s e t of
(a)
E'
5
= P(y) = 0
0
for
and t h e r e f o r e by
i s d e t e r m i n i n g f o r t h e e l e m e n t s of T h i s c o m p l e t e s t h e p r o v e of
Ly
3
PM(kE')
for
(b).
such t h a t i f
cp E E =
302
L U I Z A AMALIA MORAES
= (E')'
= 0,
cp/K
and
cp s 0.
then
W e claim t h a t
u n i t a r y b a l l of a c o n t i n u o u s norm on
E
cp
exists
=?
#
cp
hypothesis
#
a E d:
E
acp
such t h a t
0,
true, there
f o r e v e r y a E C.
KO
and c o n s e q u e n t l y
Contradiction.
0.
I f t h i s i s not
E.
vp/K = 0.
By
T h i s completes t h e prove of
(a).
4.
PROPOSITION If
cp
IIacplIK d 1 f o r e v e r y
So
(c)
= E,
(E')'
i s the
KO
Let
be a s t r i c t i n d u c t i v e l i m i t of FM-spaces En.
E
h a s a c o n t i n u o u s norm,
E
E'
i s a holomorphically b o r n o l o g i c a l
space. From P r o p o s i t i o n 54 of [ l ] i t i s enough t o show t h a t
PROOF.
= H(U)
%(U)
U c E'
f o r every
and
i s a holomorphically
E'
in-
f r a b a r r e l e d space.
then
E,
Since
WM(U) = # ( U ) :
Let u s show
1)
p
Lemma 3
i s t h e Minkoswki f u n c t i o n a l of K = Vo
i s Montel,
E'
convex b a l a n c e d open s u b s e t of f IK
n
W = 0
then
As
f e 0.
V = [ c p E E : p(cp) c
i s a compact s u b s e t of
i s determining f o r every
K
i s a c o n t i n u o u s norm on
p
If
= l i m EL
f
and f r o m
E'
where
WM(W)
t h a t i s , if
E', E'
E
f
11.
is a
W
E WM(W)
and
i s an open and compact
nn s u r j e c t i v e l i m i t ( s e e Example 2.10 [ 5 ] ) we may assume w i t h o u t l o s s U = T Tm- ~ ( W )
of g e n e r a l i t y t h a t Eh).
s u b s e t of
Now l e t
f E HM(U).
s u r j e c t i v e r e p r e s e n t a t i o n of
Eh
i.e.
x,
x+y E U
some
n E IN
t h e r e e x i s t s an
n,
whenever
yn
#
0
and
n
of
n
2
aM(V)
r).
For f i x e d
where
V
E'
i s an open and compact
If
f(zn+yn)
#
i n o r d e r t o show
f(x+y) = f(x)
for a l l
does n o t f a c t o r t h r o u g h
f
zn,
f(zn).
E'
i s an open
W
f a c t o r s t h r o u g h some
f
there e x i s t s
n,
(where
such t h a t
v e r y s t r o n g l y convergent t o z e r o i n ever
m
by DFM-spaces,
rrn(y) = 0 .
t h e n f o r each
nn(yn) = 0 ,
E'
As
i t s u f f i c e s t o show t h a t
f E w(U)
that
f o r some
Note t h a t
(since
z cf(z+yn)
-
zn+yn E U
rrr(yn)
f(z)
where
(Y,)
= 0
is
when-
d e f i n e s an element
i s some convex b a l a n c e d neighbourhood of z e r o
303
HOLOMORPHIC FUNCTIONS ON HOLOMORPHIC INDUCTrVE LIMITS
E'
in
,
and hence t h e r e e x i s t s
f(xn+yn)
#
f(xn).
n
For
c o n s t a n t e n t i r e f u n c t i o n on such t h a t ever
C,
closure i s contained i n
and
U
in
i.e.
EA,
E'
Suppose
.
Hence
E'
(
We want t o show
1, E C
> n. U
How-
whose
(Xn+'nyn)nzm
must f a c t o r
f
i s a holomorphically
i s a T -bounded
(fU)a€A
i s a non
E #(U).
f
Let u s prove t h a t
space:
If(xn+knyn)I
i s unbounded on
f
f E aM(U).
contradicting the f a c t that
2)
i.e.
such t h a t
f(xn)
i s a r e l a t i v e l y compact s u b s e t of
(xn+'nYn)n2m
t h r o u g h some
-
and hence t h e r e e x i s t s
> n + If(xn)l,
lgn(Xn)l
( a < 1)
(1-a)U
gn(X) = f ( x n + X y n )
m,
2
n
xn E K
fU)aEA
infrabarreled
s u b s e t of
#(U),
i s l o c a l l y bounded.
U
open
A s i n part
1) of t h i s p r o o f , we may assume w i t h o u t l o s s of g e n e r a l i t y t h a t U = n-l(W)
f o r some
m
claim t h a t
m
where
W
i s a n o p e n s u b s e t of
f a c t o r s uniformly through
(fa)aEA
t h e r e e x i s t s nEN such t h a t f
= f
o n
EL
EL.
f o r some
We
n ie.,
f o r a l l U E A where FU€#(nn(U)).
a a n If n o t , we c a n , a s i n p a r t 1) of t h i s p r o o f , f i n d a s e q u e n c e (1-a)U
(xn+Xnyn) C
which i s r e l a t i v e l y compact a n d a s e q u e n c e ( f
)
%nm such t h a t
( f a a€A
(fan(xn+Xnyn)I 2 n. i s ro-bounded,
t h r o u g h some
(F
)
a aEA
[4]
EL.
Now
(fa)(*EA limit).
and h e n c e
E'
i s lo-bounded i n
we c o n c l u d e t h a t
This contradicts the f a c t that
(fa)aEA
i s a compact s u r j e c t i v e l i m i t a n d h e n c e
a(nn(U)).
('a)aEA
i s l o c a l l y bounded on
U
(since of
We would l i k e t o p r o v e t h a t i f
l o g i c a l space, then
En E
under such h y p o t h e s i s
T h e r e f o r e from P r o p o s i t i o n
such t h a t
E'
E'
i s an open s u r j e c t i v e
Proposition E
4.
i s a s t r i c t inductive
i s a h o l o m o r p h i c a l l y borno-
h a s a c o n t i n u o u s norm.
En
6
i s l o c a l l y bounded a n d h e n c e
T h i s completes t h e proof
l i m i t of FM-spaces
f a c t o r s uniformly
A s we know t h a t
h a s a c o n t i n u o u s norm f o r e v e r y
n
(see
P r o p o s i t i o n 2 [ 8 ] ) , t h e f i r s t i d e a t o d o t h i s was t o p r o v e t h a t i f
304
E En
L U I Z A AMALIA MORAES
i s a s t r i c t i n d u c t i v e l i m i t of h a s a c o n t i n u o u s norm, t h e n
FM-spaces E
En
h a s a c o n t i n u o u s norm.
t u n a t e l l y F l o r e t proved, giving a counterexample,
a r e FN-spaces.
that this state-
These counterexamples can be found i n
The o r i g i n a l p r o b l e m r e m a i n s a n open p r o b l e m . that if
[6].
We know now
i s a s t r i c t i n d u c t i v e l i m i t of FM-spaces
E
Unfor-
A c t u a l l y i t i s f a l s e e v e n i n t h e c a s e when t h e
ment i s n o t t r u e .
En
such t h a t every
En,
at least
one of t h e f o l l o w i n g two s t a t e m e n t s must be f a l s e :
i s a holomorphically b o r n o l o g i c a l space
(1) E '
E
3
has a
c o n t i n u o u s norm. PM(%') = P(%')
(2)
k
f o r every
=,
HM(U) = H ( U )
for e v e r y
U C E'. ( 2 ) i s correct,
If
the following assertions are equivalent:
n.
(a)
En
h a s a c o n t i n u o u s norm f o r e v e r y
(b)
E'
i s a holomorphically bornological space.
I n [ 3 ] D i e r o l f and F l o r e t prove t h e f o l l o w i n g : LEMMA
5.
Let
F
r e s p e c t t o a c o n t i n u o u s norm norm
q
on
F
p
E.
on
can be extended t o
E
which i s c l o s e d w i t h
E
be a l i n e a r s u b s p a c e of
Then e v e r y c o n t i n u o u s a s a c o n t i n u o u s norm.
T h i s lemma a l l o w s u s t o s t a t e t h e PROPOSITION
6.
Let
such t h a t e a c h
En
be a s t r i c t i n d u c t i v e l i m i t of FM-spaces E
E
i s norm-closed
in
Then
En+la
E'
n
i s a holo-
morphically bornological space.
I t i s a c o n s e q u e n c e of Lemma 5 a n d P r o p o s i t i o n 3 f 8 ] .
PROOF.
LEMMA
7.
If
E
such t h a t each
i s t h e s t r i c t inductive l i m i t of Frbchet-spaces
En
h a s a c o n t i n u o u s norm.
If
E
h as a n uncondi-
t i o n a l b a s i s t h e n t h e r e e x i s t s a c o n t i n u o u s norm on PROOF.
See F l o r e t [ 61
.
En
E.
305
HOLOMORPHIC FUNCTIONS ON HOLOMORPHIC INDUCTIVE LIMITS
PROPOSITION 8.
En
E
and
If
i s t h e s t r i c t i n d u c t i v e l i m i t of FM-spaces
E
has an unconditional b a s i s ,
then the following are
equivalent: (a)
En
(b)
E’
PROOF.
3
(a)
h a s a c o n t i n u o u s norm f o r e v e r y
n.
i s a holomorphically bornological space.
It i s a c o n s e q u e n c e of Lemma 7 a n d P r o p o s i t i o n
(b):
3
mi. (b)
It i s a c o n s e q u e n c e of P r o p o s i t i o n 2 [8].
(a):
3
SECTION 2 .
HOLOMORPHIC INDUCTIVE LIMITS
PROPOSITION
9.
be t h e h o l o m o r p h i c i n d u c t i v e l i m i t of
E
Let
(pi)
by
l o c a l l y convex s p a c e s
iEI
(Ei)iEI
.
the
The f o l l o w i n g s t a t e -
ments a r e t r u e : If
i s a holomorphically bornological space f o r every
Ei
i E I,
If
i s a holomorphically b a r r e l e d space.
E
then
i s a holomorphically
Ei
then
If
i s a h o l o m o r p h i c a l l y Mackey s p a c e f o r e v e r y i E I ,
Ei
(a) such t h a t
E Let
f
arbitrary.
f: U
i s continuous.
therefore
i n f r a b a r r e l e d space.
-+ F
be a a l g e b r a i c a l l y holomorphic mapping
i s bounded on e v e r y compact s u b s e t of
fopi
fopi
i s a holomorphically
i s a h o l o m o r p h i c a l l y Mackey s p a c e .
Then
l i n e a r , and
space,
E
i n f r a b a r r e l e d space f o r every
i E I,
then
pi
i s a holomorphically b a r r e l e d space f o r every i E I,
Ei
If
i s a holomorphically bornological space.
E
then
fapi
U.
i s a l g e b r a i c a l l y holomorphic a s
Let
pi
i s bounded on e v e r y compact s u b s e t o f Since
E #(Ui;F)
f E g(U;F)
Ei
T h i s completes t h e proof
of
E
is
as
i s a holomorphically bornological
f o r every (as
Ui
i E I
i E I
a n d for e v e r y
F
and
i s a holomorphic i n d u c t i v e l i m i t ) .
(a).
LUIZA AMALIA MORAES
30 6
From P r o p o s i t i o n 35 of r l ] i t f o l l o w s t h a t i t s u f f i c e s t o
(b)
U C E
show t h a t f o r e v e r y
X
i s l o c a l l y bounded i f
U,
compact s u b s e t of
on e v e r y f i n i t e d i m e n s i o n a l
i s bounded
X
i.e.,
pi:
As
Tof-bounded.
Ei
i s l i n e a r and c o n t i n u o u s ,
E
-t
U.
f i n i t e d i m e n s i o n a l compact s u b s e t of
= [fopi
: f
E X] c
i E I).
every
x E U
for
and
f E
lm(X)
g: U +
X.
E
H(Ui;lm(X)):
If
S
We c l a i m t h a t
gopi:
Ui
lm(X)
-t
hence
s i n c e X i C W(Ui)
p h i c . On t h e o t h e r h a n d ,
i s bounded
gopi(K)
S o , gopi
i s G-holomor-
i s l o c a l l y bounded, i t i s i E #(Ui;lm(X))
f o r every i E I . A s E i s a holomorphic i n d u c t i v e l i m i t , g
t h e image of e v e r y compact s u b s e t of U
and c o n s e q u e n t l y ,
i s To-bounded;
C W(Ui)
Xi
space,
X
f E
and
bounded, (d) that
as
g: U
-t
lm(X)
belongs t o
Let
U
Ui
d e f i n e d by
W(U;l”(X))
a n d c o n t i n u o u s and
f o r every
6 E
$ o f E #(U)
Ei
pi:
+ E
Xi
i € I
i s continuous,
i s a compact s u b -
= pil(U)
: f €
= [fopi
i E I.
X)
Now we p r o v e a s
= f(x)
g(x)(f)
X
and t h e r e f o r e
for
x E U
is locally
of ( c ) .
b e an open s u b s e t of
E W(U)
is locally
i s a holomorphically i n f r a b a r r e l e d
T h i s completes t h e proof
g o f
As
f o r every Ei
gEH(U;lm(X))
(a).
i s l o c a l l y bounded f o r e v e r y
i n (b) t h a t
X
i s l o c a l l y bounded, t h a t i s ,
bounded. T h i s c o m p l e t e s t h e p r o o f of (c) Let X C H ( U ) be l o - b o u n d e d .
s e t of
is al-
pi E
go
i s l o c a l l y b o u n d e d . S o , we h a v e g o p
and c o n s e q u e n t l y
is
i s a f i n i t e d i m e n s i o n a l v e c t o r s u b s p a c e of E
a n d K i s a compact s u b s e t of S fl U i ,
c l e a r t h a t gopi
Xi
g ( x ) ( f ) = f(x)
d e f i n e d by
i s a f i n i t e d i m e n s i o n a l s u b s e t of U.
as Pi(K)
is a
i E I
So, f o r every
g e b r a i c a l l y h o l o m o r p h i c and l o c a l l y bounded;
m e e t i n g Ui
t h e image
-1 = p i (U)
Ui
be
i s a holomorphically b a r r e l e d space f o r
Ei
Consider
C W(U)
i s Tof-bounded a n d t h e r e f o r e
#(Ui)
l o c a l l y bounded ( s i n c e
X
is ~~~-bounded L .e t
of e v e r y f i n i t e d i m e n s i o n a l compact s u b s e t o f
Xi
X c #(U)
we h a v e t h a t e a c h c o l l e c t i o n
F‘.
E.
Consider
Since
w e have t h a t
pi:
f: U + F
Ei + E
Jlofopi
such
is linear
E W(Ui)
where
307
HOLOMORPHIC FUNCTIONS ON HOLOMORPHIC INDUCTIVE LIMITS
Ui = pil(U)
is a n open subset of
ly Mackey space for every
i E I
Q E F ’ , we conclude that
fapi
is,
fopi
E FUi n W(Ui;?)
Ei
.
Ei
As
and
E W(Ui)
$ofopi
for every
i E I.
Now,
i E I
E
f E Fu
n
that
is a holo-
W(U;G) =
This completes the proof of (d).
If
REMARK.
for every
E H(Ui;F) for every
morphic inductive limit and so we infer that
= H(U;F).
is a holomorphical-
E
E
is a DFM-space, m
has a fundamental sequence of
compact sets,
(Bn)n,l,
and increasing.
denote the vector space Bn and endowed with the norm generated by the Minkowski
En
spaned by
functional of
which we may suppose are convex, balanced
For each
Bn.
n
For every
let
n E
E
(N,
E
is a Banach space and
Bn s o , it is a holomorphically barreled space. morphic to the inductive limit
The space
E
is iso-
lim E
in the category of locally n Bn convex spaces and continuous linear mappings (see $ 2 of [hi). We
E = lim E is a holomorphic inductive limit. This n Bn conjecture is equivalent to that of Dineen in the p. 163 of [4]. conjecture if
Matos introduced, in bornological space.
[7], an other notion of holomorphically
We will call S-holomorphically bornological
the spaces that are holomorphically bornological in the sense of Matos
[7] and holomorphically bornological the spaces that are ho-
lomorphically bornological in the sense of Barroso, Matos and Nachbin.
Every S-holomorphically bornological space is holomorphi-
cally bornological.
In [4] Dineen proves that DFM-spaces are ho-
lomorphically bornological and asks if they are S-holomorphically bornological (see conjecture, p . 163 of
[4] )
.
We are going to
prove that the holomorphic inductive limit of S-holomorphically bornological spaces is a S-holomorphically bornological space. Let
BE
denote the family of all closed absolutely convex
bounded subsets of
E.
For each
B E OE,
let
EB
denote the
308
AMALIA
LUIZA
B
v e c t o r s p a c e s p a n n e d by
and endowed w i t h t h e norm g e n e r a t e d by
t h e Minkowski f u n c t i o n a l of DEFINITION 10. morphic i n
B E EE
U
A mapping if
MORAES
B.
U
from
f
i s c a l l e d S-holo-
F
into
i s f i n i t e l y holomorphic i n
f
flu
t h e mapping
n
i s continuous
EB
U
and for e v e r y
(or, e q u i v a l e n t l y ,
h o l o m o r p h i c ) r e l a t i v e t o t h e normed t o p o l o g y .
Let
HS(U;F)
U
mappings f r o m DEFINITION 11.
d e n o t e t h e v e c t o r s p a c e of a l l S - h o l o m o r p h i c
into
F.
The s p a c e
U
space i f f o r every
i s a S-holomorphically b o r n o l o g i c a l
E
and
H(U;F) = H S ( U ; F ) .
it i s true that
F
For b a s i c p r o p e r t i e s , s e e [ 7 ] . PROPOSITION 1 2 .
Let
b e t h e h o l o m o r p h i c i n d u c t i v e l i m i t of
E
l o c a l l y convex s p a c e s
(Pi)iEI.
by
( E ii ) EI
morphically b o r n o l o g i c a l space f o r ever y
If
Ei
i E I,
the
i s a S-holo-
E
then
is a
S-holomorphically b o r n o l o g i c a l space. PROOF.
Let
f: U
-I
F
be a G-holomorphic
on t h e s t r i c t compact s u b s e t s of
1)
If
E = 1 2 Ei
mapping which i s bounded
U.
i s a i n d u c t i v e l i m i t and
is a strict
Ki
p i compact s u b s e t of E:
Ei,
by d e f i n i t i o n ,
exists Since
Bi E BE
pi
Ki
then
pi(Ki)
i s a s t r i c t compact s u b s e t of
such t h a t
i
K i c (Ei)Bi
i s l i n e a r and c o n t i n u o u s ,
t o prove t h a t
pi(Ki)
t h e norm.
E P~
As
e v e r y sequence
(x,)
i s compact i n
n E IN.
pi(Bi) E
c pi(Ki)
Then
(y,)
Ei
i s compact i n
E
aE
there
iff (Ei)Bi.
and we a r e g o i n g
i n t h e t o p o l o g y of
Pl(B1)
i s a normed s p a c e ,
Pi(Ki)
i s compact i f f
a d m i t s subsequence which converges
( i n t h e norm t o p o l o g y ) t o a p o i n t o f for e v e r y
i s a s t r i c t compact s u b s e t o f
pi(Ki).
i s a sequence i n
Let
Ki
y,
-1 = P i (x,)
and,
as
Ki
is
309
HOLOMORPHIC FUNCTIONS ON HOLOMORPHIC INDUC'IIVE LIMITS
(y ) = subsequence of nk which converges in (Ei)Bi to y o E K i , i.e., l/Y -YollBi+ 0 "k k -+ m . It follows that given c > 0, there exists No such
a compact subset of (y,) as
k > No
that f o r every
> 0
inf(X
-yo) E
pi(y
llYnk-YollBi < c
.
such that
'c ,k
But this implies
we have
XBi) < E
: ynk-y0 E
there exists
there exists
(Ei)Bi,
E ,k
9
c > 0, f o r each
So, given 0< h
i.e.
< e
and
k > No
E h E ,kBi *
Yn -Yo k
X c ,kpi(Bi)
and consequently
"k
P~(Y
nk
-
)
inf{X > 0 : pi(y k > No.
every
)
nk So,
pi(Ki)
C E
pi(yo) E
f: U + F
E,
such that
(as
by 1). Ki
is continuous).
pi
If
E i , Pi(Ki) c U fi = fopi
Ei
P q q -
*
is a strict
(u) = ui
f E U(U;F).
is an open
Ki C pil(U)
is a
is a strict compact sub-
is bounded on every K i c pyl(U)
U,
Ei
(as
by hypothesis).
f
is bounded
fiE
So,
is a S-holomorphically bornological space
i s continuous f o r every
fi
ACKNOWLEDGEMENTS.
E
in
pi(Ki)
i s a strict compact subset of
tive limit is holomorphic and s o , implies
yo E Ki)
(x,)
G-holomorphic and bounded on the
pi
So,
and, as
by hypothesis,
i s a subsequence of
It i s clear that
on the strict compact subsets of E HS(Ui;F)
))
U.
strict compact subset of set of
and this is true f o r
< c
E.
compact subsets of
Ei
It follows
*
and is compact, i.e.,
We consider now
subset of
hPi(Bi)]
(xnk) = (pi(y
P1(B1)
compact subset of 2)
-
,kPi(Bi)
nk pi(yo) E pi(Ki) (as
which converges to So,
X,
Pi(y0) E X c ,kPi(Bi) c
f
i E I.
E U(Ui;F)
But the induc-
for every
i
E I
This completes the proof of Proposition 8.
I would like to express my thanks to Professor
Leopoldo Nachbin and to Professor Mario Matos f o r some useful discussions concerning this paper.
This research was supported in
part by FINEP, to which I express may gratitude.
LUIZA
310
AMALIA
MORAES
REFERENCES
1.
J.A. BARROSO, M.C. MATOS and L. NACHBIN,
O n holomorphy versus
linearity in classifying locally convex spaces. Infinite Dimensional Holomorphy and Applications, Ed. M.C. Matos, North Holland Math. Studies, 12, 1977, p. 31-74. 2.
P.J. BOLAND and S. DINEEN, Duality theory for spaces of germs and holomorphic functions on nuclear spaces. Advances in Holomorphy.
Ed, J.A. Barroso, North Holland Math. Studies,
34, 1979, P. 179-207. 3.
S. DIEROLF and K. FLORET,
Normen.
4.
S. DINEEN,
ober die Fortsetzbarkeit stetiger
Archiv. der Math., 35, 1980, p. 149-154. Holomorphic functions on strong duals of FrBchet-
Monte1 spaces. cations,
Infinite Dimensional Holomorphy and Appli-
Ed. M.C. Matos, North Holland Math. Studies, 12,
1977, p. 147-166. 5.
S. DINEEN,
Surjective limits of locally convex spaces and their
application to infinite dimensional holomorphy.
Bull. SOC.
Math. France, 103, 1975, p. 441-509.
6. K. FLORET,
Continuous norms on locally convex strict inductive
limit spaces.
7.
Preprint.
M. MATOS, Holomorphically bornological spaces and infinite dimensional versions of Hartogsf theorem. J. London Math. SOC., 2,
17, 1978, P. 363-368.
8. L.A. MORAES,
Holomorphic functions on strict inductive limits. Resultate der Math., 4, 1981, p. 201-212.
Universidade Federal do Rio de Janeiro Instituto de MatemAtica Caixa Postal 68.530 21.944 Rio de Janeiro, RJ, Brasil
-
Functional Analysis, Hobmorphy and Approximation Theory 11, G I . Zapata (ed.) @Elsevier Science Publishers B. K (North-Holland), 1984
NUCLEAR KOTHE QUOTIENTS OF FF&CHET
SPACES
V.B. Moscatelli ( * )
The structure theory of Frechet spaces is, at present, the object of an intensive study not only because of its intrinsic interest, but also because of its applications to approximation theory and to concrete function spaces.
Within this framework, one
is led to problems concerned with the determination of what kinds of subspaces and quotients can be found in arbitrary Fr6chet spaces, and here I shall attempt to sketch briefly the history of one of these problems up to its present state.
In order to introduce the
problem, let us first explain the title. will be infinite-dimensional.
t9l
Background references are [4], [ 8 ] ,
and [ 111. We recall that a Fr6chet space
operators
uk: Ek+l + Ek
(k E N )
is the set of all sequences with the product topology. choose Banach spaces that each
("1
Of course, all our spaces
uk
Ek
(xk)
E
E
is a projective limit of
on Banach spaces, that is, such that
xk =
E
U ~ ( X ~ + ~( L)E N )
is said to be nuclear if we can
and linking maps
uk: Ek+l
3
Ek
such
can be represented as
~
The author gratefully acknowledges partial support from the Italian CNR through a travel grant.
V.B.
P
Given a set K8the space
h(P)
x(p) =
MOSCATELLI
of non-negative sequences
a = (an),
the
is defined as
mn) nE :
an15,
<
for each
m
(an) E P]
with the locally convex topology generated by the semi-norms
Pa(5,)
=
Here we assume, to have a Hausdorff topology, that for each there exists
(an) E P
A sequence basis if for each -
(sn)
with
(x,) x
am > 0.
in a topological vector space
E E
m E N
is a
E
there exists a unique scalar sequence
x = C snxn in E. A basic sequence is a sequence n which is a basis for the closed subspace it generates. such that
Now let
E
be a nuclear Frechet space with a basis
(x,).
Then, by the fundamental Basis Theorem o f Dynin-Mitiagin [ 6 ] , (x,)
is an absolute basis in the sense that the above series con-
verges absolutely for each of the semi-norms defining the topology k that E is of E. From this it follows, putting an = pk (xn) , k isomorphic to the K8the space X(P), where P = ((a,)). The matrix
P
is a representation of the basis
be taken to satisfy
0 5
:a
5
ak+l n
(x,)
and it can always
and the following condition,
known as the Grothendiek-Pietsch criterion:
(*I
for each
k
there is a
j
such that
ak n n an J
C -<
m.
Thus we see that the collection of all nuclear Frechet spaces with x(P)
basis is the same as the collection of K6the spaces
with
P
countable and satisfying ( * ) . Finally, if there exists a continuous norm on that space
E
admits a continuous
x(P),
norm.
E
we say
In the case of a nuclear K6the
this is the same as assuming that
a:
>
0
for all n.
NUCLEAR
KBTHE
QUOTIENTS
OF F ~ C H E TSPACES
313
With a slight abuse of language, from now on by a nuclear K6the space I will mean a nuclear Fr6chet space which has a basis and admits a continuous norm, and the problem under consideration is : Which Fr6chet spaces have nuclear K6the quotients?
REMARK 1.
The continuous norm business is crucial hare.
Already
in 1936 Eidelheit [7] showed that any non-normable Fr6chet space has a quotient isomorphic to
w
(the topological product of
countably many copies of the real line) and, of course, not have a continuous norm. dual
E'
The proof is simple:
as the union of an increasing sequence
w
does
represent the (EL)
of Banach
EA 4 EA+l. Pick elements x' E E;+l-EA; the required quotient is then E/r span(xL)] a spaces, where we can assume
.
REMARK 2.
Nuclearity is also crucial in the sense that the problem
is likely to be much more difficult without it.
Indeed, the answer
is unknown even in the Banach space case and it is a celebrated open problem to know whether every Banach space has a separable quotient
.
Thus, nuclearity rules out Banach spaces, but the above remark points at the difficulty that might lie at the heart of the problem and, indeed, this has been solved s o far only for separable Fr6chet spaces (see (4) below). The problem may be raised, of course, for subspaces as well as quotients and it is instructive to look at the subspace case. Again, nuclearity rules out Banach spaces (but it is an old and classical result that every Banach space has a subspace with a basis) and the subspace problem for Fr6chet spaces was ultimately solved about twenty years ago by Bessaga, PeZczyfiski and Rolewicz
[ 21
, [ 31, who
showed that
314
V.B. MOSCATELLI
(1) a non-normable Frgchet space
if and only if X x w,
with
X
E
E
has a nuclear Kbthe subspace
is not isomorphic to a product of the form
a Banach space (possibly
Note that all closed subspaces of
{O]).
X x w
(X
w.
either of the same form, or Banach or isomorphic to of proof can quickly be summarized as follows.
Banach) are
If
A
(p,)
method is an
increasing sequence of semi-norms defining the topology of first one finds a separable subspace norms
pn
F C E
on which the semi-
are mutually non-equivalent norms.
inductively basic sequences
: k E N)
(x:
(F,P,+~)" such that, denoting by (F,P,+~)",
all embeddings
Xn
Xn + Xn-l
E,
Next, one chooses
in each Banach space
their closed linear spans in are nuclear.
Finally, one
takes suitable linear combinations of elements from the set n (xk : n,k E N) to construct a basic sequence (xk) in F (hence in
E)
whose closed linear span is the required nuclear K6the
subspace. Now let us go back to our problem.
To work directly with
quotients is generally difficult and s o one is tempted to work with subspaces in the dual clude by duality.
E' of a Frbchet space E
and then con-
What I mean is that one is led to represent
as the union of an increasing sequence of Banach spaces to look for a subspace
F
of
E'
EL,
E'
then
on which the EL-norms form a de-
creasing sequence of mutually non-equivalent norms and, finally, to try to construct a basic sequence
(xk)
in
F
this approach does not work because the sequence ed is basic i n each Banach space (strong topology).
E;
as above. (xk)
Well,
thus obtain-
but might not be basic in E'
Indeed, there are examples to the contrary due
to Dubinsky and we refer to [ 5 ] for this as well as for related pathologies.
Of course, this is not surprising, for it occurs all
the time when one deals with inductive limits.
We note however
NUCLEAR K~THE QUOTIENTS
OF
FRFCHET
SPACES
715
that the approach through the dual space works, but with entirely different methods, if the original space is already nuclear (cf
.[5 3 ,
leading to the positive result that (2) Every nuclear Fr6chet space not isomorphic to
w
has a nuclear
Kbthe quotient. Now let us see what can be said on the negative side.
There
is a class of Fr6chet spaces which can be ruled out without any it is the class of those Fr6chet spaces that
assumption whatsoever:
are now called quojections. class of Fr6chet spaces
E
I introduced this class in [lo] as the that are projective limits of a sequence
of surjective operators on Banach spaces.
Obviously, a countable
product of Banach spaces is a quojection, but there are a lot of quojections which are not products (these were called "twisted" in [lo]).
Quojections fail to have nuclear Kbthe quotients
in a very
strong way, for it is not difficult to show that
(3)
If
E
is a quojection and
F
is a quotient of
E,
then:
(a)
F
is nuclear if and only if it is isomorphic to
(b)
F
admits a continuous norm if and only if it is Banach.
By (3)(a),
w
W;
is the only nuclear quojection, so in the
light of (2) and ( 3 ) we can ask: Are quojections the only Fr6chet spaces without nuclear Kbthe quotients? The answer is unknown and in general no more can be said at present (remember Remark 2).
However, there is an important posi-
tive result obtained only recently by Bellenot and Dubinsky [l] under the assumption of separability. generalizes (2):
I t is the following, which
316
V.B.
(4)
A separable Frechet space
and only if Banach spaces
E
has a nuclear Kbthe quotient if
is not the union of an increasing sequence of
E'
EL
MOSCATELLI
with each
being a closed subspace of E' n+l*
EL
F
What we are saying here is that there is a subspace E'
and an increasing sequence
the dual norms
(ph)
(p,)
of semi-norms on
E
of
such that
form a (clearly decreasing) sequence of
F.
mutually non-equivalent norms on
Unfortunately, the proof of
(4) is quite technical and its heavy use of separability points at the difficulty that may be encountered in trying to solve our problem in general.
In proving ( b ) , first one goes over to a
quotient with a continuous norm stated in (4). quence
(d,)
p
and whose dual has the property
Then separability comes in, and we may choose a se-
which is dense in this quotient.
linear span of
(d,),
Calling
the
Eo
we then use the B e s s a g a - P e $ c z y f i s k i - R o l e w i c z
method mentioned above to construct a biorthogonal system
(xn,fn)
such that:
Condition (ii) enables us to extract a subsequence (fn ) j
(fA) such that, if quotient map, then
N =
n
fnl(0)
j
J
)
6(xn
and
6 : Eo
-t
Eo/N
C
is the
is a basis in the completion
(Eo/N)
j
By construction, the latter space is a Kbthe quotient of condition (iii), when reflected on the
),
@(xn
E
-.
and
ensuresnuclearity.
j
Let us remark that the condition of (4) is on the dual of
E.
E'
We know that the dual of every quojection satisfies the
condition.
I s the converse true?
This is at present unknown and
an answer to it would settle our original problem in the separable
NUCLEAR K~~THEQUOTIENTS
317
What we can say is that if the condition of (4) holds, then
case. E’x
OF F ~ C H E TSPACES
(the space of bounded linear functionals on
E’)
is a quojec-
tion (easy to prove) and therefore we can conclude that
(5) Within the class of separable, reflexive Frechet spaces, quojections are exactly those spaces without nuclear KOthe quotients,
REFERENCES 1.
S.F. BELLENOT and E. DUBINSKY, Fr6chet spaces with nuclear KOthe quotients, Trans. Amer. Math. SOC. (to appear).
2.
C. BESSAGA and A. PEgCZYiSKI, On a class of B -spaces, Bull. Acad. Polon. Sci., V . 4 (1957) 375-377.
3.
C. BESSAGA, A . P E g C Z d S K I and S . ROLEWICZ, On diametral approximative dimension and linear homogeneity of F-spaces, Bull. Acad. Polon. Sci., IX, 9 (1961) 677-683.
4.
E. DUBINSKY, The structure of nuclear Fr6chet spaces, Lecture Notes in Mathematics 720, Springer 1979.
5. E. DUBINSKY, On (LB)-spaces and quotients of Frechet spaces, Proc. Sem. Funct. Anal., Holomorphy and Approx. Theory, Rio de Janeiro 1979, Marcel Dekker Lecture Notes (to appear).
6. A.S. DYNIN and B.S. MITIAGIN,
Criterion for nuclearity in
terms of approximative dimension, Bull, Acad. Polon. Sci., 111, 8 (1960)
535-540.
7. M. EIDELHEIT, Zur Theorie der systeme linearer Gleichungen, Studia Math., 6 (1936) 139-148. 8.
H. JARCHOW,
9. G. 10. V.B.
KtlTHE,
Locally convex spaces, Teubner 1981.
Topological vector spaces I, Springer 1969.
MOSCATELLI,
Fr6chet spaces without continuous norms and
without bases,
Bull. London Math. SOC., 12 (1980) 63-66.
318
11.
V.B.
A. PIETSCH,
Nuclear locally convex spaces, Springer 1969.
Dipartimento di Matematica Universita
73100
-
Lecce
C.P.
MOSCATELLI
193
- Italy
Functional Analysis, Holomorphy and Approximation Theory 11, G.I.Zapata ( e d . ) 0 Elsevier Science Publishers B. K (North-Holland), f 984
A COMPLETENESS CRITERION F O R INDUCTIVE LIMITS OF BANACH SPACES
Jorge Mujica(*)
INTRODUCTION By an (LB)-space
X = lim X +
j
we mean the locally convex in-
ductive limit of an increasing sequence of Banach spaces X =
X
*where j
0
U X
.ko
and where each inclusion mapping
Xj
-t
Xj+l
is contin-
j
It is often of crucial importance to know whether a given
UOUS.
(LB)-space is complete o r not, but there are very few criteria to establish completeness for (LB)-spaces, and in many situations these criteria do not apply.
In Theorem 1 we show that if there exists a
Hausdorff locally convex topology
T
on the (LB)-space
with the property that the closed unit ball of each then
X
is complete.
that the space
H(K)
compact subset
K
X = lim X +
j
X j is ‘T-compact,
As an easy consequence of Theorem 1 we prove of all germs of holomorphic functions on a
of a complex Frechet space is always complete.
This result had already been established by Dineen more complicated way.
[4] in a much
In Theorem 1 we also give a sufficient con-
dition f o r an (LB)-space to be the strong dual of a quasi-normable Fr6chet space.
As an application of this criterion we show that if
K is a compact subset o f a complex quasi-normable Frechet space, then H(K)
is the strong dual of a quasi-normable Fr6chet space.
This improves a previous result of Aviles and the author El].
(*)
This research, partially supported by FAPESP, Brazil, was performed when the author was a visiting lecturer at the University College Dublin, Ireland, during the academic year 1980-1981.
JORGE M U J I C A
I would like to thank Richard Aron and S6an Dineen for many helpful discussions that we had during the preparation of this I would also like t o thank Klaus-Dieter Bierstedt, for a
paper.
problem we had discussed some time ago was one of the principal motivations for this research,
That problem, that up to my knowledge
Is every regular (LB)-space
still remains open, is the following: comple te7
1. T m M A I N RESULT
A basic tool in this paper is Berezanskiifs inductive topo-
convex space then we will denote by
Yf
Y
If
logy on the dual of a locally convex space.
is a locally Y‘
the dual
Y, endow-
of
Y! =
ed with the locally convex inductive topology defined by
,
= l$m(Y’)
where
varies among all neighborhoods of zero in Y.
V
VO
This inductive topology is stronger than the strong topology. Berezanskii
f2] o r Floret
ductive topology on
.
f7]
See
KCIthe has also studied this in-
Y‘ in the case where
Y
is a metrizable local-
l y convex space, and he has proved that in that case the space is always complete.
See KCIthe f 9 , p.4001.
used without further reference. topology
T~
We should remark also that the
introduced by Nachbin on the space
continuous m-homogeneous polynomials on Berezanskii topology in the case
THEOREM 1. (a)
X
Let
X = lim X +
j
rn = 1.
Y
of all
reduces to the See Dineen [ 5 , p.511.
be an (LB)-space.
with the property that the closed unit ball
ticular
P(?)
If there exists a Hausdorff locally convex toplogy
T-compact, then
X
X =
Yi
is complete.
Yf
This result will be
K
j
of each
for a suitable Fr6chet space
Y.
on
7
X
j
is
In par-
A COMPLETENESS CRITERION
If in addition
(b)
X
has a base of 7-closed, convex, balanced
neighborhood of zero, then
Y
and
321
X
is actually the strong dual of
Y,
is a distinguished Frechet space. The statement and proof of part (a) are nothing but an
PROOF.
adaptation of a characterization of dual Banach spaces, due to N g [ll, Th.11.
Ng's result is, on the other hand, a variant of an
old result of Dixmier [ 6 , Th. 191.
X + (X,T)
First of all we observe that the identity mapping is continuous. forms on
X
denote the vector space of all linear
whose restrictions to each set
Y
If we endow sets
Y
Now,let
are 7-continuous.
j
with the topology of uniform convergence on the
Y
then it is clear that
Kj,
K
is a Frechet space (it is
actually a closed subspace of the strong dual J: X
-+ Y*
ping.
(Y* = algebraic dual of
Since
Y
3
( x , ~ ) 'and
separates the points of Let
'
and
a
(X,Y)
and
every
j,
X
since
Y) r
x).
of
X;
Let
denote the evaluation mapis Hausdorff, we see that Y
and hence the mapping
is injective.
J
denote the polars with respect to the dual pairs
(Y,Y*), respectively. we see that
J
maps
X
Since clearly
J(K.) J
C KO*
j
for
Yf. Now,the
continuously into
mapping J:
(K.,T) J
-t
(Y',U(Y',Y)) Y.
is clearly continuous, by the definition of Hence dense in J(Kj)
J(Kj)
(J(Kj))**,
= (J(K.))*'. J
is
u ( Y ' ,Y)-compact.
Since
J(Kj)
is U(Y',Y)-
by the Bipolar Theorem, we conclude that Thus,since clearly
(J(Kj))*
= KQ
,
we conclude
that J(K.) J
and hence
J
= (J(K.)*' J
= K
0 .
j
is a topological isomorphism between
T o show (b) let
U
X
and
Y i .
be a 7-closed, convex, balanced neigh-
722
JORGE MUJICA
borhood of zero in Now, since
I c Y
J
,
0 0
X.
Then
U = U
X
onto
Y' we see that
maps
by the Bipolar Theorem. = @*
J(4)'
for each
and hence J(U) = J(Uoo) = Uo*.
Since
Uo
YL
0-neighborhood in tinuous.
Y
is clearly bounded in
we conclude that
and hence the mapping
5-l: Y;
Yf + YL
Since the identity mapping
J(U)
+ X
is a is con-
is always continuous,
the proof is complete. T o verify the second condition in Theorem 1, the following
lemma will be useful. LEMMA 1.
X = lim X
Let
+
j
be an (LB)-space, and assume that there
exists a Hausdorff locally convex topology
7
on
X
with the fol-
lowing properties: (i) (ii)
The clsed unit ball
K
j
for each 0-neighborhood
of each
U
X
X
in
is
j
there exists a sequence
of 7-closed, convex, balanced 0-neighborhoods
vj
n
K. c J
compact;
Vj
in
X
such that
u.
Then
X
has a base of 7-closed, convex, balanced neighbor-
hoods of zero. PROOF.
Let
U
X.
be a 0-neighborhood in
We choose a sequence
m
(ej)
of positive numbers with
C
Cj
4
1
such that
j=O
m
where
C
j=o
m
E .K
denotes the set
~j
u n=O
n
C
j=o
By (ii) we can find a sequence of 0-ne ighb orhood s
Vj
in
X
EjKj
closed, convex, balanced
such that
c -1 2u .
A COMPLETENESS CRITERION
323
Define n
m
(3)
V =
fl
( c
n=O
V
Then
Since
i s a 0-neighborhood
X.
show t h a t
V t U.
in
3'
i s b a r r e l l e d we c o n c l u d e t h a t
X
and choose
n
V
is suffices t o
To c o n c l u d e t h e proof
z E V
Let
Vn).
i t i s convex and b a l a n c e d , and
K .
absorbs every
by ( i ) i t i s r - c l o s e d .
+
EjKj
j=O
such t h a t
z
E
nKn.
BY ( 3 ) we can w r i t e
n z = x+y,
(4)
with
x E
C
e .K.
j=O
and
J J
y
E
Vn
.
Hence
E
y = z-x
(5)
nKn
n
+
C j=O
c .K. J J
C
(n+l)Kn
m
C c 1 *: 1 and s i n c e we may assume, w i t h o u t l o s s of g e n e r J j=O a l i t y , t h a t t h e sequence (Kj) i s increasing. Thus from ( b ) , ( 5 ) since
and ( 2 ) we conclude t h a t
Y
E vn n
(n+l)Kn c
1 2 u
and t h e r e f o r e n
by
(4)
,
( 6)
and (1). Thus
V c U
and t h e proof
i s complete.
2. APPLICATIONS TO COMPLEX A N A L Y S I S If
i s a compact s u b s e t o f a complex F d c h e t s p a c e
K
then t h e space
#(K)
of a l l germs of holomorphic f u n c t i o n s on
E K
i s d e f i n e d a s t h e l o c a l l y convex i n d u c t i v e l i m i t
where
(Uj)
borhoods of
i s a d e c r e a s i n g fundamental sequence o f open n e i g h K
and where
#"(Uj)
d e n o t e s t h e Banach s p a c e of a l l
324
JORGE MUJICA
U
bounded holomorphic functions on
j'
with the norm of the supremum.
This (LEI)-space has received a good deal of attention in recent years and we refer to the survey article of Bierstedt and Meise [ 3 ] or to the recent book of Dineen c5] for background information and open problems concerning #(K)
The problem of completeness of
#(K).
remained open for several years until it was finally solved
by Dineen [4, Th. 81, who proved that the space complete.
is always
#(K)
Dineen's proof is quite complicated, but we can now
obtain Dineen's result as an easy consequence of Theorem 1. THEOREM 2. E. #(K)
Then
Let
#(K)
be a compact subset of a complex Fr6chet space
K
= Yi
Y.
for a suitable Frechet space
In particular,
is complete.
Let
Before proving Theorem 2 we fix some notation.
#(U)
U
denote the space of all holomorphic functions on an open subset of a complex locally convex space then we let expansion of set
f(n)(x) f
11 f(n) "A,B =
denote the
If
nth
x.
sup
I f(n) ( x ) ( s ) I
f E #(U)
x E U
and
term in the Taylor series
f E #(U),
at
xEA
If
E.
A C
U
and
B
C
E
then we
.
sE B
PROOF OF THEOREM 2.
Let
of open neighborhoods of pology on
W(Uj),
(Uj) be a decreasing fundamental sequence Let
K.
denote the compact-open to-
T~
and by abuse of notation let
locally convex inductive topology on
As Nicodemi [12] has remarked, for the seminorms uous on
(#(K),T~)
f +
I f(n)(x)
for all
#(K)
(#(K),T~) (s)
I
n E N,
T~
also denote the
which is defined by
is a Hausdorff space,
are well-defined and continx E K
and
s E E.
other hand, by Ascoli theorem, the closed unit ball of compact in
(#(Uj),To),
and hence in
(#(K),T~).
On the Wa(Uj)
is
An application
A COMPLETENESS CRITERION
325
of Theorem 1 completes the proof. REMARK.
(#(K),T~)
The locally convex space
has recently been
studied by the author [lo] in great detail and it turns out that
Y
the Frechet space of
(H(K)
,To)
that appears in Theorem 2 is the strong dual
-
Avil6s and the author [l, Th.21 have shown that
#(K)
sa-
K
is a
tisfies the strict Mackey convergence condition whenever compact subset of a complex quasi-normable Frechet space.
This
result can be improved as follows: THEOREM 3.
Frechet space
be a compact subset of a complex quasi-normable
K
Let E.
Then
#(K)
is the strong dual of a quasi-
normable Fr6chet space. We refer to Grothendieck f 81 for information concerning quasi-normable spaces and the strict Mackey convergence condition. T o prove Theorem 3 we need the following lemma, which is essentially a reformulation of the proof of [l, Th.21. Let
LEMMA 2.
Frgchet space
K
be a compact subset of
E.
a
complex quasi-normable
Then there exists a decreasing fundamental se-
quence of open, convex, balanced 0-neighborhoods that, if we let
Xj
U
denote the closed unit ball of
for each 0-neighborhood
L
in
convex, balanced 0-neighborhoods
#(K)
k
j
in
E
such
Hm(K+Uj),
then
there exists a sequence of in
#(I()
with the following
properties: (i) (ii) PROOF.
each IJ
k
n X J. c
Since
E
is closed in
(#(K) , T ~ ) ;
L
j.
for every
is metrizable and quasi-normable, we can induct-
ively find a fundamental sequence of open, convex, balanced O-neighborhoods
U . in J
E
such that:
326
JORGE MUJICA
~ c uj ~
(a)
2
(b)
for every
set
B
E
in
IJ
j;
6
and for every
j
there exists a bounded
0
7
2Uj+l C B + 6U
such that
xj
Let
+ ~ for every
j ’
denote the closed unit ball of
is a 0-neighborhood in
Since
we can find a sequence of positive
#(K)
3 c j$t
Hm(K+2Uj).
c IJ
numbers
E
f E X j .
Then using (a) and the Cauchy integral formulas, we can
j
such that
Fix
j.
and fix
j
N E N:
write, for each
Now, since
for every
fE X j
the Cauchy integral formulas imply that
If(n)(x)(s)\
L
x E K
for all
1
and
s
E 2U j
and hence that (2)
I(f(n)(x))(k)(s)(t)I
for all
1
4
Next we note that by (b), given ed such that each
s E
2U
j+l
s = b + 6t,
Hence, for each
x E K,
x E K
s,t E U j ’
and
6 > 0 there exists B
C E
bound-
can be written in the form
with
b E B
t E Uj.
and
(2) implies that
and we conclude that
m
First we choose
N E N
such that
C 2’n n=N _. .
m
choose
0< 6 < 1
such that
C bn k=1
5
N
4
0 j+l
.
If
and next we
B
is the bounded
A
set associated with
COMPLETENESS CRITERION
327
in (b) then from (1) and ( 3 ) we conclude
6
that N-1
II f11K+2U j+l
(4)
J+1
n=O
*
If we define
then Ir .
is a convex, balanced 0-neighborhood in
J
closed in
#(K),
is
l.rj
(#(K),'T~) and by (4)
L
. n x J. c
J
30 j+lx j+l c b.
The proof of Lemma 2 is now complete. PROOF OF THEOREM
3.
Let
neighborhoods of zero in
(Uj) be the fundamental sequence of E
given by Lemma 2.
know that the closed unit ball of each
gm(K+Uj)
Since we already is compact in
(~(K),T~), then from Lemma 1 and Lemma 2 we conclude that
g(K)
has a base of convex, balanced neighborhoods of zero, each of which is closed in #(K)
(#(K),T.).
Then we conclude from Theorem 1 that
is the strong dual of a Fr6chet space
[l, Th.21
#(K)
we conclude that complete
.
Y.
But since by
satisfies the strict Mackey convergence condition, Y
must be quasi-normable.
The proof is now
JORGE MUJICA
REFERENCES
1.
P. AVILXS and J. MUJICA, Holomorphic germs and homogeneous polynomials on quasi-normable metrizable spaces, Rend. Mat. 10
2.
(1977)s 117-127.
J.A. BEREZANSKII,
Inductively reflexive locally convex spaces,
Soviet Math. Dokl. 9 (1968), 1080-1082.
3.
K.-D. BIERSTEDT and R . MEISE,
Aspects of inductive limits in
spaces of germs of holomorphic functions on locally convex spaces and applications to a study of
(H(U),T~), in
Advances in Holomorphy (J.A. Barroso, ed.),
North-Holland,
Amsterdam, 1979, p. 111-178.
4.
S. DINEEN,
Holomorphic germs on compact subsets of locally
convex spaces, in Functional Analysis, Holomorphy and Approximation Theory (S. Machado, ed.),
Lecture Notes in Math. 843,
Springer, Berlin, 1981, p . 247-263.
5.
S. DINEEN,
Complex Analysis in Locally Convex Spaces,
North-
Holland, Amsterdam, 1981.
6. J. DIXMIER, S u r un Theoreme de Banach, Duke Math. J. 15 (1948), 1057-1071.
7. K. FLORET,
h e r den Dualraum eines lokalkonvexen Unterraumes,
Arch. Math. (Basel) 25 (1974), 646-648.
8.
A.
GROTHENDIECK,
Sur les espaces (F) et (DF), Summa Brasil.
Math. 3 (1954), 57-123.
9. G. KdTHE, Topological Vector Spaces I, Springer, Berlin, 1969. 10.
J. MUJICA,
A new topolo#w
on the space of germs of holomorphic
functions (preprint). 11.
On a theorem of Dixmier,
K.F. NG, 279-280
0
Math. Scand. 29 (1971),
A COMPLETENESS CRITERION
12.
0. NICODEMI,
329
Homomorphisms of algebras of germs of holomorphic
functions, in Functional Analysis, Holomorphy and Approximation Theory (S. Machado, ed.), Lecture Notes i n Math. 843, Springer, Berlin, 1981, p . 534-546.
Department of Mathematics University College Dublin Belfield, Dublin 4 Ireland and Instituto de Matemgtica Universidade Estadual de Campinas Caixa Postal 6155
13100 Campinas, SP (current address)
-
Brazil
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Functional Analysis, Ho lo rno rp hy and Approximation Theory II, G I . Zapata ( e d . ) 0 Ekevier Science Publishers B. V. (North-Holland), 1984
ABOUT THE CARATHEODORY COMPLETENESS OF ALL REINHARDT DOMAINS Peter Pflug
It is well known that in the theory of complex analysis there are different notions of distances on a bounded domain
G
in
Gn,
for example, the Caratheodory-distance dealing with bounded holon
morphic functions, the Bergmann-metric measuring how many Lc-holomorphic functions do exist or the Kobayashi-distance describing the sizes of analytic discs in
G.
A survey on these notions, also ge-
neralized to infinite dimensional holomorphy, can be found in the book of Franzoni-Vesentini [ 31
.
The main problem working with these distances is to decide which domain
G
is complete w.r.t.
one of these distances.
There
is a fairly general result for the Bergmann-metric due to T. Ohsawa and P. Pflug [ 6 , 7 ]
which states that any pseudoconvex domain with
C1-boundary is complete w.r. t. the Bergmann-metric.
O n the other
hand it is well known that the Caratheodory-distance can be compared with the other two, in fact, it is the smallest one, but there is n o relation between the Bergmann-metric and the Kobayashi-metric [Z]. Thus the question remains which domains are complete w.r.t. Caratheodory-distance or, at least, w.r.t.
the
the Kobayashi-distance.
In this short note it will be shown that any bounded complete Reinhardt domain
G
which is pseudoconvex is complete in the sense
of the Caratheodory-distance; in fact, it will be proved that any Caratheodory ball is a relatively compact subset of
G.
Using the
above remark on the comparability of the distances it is clear that
332
PETER PFLUG
those domains are also complete w.r.t.
the two other distances.
First, some definitions should be repeated. DEFINITION 1. A domain
zo E G
domain if for any
for
1 4 i
L
n)
G C Cn
is called a complete Reinhardt {z E Cn:
the polycylinder
has to be contained in the domain
lzil
G
lzil
G.
It is well known that a complete Reinhardt domain pseudoconvex iff
S
G
is
is logarithmically convex which means the set
loglGl := {x E Rn: for
,...,loglznl)]
x 3 z E G: x = (loglzll
is convex in the u s u a l sense, DEFINITION 2. in
Let
G
be a domain in
Cn
then, for points
z’, z”
G, CG(Z’ ,z”) := sup { 1 F log 1 1
+
-
If(Z”)I
.
lf(z/’)l
*
f:G+E holomorphic with f(z‘)=O] is called the Caratheodory distance between in the future
E
It is easy to check that
on a bounded
E =
denotes the unit disc
G; hence
(G, CG(
,
CG(
,
))
)
z’
and
{ I E C:
z ” ; here and
1x1
< 13.
is, in fact, a distance
is a metric space.
Asking
whether this space is complete it suffices to establish that any Caratheodory ball
{z E G: CG(z,zo)
latively compact subset of dory-completeness of
G;
< M]
around
zo E G
is a re-
this is called the strong Caratheo-
G.
It is well known that a Caratheodory-complete domain (i.e.
(G, CG(
,
))
is complete) has to be Hm(G)-convex
and a domain of
bounded holomorphy [ 81 ; the converse, in general, is false.
In fact
m
there exists a H (G)-convex domain of bounded holomorphy which is not Caratheodory-complete.
On the other hand it should be repeated
ABOUT THE CARATHEODORY COMPLETENESS OF ALL REINHARDT DOMAINS 333
m
that any pseudoconvex domain with a smooth boundary is H (G)-convex and also a domain of holomorphy fl]. following problem:
This remark may induce the
is any pseudoconvex domain with smooth boundary
Caratheodory-complete o r , at least, complete w.r.t. distance? THEOREM.
the Kobayashi-
Here only a simple partial result can be presented, G , . which is
Any bounded complete Reinhardt domain
pseudoconvex, is strongly Caratheodory-complete. Without loss of generality we can assume that
PROOF.
tained in the unit-polycylinder.
is con-
Then assuming the proposition is
(z"] C G
false there exists a sequence
G
with
zv
-t
zo
E G
such
n
that
CG(zv,O)
4
M <
v
E
First,
IN.
f 0
zz
is
v=1
.
assumed
xo := ( l o g 1 z y l
Then
for all
m
convex set
loglGl.
,... ,log1 z z ] )
belongs to the boundary of the
L: Rn + R
Hence a linear functional
can be
found such that
n
L(x) =
c
c
Sixi < L(X0) =:
i=l for all
x E loglGl
Reinhardt domain tive
G
with
C s 0.
Using the completeness o f the
it's clear that the numbers
5,
are nonnega-
. Assume that
should be zero
-
to find integers
-
15
L
are positive the remaining ones ik then, compare [ 4 ] , it is possible, for any N E LN", B 1 , N 9' * ' "k,N
.1 1 5 -
N%
JV
-
Zil, . . . , 5
.
with
Provided
N
1
5
kN
5
Nk
and
large enough it follows:
pv,N > 0 . Defining fN(z) = e
' %. ZBl,N i1
... zBk,N ik
and
334
PETER PFLUG
w e have o b t a i n e d holomorphic f u n c t i o n s following inequalities f o r
fN(.)I
z
E G,
gNx G near
z
+ E z
0
for which t h e
,
can be proved:
k. .
= -C%
+
c
v=1
Bv,N
iV
= -kJC-L(l0glz1l 2
-N
k
loglz ,..,,log
(C-L(loglzll,
...,l o g
or
llfNll- i t i s enough t o l o o k f o r t h o s e z E G w i t h : G n k zv f 0 and c B ~ log12 , ~ 2 C-$ which i m p l i e s v= 1 v= 1 iv
To e s t i m a t e
I
7
Using t h e c h o i c e of t h e
B
and
1s
l a r g e enough t h e f o l l o w i n g
N
i n e q u a l i t y can be found:
Hence one h a s r e c e i v e d for a l l I f N ( z ) I z exp
E G:
z
k"$ = -
exp
%N Combining t h e above e s t i m a t e s one ends up
k*a . N (N
and
v
l a r g e enough)
with:
k
V
l g N ( z v ) I 2 expC-N ( C - L ( l o g l z l l , * . * t l o g l z i I 1)
kM" ka -N -T '
from which f o l l o w s e2M-1 1>e2M+1
IgN(zv)I
7
I n t h e r e m a i n i n g c a s e z o can be assumed a s n f 0. with v=1
'*
zo = ( z y ,
...,z:,O,. ..o)
ABOUT THE CARATHEODORY COMPLETENESS O F ALL REINHARDT DOMAINS 335
Then
{Z'E
GI:=
z'E n ( G ) } ,
Q ! ' :
n: G
where
a'!
-t
denotes the usual
projection, can be easily recognized as a bounded pseudoconvex complete Reinhardt domain in
z0' := n(zo).
with boundary point
CL
Hence it follows: CG/(0,n(zv))
CG(O,zv)
S
5
M
which contradicts the case discussed before. Therefore the proof of the theorem is complete.
It should be mentioned that by a lemma due to E . Low (reported by K . Diederich) that the strong Caratheodory-completeness m
implies the sequential H -convexity one has the following consequence : COROLLARY.
complete Reinhardt domain sequence
{zl"]
E
with
f: G
-I
(z"}
Any sequence of points with
G
z
V -t
in a bounded pseudoconvex zo E aG
contains a sub-
such that there exists a holomorphic function lv lim)f(z ) I = 1 and f(0) = 0 .
For the convenience of the reader a proof will be presented. PROOF.
I t is easy using the strong Caratheodory completeness of G
to find a subsequence with
fv(0) = 0
Setting
-
fv(z) :=
zv
and holomorphic functions fv:G
-I zo
fv(zv) > 1
and
fv( z) +I - fv0-l
- ev
where
1 7 fk(z).
C k=l
2
Then an easy exercise proves that for any
v E N,
lfv(z)I
5;
0< R < 1
R
22v.
one can define m
all
K C G
-K
such that, f o r all
is valid.
gives a holomorphic map
F:
G
-I {
-
compact
zE K
and
This remark implies that the
above series converges uniformly on compact subsets of F
E
EV
F ( z ) I=
there exists a number
->
-t
z
G.
E C : Rez > O} =: H,
Hence for
PETER PFLUG
336
which F(0) = 1
and
IF(z
v
)I
fv( 2")
2
2
7
zV
V*
hold. By
?(z)
=
a holomorphic function
?:
G
+ E
is constructed
with: 1F(0) = 0
and
l?(zv)I
2
1
d l
IF(ZV)I
1+
1 v-tm IF(ZV>
I
which ends t h e proof. Another application is concerned with the Serre-problem: COROLLARY.
A locally trivial holomorphic fibre bundle with Stein
base, whose fiber is a bounded pseudoconvex complete Reinhardt domain, is already a Stein space.
ABOUT THE CARATHEODORY COMPLETENESS OF ALL REINHARDT DOMAINS 337
REFERENCES
1.
CATLIN, D.:
Boundary behaviour of holomorphic functions on
pseudoconvex domains; Journal Diff. Geometry 15, 605-625
(1980). 2.
DIEDERICH, K. and E. FORNAESS:
Comparison of the Bergmann and
the Kobayashi metric; Math. Annalen 254, 257-262 (1980).
3.
FRANZONI , T. and E
. VESENTINI :
Holomorphic and invariant
distances; Notas de Matemdtica 69 (1980).
4. GAMELIN. T.W.:
Peak points for algebras on circled sets;
Math. Annalen 238, 131-139 (1978).
5. KOBAYASHI, S.:
Geometry of bounded domains; Trans. Amer. Math.
SOC. 92, 267-289 (1959).
6. OHSAWA, T.:
A remark on the completeness of the Bergmann
metric; Proc. of the Japan Academy
7. PFLUG, P.:
57, 238-240 (1981).
Various applications of the existence of well grow-
ing holomorphic functions; in Functional Analysis, Holomorphy and Approximation Theory ed. by J.A. Barroso (1982).
8.
SIBONY, N.:
Prolongement analytique des fonctions holomorphes
born6s; SQminaire Lelong, Annee 1972-1973, 44-66 (1974). ADDED IN PROOF.
For dimension n=2 our result has been proved independently also by J.-P. Vigu6 in an article
“La distance de Caratheodory n f e s t pas intgrieure” which will appear in Resultate der Mathematik.
Universittit Osnabriick -Abte i lung Vecht a F achb ereich
-
Naturwissenschaften/Mathematik
Postfach 1349
-
D-2848
Vechta
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Functional Analysis, Holotnorpliy and Approximation 7heory 11, G.I. Zapata ( e d . ) 0 Elsevier Scieiice Publishers 5.V. (North-Holland), 1984
339
BE ST SIMULTANEOUS APPROXIMATION
JOZO B .
(F,I *
Throughout this paper dean valued division ring, and dean normed space over DEFINITION 1.
Let
We denote by x
in
M.
I)
is a non-trivial non-archime-
(E,ll*ll) is a non-zero non-archime-
(F,l * I ) .
M c E
A best approximation of
Prolla
be a closed linear subspace, and x
PM(x)
M
is any element
y E M
x E E.
such that
the set of all best approximations of
There are two problems to be considered, once
x
and
M
are given: (1)
PM(x) f $,
When is
i.e., the existence of best approxima-
tions ; (2)
When
PM(x)
contains no more than one element, i.e. the
uniqueness of best approximations. Let us start with the second problem, which in the non-archimedean case has a very simple solution: no uniqueness.
[31
M f [O},
there is
More exactly, we have the following result (Monna
1:
THEOREM 2. x E E, r =
when
x @ M,
11 x-y((,
M c E
Let
if
then
be a closed linear subspace.
y E pM(x)
t E PM(x).
and
t E M
with
For every
)It-yll < r,
where
3 40
JOKO B . PROLLA
PROOF.
Since
x $? M ,
and
E M,
y
r = IIx-t/l is
the distance
>O.
By the strong triangle inequality,
I1 x-tll t E E
for all
= IIX-YII =
Ilt-yll
such that
< r.
t E PM(x).
Hence
It remains the problem of existence of best approximations. DEFINITION 3. if pM(x)
A closed linear subspace
M
is called proximinal
C E
contains at least one element for all
x E E.
The definition above poses two problems: Let
(i)
M c E
be a closed linear subspace.
conditions on x E E (ii)
M
so
that
M
Give sufficient
is proximinal, i.e., every
has at least one best approximation in
Give sufficient conditions on
E
so
M.
that every closed
linear subspace is proximinal. Concerning problem (i), one has the following result of Monna
[ 3 ] , who introduced the notion of orthogonal projection for non-archimedean spaces. DEFINITION 4.
A continuous linear map
P:
E
-t
is called a continuous linear projection from
E
such that
E
onto
P
2
= P
P(E).
It follows that, f o r any non-zero continuous linear projection
P
one has
1
5
IIPII.
A continuous linear projection
projection onto approximation of
M = P(E), x
in
if foF all
P
is called an orthogonal
x E E,
Px
is a best
M.
It follows that, for any orthogonal projection
IIPIl
5
1.
Indeed, for any
x E E:
P
one has
BE ST SIMULTANEOUS APPROXIMATION
approximation of
P
jection
[&I,
Monna
x
M.
in
Hence for any non-zero orthogonal pro-
IIPlj = 1.
one has
341
Let us prove the converse.
(See
p.478.)
THEOREM 5 .
Every continuous linear projection of norm one is an
orthogonal projection. PROOF.
M = P(E)
Clearly
y = Py.
Hence
Px = x
M
is the set of all
is closed.
Let
is a best approximation of y E M
for every
IIx-Px/l S dist(x;M).
and
is a best approximation of
COROLLARY 6. M = P(E) Then
Let
x
in
x E M,
If M.
If
M
Px E M ,
Since
x
&'
then M,
then
x
in
Ilx-Px/l = dist(x;M),
M.
be a closed linear subspace of
for some continuous linear projection
M
such that
we have
that is Px
x E E.
y E E
P
E
with
such that ) ) P I / = 1.
is proximinal.
COROLLARY 7.
Every spherically complete linear subspace is proxi-
minal. PROOF.
By [6], 6.10, given a spherically complete subspace M
there is a continuous linear projection COROLLARY 8.
Assume that
(F,]- 1
)
Apply Theorem 6.12, [ 61
onto
M
with
IIPII = 1.
is spherically complete.
every finite-dimensional subspace of PROOF.
P
E
# (01,
Then
is proximinal.
.
Let us now consider the problem of best simultaneous approximation. DEFINITION and
B
7. Let
(E,11
*)I )
be a normed space over
be a bounded subset of
E.
(F,I
I ),
G
C
Define the relative Chebyshev
E,
342
PROLLA
JOXO B.
r a d i u s of
(with respect t o
B
G)
radG(B) = i n f sup IIg-flI. &G f€B If
G
= E,
t h e n we w r i t e radE(B) = r a d ( B )
and c a l l i t t h e Chebyshev r a d i u s of The elements
go
E
where t h e infimum i s a t t a i n e d a r e c a l -
G
l e d r e l a t i v e Chebyshev c e n t e r s of
d e n o t e by
centG(B) G = E,
If
B.
(with respect t o
B
t h e s e t of a l l such
and we
G),
go € G .
t h e n we w r i t e centE(B) = c e n t ( B )
and c a l l i t t h e s e t of Chebyshev c e n t e r s of We say t h a t
in
E
Since
if
B = {f}
h a s t h e r e l a t i v e Chebyshev c e n t e r p r o p e r t y
G
centG(B) f
$
f o r a l l non-empty bounded s e t s
i s bounded, any subspace
Chebyshev c e n t e r p r o p e r t y i n When
G = E,
B C E,
subset erty in
E,
i.e.
B.
and if
E
cent(B) E
we say t h a t
G
C
E.
which h a s t h e r e l a t i v e
i s proximinal i n
# $
B
E.
f o r e v e r y non-empty bounded
h a s t h e r e l a t i v e Chebyshev c e n t e r propE
admits Chebyshev c e n t e r s .
The f o l l o w i n g r e s u l t g e n e r a l i z e s C o r o l l a r y 7 , and w i l l be given a d i r e c t proof.
THEOREM 8.
Every s p h e r i c a l l y complete l i n e a r subspace o f a non-ar-
chimedean normed space h a s t h e Chebyshev c e n t e r p r o p e r t y . PROOF.
Let
(E,ll
*)I )
b e a non-archimedean normed space and l e t
b e a s p h e r i c a l l y complete subspace. bounded s u b s e t .
For each
g E G,
Let put
B c E
b e any non-empty
G
BE ST SIMULTANEOUS A P P R O X I M A T I O N
C
Consider t h e f a m i l y
C
The f a m i l y and
Since
say t h a t
for a l l
G,
then f o r a l l
g E G.
for a l l
for a l l
~ ~ g o -S g p~( g~ ) ,
g
f E B,
g E G. g E G.
go E G
This i s equivalent t o
Now
Hence
sup
11 g o - f / ( <
fEB T h i s proves t h a t COROLLARY 9 .
Indeed, i f
i s s p h e r i c a l l y complete, t h e r e i s some
G
go E B ( g ; p ( g ) )
such t h a t
given by
G
has the binary i n t e r s e c t i o n property.
belong t o
g'
of c l o s e d b a l l s on
343
go
i n f p ( g ) = i n f sup IIg-fll. gfG &G fEB
E centG(B).
Every f i n i t e dimensional subspace of a non-archimedean
normed space over a s p h e r i c a l l y complete valued d i v i s i o n r i n g (F,I
01)
h a s t h e Chebyshev c e n t e r p r o p e r t y . COROLLARY 10.
Every s p h e r i c a l l y complete n.a.
normed space admits
Chebyshev centers.. Let
X
c l o s e d s u b a l g e b r a s of
C(X;R)
tended by Smith and Ward b r a of if
C(X;IR)
A c C(X;R)
I t i s w e l l known t h a t
be a compact Hausdorff space.
[TI,
a r e proximinal.
T h i s r e s u l t was ex-
who proved t h a t every c l o s e d subalge-
h a s i n f a c t t h e Chebyshev c e n t e r p r o p e r t y i n C ( X ; R ) :
i s a c l o s e d s u b a l g e b r a , and
B C C(X;R)
i s any non-
344
J O X O B. PROLLA
empty bounded subset, then
# $.
centA(B)
(See Theorem 1,
[TI).
O n the other hand, if one considers vector-valued continuous func-
tions, i.e., if
X
is a normed space, then for suitable space
C(X;E)
admits centers.
real Hilbert space, then Ward
f81.)
E
is as before a compact Hausdorff space and E
(over the reals), the
For example, if
C(X;E)
E
an arbitrary
admits centers.
(See Theorem 2 ,
Another result true for vector-valued functions is the every Stone-Weierstrass subspace of
following:
minal, for suitable
E.
C(X;E)
is proxi-
For example, this is true if
E
is a
or if
E
is a
R
Lindenstrauss space over
(see Blatter fl]),
uniformly convex Banach space over
R
or
C
(see Olech f 5 ]
,
The-
orem 2). Let
D E F I N I T I O N 11.
(E,11
011
X
be a compact Hausdorff space and let
be a normed space over a non-archimedean non-trivially
)
valued division ring
(F,
I * I ).
A closed vector subspace W C C(X;E)
is called a Weierstrass-Stone subspace if there exists a compact Hausdorff space
Y
and a continuous surjection W =
Clearly,
W
{ g a r :
H:
X
3
Y
such that
g E C(Y;E)].
contains the constants.
The results of [ 6 ] , $ 5 , allow a characterization of the Weierstrass-Stone subspaces, since
(F,I
non-trivially valued division ring.
-1)
is a non-archimedean
Indeed, let
W C C(X;E)
be a
Weierstrass-Stone subspace and let A = {ban; b Then
A
is a subalgebra of
such that dulo
X/A.
fact that
(n-'(y);
y E Y]
Moreover,
W
W
E C(Y;F)}.
C(X;F),
containing the constants and
is the set of equivalence classes mois an A-module.
is closed, if follows that
B y [ 6 ] , Th. 5 . 5 ,
f E C(X;E)
and the
belongs to
W
BEST SIMULTANEOUS APPROXIMATION
if, and only if,
345
n-1(y),
is constant on each equivalence class
f
Y E Y. Conversely, let f E C(X;E)
where Let
A
be the vector subspace of all
W C C(X;E)
which are constant on each equivalence class modulo X/A, is some subalgebra containing the constants.
C(X;F)
C
be the quotient space of
Y
be the quotient map. fE W
Clearly, each
Y
Then
X
modulo
factors through
c {goT, g E C ( Y ; E ) ] .
and let
X
TT:
-i Y
is a compact Hausdorff space.
definition of the quotient topology, W
X/A
f =
i.e.,
TT,
g E C(Y;E).
B y the
g a n .
Hence
Conversely, each function of the form
gon
is clearly continuous and constant on each equivalence class modulo X/A. PROPOSITION 12. and let
W
Let
X, E
and
(F,I
-1)
be as i n Definition 11,
be a Weierstrass-Stone subspace.
C C(X;E)
A 8 E
and
are given by Definition 11.
and
TT
PROOF.
f E W
Y
Clearly,
is an A-module, and by A 8 E.
belongs to the closure of
valence class
n -1(y),
constant value on
A 8 E
Then
Y.
throughout
g =
y E Y
of on
n
(ban) 8
v
v
where
C(X;E),
and any
-1
y),
DEFINITION 13. cp
Indeed, given any equi-
> 0, consider the
6
A % E
b = 1
and is equal to
f
-1 TT (y): IIg(x)-f(x)/l
ping
5.5, [ 61 , each
and the constant function
belongs to
is
A = {ban; b E C ( Y ; F ) ] ,
the closure of
in
W
Then
from
Let X
X
Z
and
= 0 <
E
be two topological spaces.
into the non-empty subsets of
Z
A map-
is called a
carrier. A carrier
cp
from
X
into the non-empty subsets of
said to be lower semicontinuous if
[x E X ; cp(x) fl G
# $1
Z
is
is open
346
in
JOXO B. PROLLA
X
for every open subset
G C 2.
f: X -+ Z
continuous mapping
A
rp
tion for a carrier
is called a continuous selec-
f(x) E cp(x)
if
x E X.
for all
The following result is an obvious consequence of Michael
[ 21
,
Theorem 2, page 2 3 3 .
14. Let X
THEOFEM
be a 0-dimensional compact T1-space and let
( E , \ \ * l l ) be a Banach space over a non-trivially valued division
ring
(F,I.l).
Every lower semicontinuous carrier
E
the non-empty, closed subsets of THEOREM 1 5 .
Let
X
from
CQ
X
into
admits a continuous selection.
be a 0-dimensional compact T1-space and let
be a non-archimedean Banach space over a non-trivially
(E,\\-ll)
valued division ring
-1)
(F,I
non-empty compact subset
such that
cent(K)
# $
for every
K C E.
Then every Weierstrass-Stone subspace
W C C(X;E)
is p r o -
ximinal. PROOF.
Let
n: X
-I
Y
be a continuous surjection of
Y
compact Hausdorff space
Let
f E C(X;E)
since
W
X
onto
a
such that
be given with
f
&’
W.
Then
from
Y
6 = dist(f;W) > 0,
is a closed subspace. rp
Let us define a carrier closed subsets of
E.
For each
rp(Y) =
cs
E E;
y E Y,
SUP XE ll
It is clear that pact subset of
X
cp(y)
-
define Ilf(x)-sl/
B y hypothesis, there exists some
63.
(Y)
is closed.
and therefore
into the non-empty
Now
K = (f(x); so E E
n-’(y)
x E n”(y))
such that
is a comis compact.
BE ST SIMULTANEOUS APPROXIMATION
3 47
We claim that
(""1
rad(K)
Indeed, let
Since
E W
g
be given.
8 ,
I;
Then
was arbitrary,
g
rad(K)
inf I\f-g//, &W
S
and (**) is true. Now, from ( * ) and ( * * ) it follows that Hence,
~ ( y ) f
We claim that
is open in Let
#,
yo E
Y
E ep(y).
y E Y.
for all
is lower semi-continuous, i.e. that
Q
for each open subset
Y
so
be such that
G C E.
cp(yo)
n
G f
@.Let
so
E
epbo)n
G.
Then SUP
llf(x)-~oll
5
6
-1
XEl-f This means that /ls-soll
81.
c fe1(B(s0;6)). n
is closed.
with
c B(so;8),
f(i7-'(y0))
Notice that Since
(Yo)
X
B(so;S)
where
B(so;8)
is open, and that
is compact and
Y
= ( s E E;
-1 IT
(yo) c
is Hausdorff, the map
Hence there is some saturated open set
V
in
X
J O I O B. PROLLA
348
TT
U = n(V)
Then y E U,
all
rr-'(y)
U;
y
C
y E U,
cp,
(because
V C f-'(B(s0;6)).
TT
Hence
cp
and
and, for any
Hence
11 f-wll
W
for all
x E X
y E Y.
y = TT(X).
let
and therefore
for
E cp(y)
for
Let
g E C(Y;E)
w =
gow.
Then
Then
w E Pw(f).
This ends
(E,ll=\\) admits Chebyshev centers, a
better result can be proved, namely that
X
not only a
centw(B) f $
B = (f}, but for equicontinuous bounded sets Let
so
is proximinal.
When the Banach space
B
C
C(X;E).
be a 0-dimensional compact T1-space and let
be a non-archimedean Banach space over a non-trivially
(E,l/.ll)
valued division ring
(F,I
-1).
If
E
admits Chebyshev centers,
and
W C C(X;E)
f $
for every non-empty bounded subset
is a Weierstrass-Stone subspace, then
continuous at every point of PROOF.
Let
TT:
X -+ Y
C
C(X;E)
B c C(X;E)
Y
centw(B) f
which is equi-
X.
be a continuous surjection of
compact Hausdorff space
B
B(so;G)
C
is lower semi-continuous.
< dist(f ;W) ,
the proof that
Let
f(n-'(y))
But this means that
g(y) E c p ( y )
i.e.,
THEOREM 16.
and for any
14, there is a continuous selection
w E W
for
-1 (U) = V)
that is
By Theorem for
Y
is open in
t E v -1(y), y E U .
for all all
-1(yo) c V c f-'(B(s,;*)).
X
onto a
such that
be a non-empty bounded subset which is equicontin-
B E S T SIMULTANEOUS A P P R O X I M A T I O N
uous a t e v e r y p o i n t of Let CASE I :
X.
6 = radli(B).
6 > 0. Define a c a r r i e r
s e t s of
349
E
cp
i n t o t h e non-empty
Y
from
c l o s e d sub-
by p(y) =
( S
E E ; sup
sup
61.
Ilf(x)-sll
f E B xEll-l(y)
It is clear that B C C(X;E)
so
for e a c h
E E
E,
and by h y p o t h e s i s
cent(B(y))
# #,
~ ~ f ( x ) - s=o r~a ~ d(B(y)).
SUP
M B xEft-l(y) We claim t h a t
("1
rad(B(y)) I n d e e d , f o r any
=
g E
W,
SUP
IIf-gll
MB
g
Y.
such t h a t SUP
Now
y E
Since
i s bounded,
i s bounded i n exists
i s closed,
p(y)
was a r b i t r a r y ,
so
5
we h a v e
.
6.
i.e.,
there
350
J O X O B. PROLLA
and s o ( * ) is true, as claimed. Therefore
E cp(y),
so
cp
We claim that
Y,
that
n
rp(y0)
q(y)
Choose
so
n -1(yo)
finite open covering
# $,
1
i
J;
4
n,
f E B.
for all
f E
G C E.
Vl,V2,
...,Vn
of
yo E Y
Let
be such
Then
s 6
. X,
n-'(yo),
there exists a with
Vi
n n-1(yo)#
such that
vi
=)
llf(x)-f(x')ll
This is possible
Vo = V1 U V2 U...U
x E Vo x E Vi
such that
$3
11 f(x)-so]l
and
,
< 8
because the set
C C(X;E)
B
is
X.
equicontinuous at every point of Let
f
is a compact subset of
x,x' E for all
G
E ~(y,) n G.
sup sup fEB xErr-l (Yo) Since
n
Y; V ( Y )
for each open subset
G f $.
is non-empty.
is lower semicontinuous, i.e., that
EYE is open in
and
Vn.
We claim that
\lf(x)-soll
6
J:
Indeed, given x E Vo choose Vi -1 and choose t E Vi n TT (yo). Then, for all f E B.
B
llf(x)-soll
max(llf(X)-f(t)il,
6
llf(t)-soll)
*
Theref ore TI
c Vo
-1(Y,)
C
n
f-1(B(so;6)).
fEB Choose a saturated open set (This is possible because an open neighborhood of
E V
C
f-l(B(s0;8))
x
E ~"(y)
yo
for all
and
in
X
with
is a closed map), in
Y,
f E B.
llf(x)-~oll for all
V
f E B.
'IT
-1
Then
and for every
(yo) C
V
C
U = n(V)
y E U,
=6 so
is
rr-l(y) E
Hence
This means that
Vo.
E cp(y)
for
BEST SIMULTANEOUS APPROXIMATION
all
y E U,
and s o
351
is lower semicontinuous.
cp
By Theorem 14 there exists a continuous selection g(y) E q(y)
such that
and for any
x
E X,
sup IIf-wll < 6 fEB
Hence
CASE 11:
Let
w =
gon.
.
and s o
Then
fE B
y = ~ ( x ) Then for any
let
,
y E Y.
Y
-I
E
w E W
we have
w E centW(B).
6 = 0.
Now
radW(B) = 0
= 0. Therefore REMARK.
for all
gr
f
E W
implies
B = [f}
and
dist(f;W) = radW(B)=
and there is nothing to prove.
In Olech [ 5 ] the formula dist(f;W) = sup rad(f(n-'(y))) YEY
was proved for Weierstrass-Stone subspaces W C C(X;E), is compact and
E
where
is a uniformly convex Banach space (over
X
R or
C),
We will show that (*) is a consequence of the Stone-Weierstrass Theorem. THEOREM 17.
Let
X
be a compact Hausdorff space, and
( E , l l * \ \ ) be
a normed space over a non-archimedean non-trivially valued division ring
(F,I*I).
W
C
C(X;E)
v: X
-I
Y
subspace
where
For every
Hausdorff space
f E C(X;E)
and every Weierstrass-Stone
we have
is the continuous surjection of
Y
such that W = { P n ; g E C(Y;E)}.
X
onto a compact
JOXO B. PROLLA
352
Let
PROOF.
y E Y.
w E W
Then, for every inf
sup
we have
Ilf(x)-zll
z E E xEl?(y)
SUP Ilf(x)-w(x)ll xEn -I (Y) because
w
is constant on
rl
-1 (y).
rad(f(r-l(y)))
IIf-wll
5
Since 5
w
was arbitrary,
dist(f;W)
and then sup rad(f(n YE y Conversely , by Theorem 6.4,
-1
(y)))
S
dist(f;W).
f 61 , we have
dist(f;W) = sup inf Ilf(X)-w(x)ll. YEy wEW xEn-I(y) Let
y E Y.
x E X,
For each
belongs to
z E E,
W.
the constant function
Hence, for each
inf
sup
z
E E
Ilf(x)-w(x)ll
wEw xEn-l(y)
* Since
z
SUP llf(x)-zIl xE rl- Y)
was arbitrary, we have inf
sup
WEW
xErr-l(y and from this it clearly follows that dist(f
REMARK.
of
In the proof given above we used the following properties
w; (1) every (2)
w E W
for each
is constant on each
y E Y,
and
z
E E,
TT
-1
(y),
y E Y;
there is some
w E W
such
BEST SIMULTANEOUS APPROXIMATION
w(x) = z
that
(3) W
such t h a t modulo
x E fl-l(y);
for a l l
i s an A-module,
where
n"(n(x))
i s a s u b a l g e b r a of
A
C(X;F)
i s t h e e q u i v a l e n c e c l a s s of
x
x E X.
f o r each
X/A,
353
Hence t h e f o l l o w i n g r e s u l t i s t r u e : THEOREM 18.
Let
b e a compact Hausdorff s p a c e and- l e t
X
(E,\\*ll)
b e a normed s p a c e o v e r a non-archimedean n o n - t r i v i a l l y v a l u e d d i v i sion r i n g
n: X
3
Y
(F,I
01).
A c C(X;F)
Let
be t h e q u o t i e n t map of
onto t h e q u o t i e n t space
X
o f a l l e q u i v a l e n c e c l a s s e s modulo
be a s u b a l g e b r a and l e t
Let
X/A.
W c C(X;E)
Y
be an
A-module such t h a t w E W
(1) e v e r y
i s c o n s t a n t on e a c h e q u i v a l e n c e c l a s s ~ - ' ( y ) ,
Y E y;
f o r each
(2)
that
y E Y
w(x) = z
and
t h e r e i s some
E
w E W
x E n-'(y).
for a l l
t o ask t h e f o l l o w i n g q u e s t i o n :
i s an A-module, where i s such t h a t
f o r each
x E X,
A
C
g i v e n a subspace
C(X;F)
W(x) = ( w ( x ) ; w €
W
c C(W;E) which
i s a s e p a r a t i n g s u b a l g e b r a , and W)
does i t f o l l o w t h a t
C E
W
i s proximinal i n
,I] *I[ )
X
and
s a t i s f y t h e h y p o t h e s i s of t h e s e l e c t i o n Theorem 1 4 .
THEOREM 1 9 . (E
E,
i s proximinal i n C(W;E)
O u r n e x t r e s u l t shows t h a t t h e answer i s y e s i f
(E,ll-ll)
such
Under t h e h y p o t h e s i s of Theorems 17 and 18 i t i s n a t u r a l
REMARK.
W
z €
Let
X
b e a 0 - d i m e n s i o n a l compact T1-space
and l e t
be a non-archimedean Banach s p a c e o v e r a n o n - t r i v i a l l y
valued d i v i s i o n r i n g
(F,
I . I ).
Let
A c C(X;F)
be a separating
?
354
JOXO B. PROLLA
subalgebra and let is proximinal in Then
W
Let
PROOF.
W
x E X.
for every
E,
C(X;E).
is proximinal in
f E C(X;E)
> 0, because
be a closed A-module such that W(x)
C(X;E)
C
be given with
is closed.
W
x E X,
F o r each
n
there is
( S
cp(x) f $.
so
Clearly,
lower semicontinuous. Choose and
so
n
E cp(xo)
G.
<
61.
( s E E;
IIs1)
hood
of
such that
U
xo
all
x E U,
because
all
x E U,
and
By Theorem for
cp.
Then
Th. 6.4.
We claim that
REMARK.
g
be open and w E W
cp(xo)
n
cp
is
G f $.
such that
(f-w)-l(B(O;a))
w(x) € G
and
is open.
so
= w(xo)
where
x €
(f-w)-l(B(O;a)) w(x) E cp(x)
Then
n
for G, for
is lower semicontinuous.
14 there is a continuous selection
g(x) E W(x)
for all
x E X,
and s o
g E C(X;E g E W,
by
c 61
On the other hand
x E X,
E PW(f) When
and
W(x)
W
6 = dist(f;W)
centw(B)
S
dist(f;W),
is proximinal.
is not only proximinal in
Chebyshev center property in namely that
S
and therefore
11 f-gll i.e.,
is closed.
6
By continuity, there is some neighbor-
Ilf(x)-g(x)II for all
61.
cp(x)
xo E
X
define
5
B(O;6)
cp
from
< dist(f(x);W(x))
Hence
B ( o ; ~ )=
cp
such that
There is some
~~f(xo)-w(xo)~[ g 6 .
6 = dist(f;W) >
x E X,
F o r each
E.
w € W
G c E
Let
Then
E E ; Ilf(x)-s11
some
Ilw(x)-f(x)ll and
W.
L - t us define a carrier
into the non-empty closed subsets of
cp(x) = W(X)
f
f $
E,
E
but has the
a better result can be proved,
for every equicontinuous bounded subset
9
BEST SIMULTANEOUS APPROXIMATION
B
C(X;E).
C
Let
THEOREM 2 0 . A
C
355
C(X;F)
X
and
(E,\l-ll) be as in Theorem 19.
closed A-module such that property in
for every
E,
2,
centw(B) f
Then
bounded subset Let
PROOF.
B
B C
W(x)
B
C(X;E)
for every non-empty equicontinuous
be a non-empty bounded subset which is
Define a carrier E
with
(f]
cp
X.
b
from
X
Let f E W
>
6 = radW(B).
If b =
0,
and there is nothing to
0.
into the non-empty closed sub-
by
x E X,
F o r each
be a
x E X.
Hence we may assume that
sets of
C(X;E)
C(X;E).
C
is a singleton
prove.
C
has the relative Chebyshev center
equicontinuous at every point of then
W
be a separating subalgebra and let
Let
there is some
B(x)
w E W
= (f(x);
f E B}
is bounded in
E,
and
such that
Now
= inf SUP Ilf(x)-w(x)II
radW(x)(~(x))
wEW fEB inf sup
g
1) f-wll
= 6.
W E W fEB
Hence
cp(x) f
2.
Clearly,
lower semicontinuous, i.e. for every open subset
f
2
and choose
s o = w(xo)
and
so
cp(x) that
G C E.
is closed. [x E X; cp(x)
Let
E rp(xo) fl G.
SUP Ilf(xo~-w(xo)II
xo E X
We claim that
n
G
f 23 w E W
is
is open,
be such that
There is some
cp
cp(xo)
n
Gf
such that
6
f€ B Hence B(0;b) =
[ S
xo E (f-w)-l(B(O;b))
€ E ; IIs(I
L
b ] .
f E B,
for every
By continuity of
w
where
and equicontinuity
356
of
JOXO B. PROLLA
{f-w; f E B),
such that
x E U.
E G
W(X)
Then
there is some neighborhood
W(X)
x E
and
E cp(x)
n
U
(f-w)-l(B(O;6))
G,
in
xo
X
f E B
for all
x E U,
for all
of
and
and the carrier
is lower semicontinuous. By Theorem 14 there is a continuous selection for the carrier
Then
by [ 6 ] , 6.4.
g E W,
x E X,
for all
COROLLARY 2 1 . that
rp,
Let
X
and
W = { g E
is such that bounded subset W
space,
C(X;F) x E X,
COROLLARY 2 2 . space
( E , \ \ * \ l ) be as in Theorem
E
centW(B)
W
C(X;E);
# #
Z C X,
given by x E Z],
g(x) = 0,
for every non-empty equicontinuous
B C C(X;E).
is separating over W(x) = 0 ,
Let
X
and
being a 0-dimensional T 1X. On the other hand, f o r X
x E Z;
if (E
,I1 *I1 )
and
non-empty equicontinuous bounded subset PROOF.
W = C(X;E)
x E X.
Since
is a C ( X ; F ) - m o d u l e ,
C(X;F)
W(x) = E
if
x
be as in Theorem 19.
admits Chebyshev centers, then
Theorem 2 0 .
19. Assume
F o r each closed subset
c C(X;E),
is a C(X;F)-module, and
PROOF.
and s o
x E X,
and therefore
admits Chebyshev centers,
E
for a l l
On the other hand
the closed vector subspace
every
g ( x ) E W(x)
g E C(X;E)
cent(B) B
C
and
# #
Z.
If the
for every
C(X;E). W(x) = E
for every
is separating, the result follows from
rp
BE ST SIMULTANEOUS APPROXIMATION
357
REFERENCES
1.
BLATTER, J., Grothendieck spaces in approximation theory, Memoirs Amer. Math. SOC. 120 ( 1 9 7 2 ) .
2.
M I C H m L , E.,
Selected selection theorems,
Amer. Math. Monthly
63 ( 1 9 5 6 ) ~ 233-238. 3.
MONNA, A . F . ,
I.
4.
S u r les espaces lineaires normes non-archim&diens,
Indagationes Mathematicae 18 ( 1 9 5 6 ) , 475-483.
MONNA, A.F.,
Remarks on some problems in linear topological
spaces over fields with non-archimedean valuation,
Inda-
gationes Mathematicae 30 ( 1 9 6 8 ) , 484-496.
5.
OLECH, C.,
Approximation of set-valued functions by contin-
uous functions,
6.
PROLLA, J.B.,
Colloquium Mathematicum 19 ( 1 9 6 8 ) , 285-293.
Topics in Functional Analysis over Valued Divi-
sion Rings, North-Holland Publ. Co., Amsterdam, 1982.
7.
SMITH, P.W. of
8.
and J.D. WARD,
C(X),
WARD, J.D., tions,
Chebyshev centers in spaces of continuous funcPacific Journal of Mathematics 52 ( 1 9 7 4 ) , 283-287.
Departamento de Matemitica UNICAMP
-
IME'cc
Campinas, SP
Restricted centers in subalgebras
Journal of Approximation Theory 1 5 ( 1 9 7 5 ) , 54-59.
- Brazil
This Page Intentionally Left Blank
Functional Analysis, Holomorphy and Approximation Theory Il, G I . Zapata (ed.) 0 Elsevier Science Publishers B. V. (North-Holland), 1984
359
ABSTRACT FROBENIUS T H E O W M
-
GLOBAL FORMULATION
APPLICATIONS TO LIE GROUPS
Reinaldo Salvitti
The main goal of this work is to give the Global Formulation of the Abstract Frobenius Theorem in the context of Scales of Banach Spaces and to applie it in the construction of Lie Subgroups.
The
motivation of this work was the study of germs of analytic transformations of
Cn
that vanish at the origin,
studied by Pisanelli in [l]. scale
gh(n,C),
gh(n,C),
as it was
The applications of this work, in the
will appear in another paper.
1. INTRODUCTION 1.1 DEFINITION.
A Scale
X
of Complex Banach Spaces is a topolo-
gical vector space, obtained from the union of a family of Complex Banach Spaces (a)
,
<
s’<
11 \ I s
with norm
1) \Is/
Xs C Xs’, 0
(b)
Xs
L
1) ) I s
and
0 c s S 1,
for all pair
s,
s’
,
and such that: such that
s 1;
s
X = lim X s
+
,
Hausdorff and sequentially complete.
1.2 EXAMPLES
1.2.1
Xs = X
1.2.2
If
X =
for all
U
Yn
n21 1 L< s LZ, n+ 1 Banach Spaces.
1.3 DEFINITION.
Let
s,
0
< s s 1, then
X
is a Silva Space, letting
is a Banach Space. Xs = Yn
for
we get that every Silva Space are Scale of
X
and
Y
be locally convex sequentially
360
E I N A L D O SALVITTI
complete spaces,
n
C
X
open.
A map
f
f : Q + Y is LF-analytic if for each map
g
w
g:
open, the map
W C C
1.4 LIE
+
n
is analytic.
f o g
GROUP [ 21
Let
X
plex space,
S
be a Hausdorff, sequentially complete locally comC
open and endowed of a group structure such that
X
the maps
Id:
SXS (XtY)
are LF-analytics.
s+s
S
+
Pl(X,Y) = XY
3
We then call
B
x
+
x
-1
a Lie group.
The linear maps L(x):
X(x):
x
-#
x
h
+
(d;(x,e)h
x+ x h
+
pl;.(x
-1
,X)h
are the inverses of one another, for each
x E
s,
where
2
is the
group’s unit. From the associative law and differentiating with respect to the second variable we get the Lie equations for the group:
The LF-analytic map
Lh
defined by
ABSTRACT FROBENIUS THEOREM
361
is called infinitesimal transformation of the group h.
1.5
We also have
= h,
L(e)h
for each
S
at vector
h E X.
h,
LIE ALGEBRA OF THE INFINITESIMAL TRANSFORMATIONS. [l] Let
S
space and
be a Hausdorff, sequentially complete locally convex
X
open.
X
C
The vector space
W(s:X)
of LF-analytic
maps is a Complex Lie Algebra with the bracket
When
(j
is a Lie Group the set of infinitesimal transformations is
a Lie Subalgebra of
Sr(S:X).
Algebra of the group
Q.
This Subalgebra is called the Lie
It is possible to endow
X
with a Lie
algebra structure, isomorphic to that infinitesimal transformation by defining [h,k] = L'(e)kh
1.6
Let
X
Y
and
locally convex space
-
L'(e)hk.
be complex, Hausdorff, sequentially complete Let
A c X
and
B
C
Y
be open subsets and
f: AxBXX + Y
a LF-analytic map.
The map
f
satisfies the integrability condi-
tion if f
S,
is symmetric for all
1.7
X,Y)kh +
(f(X,Y k)h
f'Y (X,Y)
(h,k) E X X Y
and each pair
(x,y) E AXE.
LOCAL FROBENIUS THEOREM. [ 3 ]
Spaces.
u
"
X s and Y = ys O<s
Let
X =
be Scale of Banach
3 62
REINALDO SALVITTI
where
xo
E
and
X
and
Y
U B~(o,R)x U D ~ ( O , R ) X X-t Y o< s s 1 O<srl
f:
with
yo
b a l l of c e n t e r
0
and r a d i u s
R
of
Xs
Ds(O,R):
b a l l of c e n t e r
0
and r a d i u s
R
of
Ys
Suppose t h a t
f
restricted to
t a k e s i t s v a l u e s i n s i d e Ys / ,
with
f(x,y)h
u
0
<
s
s‘<
4
i s G - a n a l y t i c and
1,
x
Bs(O,p)
u
u
map d e f i n e d on DS(O,p)
o<s*1
.
x DS(O,R) x Xs
BS(O,R)
satisfying the i n t e g r a b i l i t y condition.
i s a unique LF-analytic
x
,
Bs(O,R):
Bs(O,p)
taking values inside
Then t h e r e
x Y,
where
Ws4l
O<s4l
R
P =
192e2C REMARK.
With t h e same system, i n Frobenius Theorem, t a k i n g
f
f: such t h a t
f
values i n
Yst,
restricted t o
Bs(wo,R)
i s G-analytic with
X Ds(z0,R)
Ilf(x,y)hlls/
x
Xs
taking i t s
C *\lhlls
and
s a t i s f y i n g t h e i n t e g r a b i l i t y c o n d i t i o n we g e t t h e e x i s t e n c e of a unique LF-analytic map
taking i t s values i n
Y,
y(x,xo,yo)
with
p =
defined i n
R 192e‘C
ABSTRACT FROBENIUS T H E O m M
363
2. INTEGRAL MANIFOLD
X, Y, 2 ,
We always denote
etc, complex, Hausdorff, se-
quentially complete, locally convex spaces. 2.1 ANALYTIC MANIFOLD
Let
M
X,
be a subset of
u
M =
with
Oi,. where:
iEI (a)
(b)
@,(OinOj)
is open in
(c)
ai0aj-1: @j(oinoj)
(d)
: ' ; 6
@i(Oi)
-+
i E I, where Each pair G = (ai,Oi)
Qi(Oi)
ai(Oi),
for all
and open
i, j
of
X
+
z E Oi
is injective, for all
and
zi = Oi(z).
is a chart of the collection
G
M.
is an atlas of
homeomorphisms and the subsets A couple of atlases, the identities
I;
is analytic for all i,j E I;
Qi(oinoj)
-+
There is a unique topology in
(a)
onto
(Qi,Oi) and
iE I
Oi
is analytic;
X
Yi
(@il)'(zi):
of
yi ;
subset of space
(e)
ai
there is a bijection
(i
(M,G)
-+
M
which makes the maps
$i
open [ 4 ] .
Oi
,.. in
and
(i
(M$)
M
and
are compatible if:
(M,;)
(M,G)
-b
are
continuous ; (b)
the maps and
and
Jjor$il
zj(zjnOi)
@i~z-lare analytic in j
respectively for all
A analytic manifold
M
in
X
@,(Oinaj)
i, j .
is a subset of
X
together
with a equivalent class of atlas of giving by the compatibility relation. 2.2
TANGENT SPACE Let
M
be a analytic manifold in
X
and
z
E M.
The
364
REINALDO SALVITTI
M
tangent space to
at
is the vector space
z
zi = @i(z),
where
The vector space
does not depend on the chart (oi,Oi)
TZ(M)
that was chosen [ 2 ] .
INTEGRABLE DISTRIBUTION
2.3
Let
U
X
space of
X
be an open subset of
and
H
a closed vector sub-
with topological supplement.
2.3.1 DEFINITION.
fi
A distribution
f: UXH
(a,h) analytic and such that for each f(z): H
U
in
-9
X
-t
f(z)h
z
E U
-9
X
is a map
f,
the map
h + f(z)h is linear and the vector spaces
H
and
f(z)H
are top-linear iso-
morphics. 2 . 3 . 2 DEFINITION.
We call
M
for all
z
Let
M
contained in
U
an integral manifold of a distribution
through
2.3.3
that contains
U
if
an integral mani-
MZ
z
and that we will say that
a9
be a distribution in
M
passes
z.
DEFINITION.
(a)
in
= f(z)H.
Whenever it is convenient we will denote by M
B
E M
TZ(M)
fold
be an analytic manifold.
We call
Let
B
integrable if for each
integrate manifold of
B
z,
U. z
passing through
E U, z.
there is an
365
ABSTRACT FROBENIUS THEOREM
(b)
A integrable distribution
each f(zo)H
2,
E U
z
there is a continuous projection
and a neighborhood of
2 E V(zo),
Wa C M a ,
a9 has the (P) property if for
V(zo) c U ,
z o ,
the restriction of
such that for each
a,
is a chart.
x v(zo)
z: V(Pb0))
V(p(zo))
. ( X , . )
is an open neighborhood of
a E V(zo),
for each
is the inverse of
the restriction of
p
restrict on
We always can consider
z = z(x,a-),
x
+
+
(X9.1
REMARK.
onto
to a neighborhood of
p
Furthermore there is an analytic map
where
p
p(zo)
z(x,a)
in to
f(zo)H
and
p(Wa) x (a)
a'
p(V(zo))
C
V(p(zo))
because
p
is continuous.
2 . 4 THEOREM.
Let
(P) property. each
z,
Let
z E V(zo
D
be an integrable distribution in
z
,
be contained in
is invertible and for each
z
x.
a
Then for
a neighborhood of -p(zo)
respect to
and suppose that f o r
H + f(zo)H
h,
v(zo)
is satisfied, where
with the
the map pof(z):
is LF-analytic.
U
U
in
z'(x,a)
h E f(zo)H, +
the map
H
+ [pof(z)]-'h in a neighborhood of f(zo)H
and
z
~f. in
the system (S),
is the differential of
z
with
366
2.5
REINALDO SALVITTI
THEOREM.
fi
Let
be a distribution in
U
and for each
suppose that there is a continuous projection
p
z
f(zo)H
onto
E U
such
that the map
H
pof(z)r is invertible for all hood of
V(zo)
+
who)
= a,
in
zo
in f(zo)H
p(zo)
in
Let
So
in
U
and
x
F
and
and
belongs to
zo
+
So
5 is in a neighborhood of
G.
y
in a
be the topological supplementary of
Under the hypotheses of the Theorem ( 2 . 5 )
analytic maps,
we have
f(zo)H.
If 2
p.
where
eo
and
U.
Under the same hypotheses ( 2 . 5 )
parallel to
z(p(z0),a) 2.8
is a neighborhood of
COROLLARY 2.
f(zo)H,
x v(zo>
V(P(.,>)
in a neighborhood of
neighborhood of 2.7
the map
is a neighborhood of p(zo)
COROLLARY 1.
for
is a neighbor-
is an integrable distribution with the (P) property, where V(p(zo))
2.6
W(zo)
.(.,a)
z(x,a):
fi
where
If the system ( S )
has analytic solution
then
h € H
z o , and for each
is LF-analytic.
E W(zo),
z
z,
f(zo)H
+
We take
V(zo),
nerality, like a product of neighborhoods of
then z o .
we define two
without lost of gep(zo) in
f(eo)H
and
ABSTRACT FROBENIUS THEOREM
in
q(zo)
4
X
F(a) = p(a)
G: V(zo)
3
X
G(a) = z(p(A),
F(zo) = z
and
G(zo)
= a
(GoF)(a)
2.10 REMARK.
Let
X
= z
A-p(A) 0
P(zo)).
f
. we have
z
be a complex Banach space. = Ix
P(zo)
(F~G)(A) = A
(ii)
F’ (G(A))G’(A) A = a = z
-
+ Z(P(z,),a)
In convenient neighborhoods of
2.9 THEOREM.
If
We define
F: V(zo)
We see that
(i)
q = I-p.
S o ,
367
Then
G’ (F(a))F’ (a) = Ix
and
.
we have
0
F’ (zo)G’ ( z o ) = Ix
G ’ (zo)F‘ ( z o ) = Ix
and
.
By the Inverse Function Theorem there is a neighborhood of where
F
and
G
In a convenient neighborhood of
2.11 THEOREM.
the bracket
[
,]
for all
f‘(z)h
fixed
.
2.13 THEOREM.
is involutive if
h,k E H
= f’(z)(f(z)k)h
is differential of
Let
we have
z E
and for all
U , where
is
[f(z)h,f(z)k] and
B
A distribution
E f(z)H
[f(z)h,f(z)k]
z
= f(zo)H.
F’(a)f(a)H 2.12 DEFINITION.
zo
are inverses to each other.
B
f
-
f’(z)(f(z)h)k
with respect to
be a distribution and for all
have a continuous projection
p
onto
f(zo)H.
h
for
z
zo
E U
If for all
we
zo
the map pof(z): is invertible, where
z
H
-P
f(zo)H
belongs to a neighborhood
V(zo)
and for
368
each
REINALDO SALVITTI
h E f(zo)H
is analytic then bracket where
we have that the map
69
is an involutive distribution if only if the
[@(z)h,$(z)k]
belongs to
f(z)H
for all
h,k E f(zo)H,
= f(z)[pof(z)]-'h.
g(z)h
2.14 THEOREM.
Let
of Theorem (2.4).
69
be a distribution with the same hypotheses
Then the second term of = f(z)[pof(z)]-lh
z'(x,a)h
satisfies the integrability condition. 2.15 COROLLARY.
On integrable distribution with the property (P)
is involutive.
2.16 THEOREM. such that
Let
a9
be an involutive distribution and
is a continuous projection onto
p
f(zo)H.
z E U Suppose
as well that H -+ f(zo)H
paf(z): is invertible for each h E f(zo)H
z
belonging to
V(zo)
and for each
fixed, the map Vbo) z
is analytic,
+
H
+
[pof(z)]-'h
Then the second term of the equation z'(x,a)h
= f(~)[pof(z)]-~h
satisfies the integrability condition for all 2.17 LOCAL FROBENIUS THEOREM
z E V(zo).
FOR A DISTRIBUTION IN A SCALE OF
BANACH SCALES
We can now apply the Theorem (1.7) to an involutive distri-
369
ABSTRACT FROBENIUS THEOmM
bution. and
u
X =
lie assume that if
Xs
is a Scale of Banach Spaces
wss 1 H
is closed vectorial subspace then
H
is also a Scale of
Banach Spaces, in fact
H = lim (H n
xS).
-4
THEOREM. and
r9
Let
u
X =
be an involutive distribution in
Xs
a Scale of Banach Spaces.
U c X
U,
open
We suppose that for
Oiss1
each
z
E U
there is a continuous projection
p
onto
f(zo)H
such that the map
H
pof(z): is invertible for
E V(zo)
z
=
u O<Sd
h E f(zo)H
Bs(zo,R)
and for each
1
the map
is LF-analytic.
Consider the system
(
PROOF.
f(zo)H
-4
z'(x,a)h
= f(z)
[pof(z)]-lh
B y (2.15) we have the integrability condition for z'(x,a) Applying (1.7) and (2.5) we have that
= f(z)[pof(z)]-'h.
19
=
is a
inte gr ab le di s tributi on. 2.18 THEOREM.
Let
Q,
S c X , be a Lie Group and
algebra and as well as a closed subset of distribution in the group
Q.
S
If 5
X.
Let
H
a Lei Sub-
L(z)H
be the
where
L
is an infinitesimal transformation of
and
q
are analytic maps
370
REINALDO SALVITTI
2.19 COROLLARY 1.
The distribution
2.20 COROLLARY 2.
We suppose that for each
continuous projection onto
f(zo)H
z E
is involutive. zo E tJ
there is a
such that the map
H + L(zo)H
poL(z): is invertible for each
L(z)H
Then the second term of the
V(zo).
equation
z' (x,a)h = L(z)[poL(z)]
-lh
satisfies the integrability condition.
3 . FROBENIUS THEOREM
- GLOBAL FORMULATION
First of all we suppose that we are always with the (2.5) hypotheses.
Then we have an integrable distribution
the (P) property.
Then f o r a neighborhood of
zo,
8
zo
in
E U,
U
with
the in-
tegrable manifolds are solutions of = f(z)[pof(z)~-~
z'(x,a)
= a
z(p(a),a)
zo E U
We will prove that for each maximal.integrable manifold
3 ZO
manifold of
19
,
passing through
there is a connected
i.e., every connected integrable
zo
.
is contained in g Z 0
3.1
A
NEW TOPOLOGY We define for
U
a new topology
of neighborhoods at each point A neighborhood of
zo
z
E U
TN
by a complete system
that will be denote by
is a subset of
U
b,.
that contains the
371
ABSTRACT FROBENIUS THEOREM
z = z(x,z )
image by
of a neighborhood of
p(zo)
in
f(zo)H,
being the solution of the system (S) with the initial
z(x,zo)
condition
= zo
z(p(zo),zo)
zo
(1) For each
,
.
we have
#
b,
because the system (S)
@
0
has solution. For each
(2)
and
z
V
E IJ
zo E V
we have
because
20
= zo
Z(P(ZO),ZO)
(3) If
V E b
*
and
then
V’2 V
V‘E b z
zO
(4) I f
and
V
0
W
I I ~ then
belongs to
.
V
n
W
belongs to
0
because the solution
z(x,zo)
is injective.
I’ZO
(5)
We will prove that if
V E IJ
there is
0E b
that
and
0t V
0 E bz
such ZO
zO
z E 0.
for all
The proof of
this will require two lemmas. 3.2 LEMMA.
Since
z1
we have that
in MZo when x
belongs to a convenient neighborhood of
-
(Fo@~)(x) F ( z l )
is taken in a convenient neighborhood of
f(zl)H.
of
is the map defined in (2.8),
p1
is a continuous projection onto
@l
is the solution of (S) replacing
z1
f(zl)H, p
and
zo
by
p1
and
respectively.
We denote
C $ ~ ( X , Z ~=)
@l(x)
and
z(x,zo) = z(x).
By the Theorem (2.11) we have F‘ (a)f(a)H
where
pl(zl)
f(zo)H
Here
F
PROOF.
has values in
z
5
= f(zo)H
belongs to a convenient neighborhood
Let
V(p(z,))
V(z,).
be a convenient neighborhood in
f(zo)h
such
372
REINALDO SALVITTI
Applying
q = I-p
to the two terms of the last equation we
have
Since
(qOFo@l)(xl)
-
(qoF)
is connected, we have, replacing
(2,)
x1
= constant, because by
w(P,(Z,))
P1(Z1),
and therefore
3 . 3 LEMMA. for all
There is a neighborhood
zl,
z l E z(V(p(zo))),
V(p(zo))
in
f(eo)H
such that
there exists a neighborhood
1=
W(P1(Z1)
1
PROOF.
By the Theorems ( 2 . 9 ) and (2.11) we have a neighborhood
V(zo)
in
f(zl)H
such that for
such that
a E V(zo) F(a)f(a)H
and
@l(w(P1(zl)
= f(zo)H
z(v(P(zo))
1.
ABSTRACT FROBENIUS THEOREM
373
(GoF)(a) = a.
(3.2) l e t
As in
and
z1 E
V(p(z,))
z(V(p(zo))).
Let
+
The maps neighborhood
and
V(0) c V ( p ( z o ) ) @1
and
f(zo)H
+
F(zl)
such t h a t
V(0) C V ( z o ) .
a r e continuous hence t h e r e i s a
Fool
W = W(pl(zl))
z ( V ( p ( z o ) ) ) c V(zo)
x1 = p ( z l ) .
W e denote
be a n e i g h b o r h o o d i n
V(0)
x1
such t h a t
in
f(zl)H
such t h a t
@,(W)
c V(zo)
and
-
( F o @ ~ ) ( W ) F ( z l ) C V(0). Then
+
C)F(zl) (Fo@~)(W
Applying
G
t o b o t h t e r m s we h a v e
L e t u s s e e what i s
Taking
V(0).
t E V(0)
G(F(zl)
+
V(0))
we h a v e
G ( F ( z l ) + t ) = G(X1+Zo-P(zo)+t) = = z ( p ( xl+zo-p(
2 o)
+ t ) ,xl+z .-P ( z o) + t -P ( X1+ZO'P ( zo)+ t1-P ( z o ) 1 = = z(xl+t,z
= z(xl+t,xl+zo-p(zo)+t-xl-t+P(zo)) Then
G(F(zl)
+
V(0) = z ( x l
=
+
0
) = z(xl+t).
V(0),zo) = z(xl + V(0))
hence
Z(Xl+V(0)) c Z W P ( Z 0 ) ) ) .
Now i t i s s i m p l e t o p r o v e t h e p r o p e r t y 5 . Let
V E bz0,
such t h a t f o r each
m,(w)
c Z(V(P(Z,))),
by t h e ( 3 . 3 ) t h e r e i s
zlE
0
there i s
(recall
w
b
0 = z(V(p(z,)))
W = W(p1(zl))
I*
z1
with
c
V
374
REINALDO SALVITTI
With those 5 properties it is defined a new topology in that we denote
3.4 THEOREM. PROOF.
IN.
(U,TN)-t U
The identity
Let
V(zo)
is continuous,
be a neighborhood of
The solution of the
zO.
system (S) is continuous hence there is a neighborhood such that
z(V(p(zo)))
borhood of
3 . 5 THEOREM. $4
in
U.
Let
c V(zo).
The set
z(V(p(zo)))
V(p(zo)) is a neigh-
(U,IN) hence the inclusion above is continuous.
in
zo
U
M
be an integrable manifold of the distribution
Then the inclusion
is continuous
(',IN)
.
T o prove this theorem we u s e two lemmas like ( 3 . 2 ) and (3.3).
3.6 COROLLARY. system (S).
For each
The map
E U
z
let
z
be the solution of the
restrict to a neighborhood of
z,
continuous one taking values in
p(zo),
is
(U,TN).
As a consequence of the last corollary there is a fundamental system of connected neighborhood in ly convex space.
(U,TN)
because
(U,TN)
is a local-
(U,IN) are open.
Then the connected components of
Now we consider in
H
the connected component 3
for zO
each
zO.
5,
It is easy to prove that
is an integrable manifold 0
of the distribution
$4
in
U.
is a
From the last theorem if MZO
connected integrable manifold which pass through connected in
(U,TN),
so
c 3,
M 20
0
.
zo
then
M,
is 0
Then 3
integrable manifold of the distribution
is the maximal zO
Q.
3 . 7 THE EXISTENCE OF A MAXIMAL INTEGRABLF MANIFOLD IN SCALE OF BANACH SCALES. The existence of a maximal integrable manifold came from the solution of the system (S) in the neighborhood of each point z o
375
ABSTRACT FROBENIUS THEOREM
of an open set
Then under, the hypotheses of (2.17)) an invo-
U.
B
lutive distribution
in a Scale of Banach Spaces is integrable
z E U
and for each point
manifold which contains
3.8 REMARK.
1.
there is a connected maximal integral
0
z
0
.
We constructed the maximal integral manifold based
on the existence of solutions of the system (S) for each where the distribution is defined. of a family of projections
,
pz
z C U 0
I t was necessary the existence
z E U.
The Theorem (3.5) shows
that the existence of a connected maximal integral manifold does not depend on the family of projections used, i.e., and
p;
,
zE U
if
pz
,
zE V
are two families of projections satisfying the
hypothesis of Theorem ( 2 . 5 ) then they produce the same connected maximal integral manifolds. 2.
If
X
is a complex Banach Space any family of projections
satisfies the hypothesis of Theorem ( 2 . 5 ) .
We will see this proof
in another paper.
4. APPLICATION TO A LIE GROUP 4.1 THEOREM.
Let
X
be a complex, Hausdorff, sequentially com-
plete, locally convex sapce and
S,
S c X, a Lie Group.
be a Lie subalgebra of
X
with topological supplementary.
sider the distribution
&
of
s
giving by
L(z)H,
S.
the infinitesimal transformation of the group that for each that
zo E S
there is a projection
p
where
Let
H
We con-
L
is
We also suppose onto
L(zo)H
such
376
Rl3INALDO S A L V I T T I
H
poL(z): is invertible for all
( @(x,a)h
z E W(zo)
where
V(p(zo))
and the system
= L(d)CPOL(@)I
has holomorphic solution
h E L(zo)H
-lh
@(.,a)
is a neighborhood in
S
z E
Then for each point
belongs t o is a Lie subgroup
of
L(zo)H.
there is a connected maximal in-
2,
tegral manifold and that one which
PROOF.
L(zo)H
-t
the unit o f the group,
q.
We have to prove only that the connected maximal integral
manifold who passes through
2
is a Lie Subgroup of
Q.
We need
two lemmas.
4.2 LEMMA. in (3.1).
Let
Q
be endowed with the new topology
Then for each
9
is continuous, where PROOF.
a E 9,
the translation
is the operation o f the group
First of all we will see that if
o f the distribution
In fact, if x E Y,
a,
Y
z(x)
19
s N defined
then
aN
is a integral manifold
N
is also a integral manifold o f
is a parametrization of points of
parameter space.
Consider
N,
8.
where
@(a,z(x)),
From the Lie equations of the group is an integral manifold we have
S.
Q
and the fact that
N
ABSTRACT FROBENIUS THEOREM
Then the tangent space at an integral manifold of Let in
(S,lN).
L($(a,z)H)
and
aN
is
S
and
be a neighborhood of a
W
zo
From ( 3 . 5 ) we know .that the inclusion
is continuous. such that
is
a9.
belong to
z
az
377
Hence there is a neighborhood
in L(zo)H
c W.
$(a,z(V(p(zo)),zo))
Since
V(p(zo))
z(V(p(zo)),zo)
is a neighborhood of
in
z
(B,SN)
the translation (S,'N) a
*
(S,'N)
+
az
is continuous.
4.3 LEMMA.
The connected maximal integral manifold
Let
x, y
belong to
Me.
From (4.2)
integral manifold which passes through and
x
xMe
X.
Since
so
x'he
is a connected x E Me,
xMe C M e
xy E Me. Analogously
-1
is a group
S.
with the operation of PROOF.
Me
x'he
2
contains
C
Me
therefore
E Me.
PROOF OF THE (4.1). e = 0.
If p
We denote
n
We may, without lost of generality, suppose
is a projection onto = p-'(V(p(e)))
L(e)H
= H
and we define two functions
inverse to each other,
f(z) = z
we have
- @(P(.))
g ( w ) = @(P(W)
+
w
-
P(Z>
P(W))
p(e) = e. f
and
g,
FEINALDO S A L V I T T I
where
@(x)
= a)(x,O). = I,
f o g
Because
and
f
g
= I
g o f
n '
are continuous there a r e
a l o c a l group [ 2 ]
(fll,U,V,W)
and
around
0,
U,V,W,W
=
n n
G,
where
= f[$(g(z),g(w))l*
$,(Z,W)
W e have
z E
but f o r
V(p(e))
Taking
g(z) = @ ( z ) .
w e have
h E H
g'(0)h = ,$'(O)h
= L(@(O))(poL(a)(O)))"h
= h
theref o r e L1(Z)h
As
(fog)(z) = z ,
=
f'
z E
(g(z))L(g(z))h.
n,
w e have
f ' ( g ( z ) ) g ' ( z ) H = H.
and s o
Then
and s o where
L1(z)H
= H
C1(z)H
= H,
z E V(p(e)). So i f
z
and
w
b e l o n g t o a neighborhood o f
2
in
H
we
ABSTRACT FROBENIUS THEOREM
have
L1(z)dil(w)H Take
C
H (*).
and
z
379
in a neighborhood of
w
2
in
H,
h E H
and
consider the systems
where
TI
is the projection,
and
[6
(k’fk)rrl
(x,~)
The solution of
z
-1
(S2
= el(z,e) =
From (*) each term of the series above is in H
i s
closed
zw
and
z
-1
belong to
We then have a local group G1 =
Also the map
g
H
and since
H.
G1 c H,
($l,ul,vl,wl).
is a local homomorphism between
Consider the local group in
S,
[6],
(Z,g(U1) ,g(vl) ,g(w1)
1
S1
and
S.
380
REINALDO SALVITTI
2
and the topological group that passes through
u
N =
We can take so
N
U1
u
=
[g(u1)ln
n2 1
n21
U
conncected then follows that
is connected too.
Since the set
2,
tegral manifold which passes through manifold which contains The set open in
N
(S,lN).
closed in
ze,
ag(U1)
a€[ g(ul)l n-l
e.
g(U,) N
Hence
N C 3,
3,
because
is open in
g(U,)
is connected
is a connected inis a connected integral
. g(U,)
= $(V(p(e)))
is
u aN is open therefore N is aE3 e/N Ze is connected N = 3,. So 3, is a to-
The set
since
pological group. Now we prove that the operations
3,x3, (Z,W)
3e z
-t
3,
-t
zw
+
+
e' z-1
are analytics. Let with
z
E 3,,
wo E
z ( t ) = z o , w(To)
and
~ ( t ) , w(T),
parametrizations
= w0
The map (tSl.1 + PO(8(Z(t),W(t)) is analytic, where
po
is a projection onto
Therefore the map Analogously the map
( z , ~ )+ zw z + z-l
L(zowo)H.
is analytic. is analytic.
ABSTRACT FROBENIUS THEOREM
381
REFERENCES
1.
PISANELLI, D.,
An example of a infinitive Lie group.
Am. Math. SOC. 6 2 (1976) no 1, 156-160 2.
PISANELLI, D.,
Grupos Analiticos Finitos de TransformagEes,
Publicaggo da Sociedade Brasileira de Matemdtica An6lise de
3.
Proc.
(1977).
-
Escola de
1977.
PISANELLI, D.,
Theoreme dtOvcyannicov, Frobenius et groupes
de Lie locaux dans une Qchelle d'espaces de Banacli,
C.R.A.
S. Paris, 277 (1973).
4.
LANG, S.,
5.
PISANELLI, D.,
Differential Manifolds,
Addison, Wesley D.C.,
1972.
Linear Connected Subgroup of a Lie Group i n a
locally convex space.
Anais da Academia Brasileira de Cicn-
cias (1979), 51(4).
6.
PISANELLI, D.,
Sulltintegrazione di un sistema di differen-
ziali totali in uno spazio d i Banach.
Academia Nationale
dei Lincei, Rendiconti della Classe di Scienze Fisiche, Matematiche e Naturali Serie VII, Vol. XLVI, fase
Instituto de Matemgtica e Estatistica Universidade de S g o Paulo
CX 20570
-
Aggncia Iguatemi
S g o Paulo, SP
-
Brasil
6, giugno 1969.
This Page Intentionally Left Blank
Functional .4nulysis, Holomorphy and Approximation Theory 11, G I . Zupata ( e d . ) 0 Ekevier Science Publishers B. K (North-Holland), 1984
383
OPTIMIZATION B Y LEVEL SET METHODS.
11: FURTHER DUALITY
FORMULAE I N THE CASE OF ESSENTIAL CONSTRAINTS
Ivan Singer
ABSTRACT Let
F
h: F
and
for some
F
W e show that if
(this happens e.g.
for
I
GS = ( $ E F"
I
a functional.
y' by the elements of
141
( $ E GS
E
-I
y'E
mation of lae of
be a non-empty subset of a real locally convex space
G
inf h(G),
h(y') < inf' h(G)
in the theory of best approxi-
G),
then, in the duality formu-
one can replace the set
$ f 0, sup $(G) <
sup $(G) < Q(y')]
+m)
by its subset
o r other similar subsets, and one can
obtain new duality theorems, which reduce the computation of inf h(G)
to the computation of the infima of
hyperplanes of
$0.
h
over some support
G.
INTRODUCTION
In the present paper we shall continue the study of the following general optimization problem, considered in [ 41 : locally convex space
F
(which will be assumed
special mention), a.subset G
of
F
real,
without any
(which will be assumed E -
empty), called the constraint set, and a functional =
Given a
h: F
-B
fi =
find convenient formulae for the number
[-m,+m],
a = inf h(G) = inf h(g)(E
G).
&G
In Q E F"
141
we have shown that the existence of functionals
(the set of all continuous linear functionals on
F), which
384
IVAN S I N G E R
G
support G
(i.e.,
JI f 0 and
sup JI(G) <
from certain level sets of
h,
inf h(G)
and which separate
closedly o r openly or nicely
imply duality formulae which reduce
(in the sense of V. Klee [l]), the computation of
+m)
to the computation of the infima of
h
on some closed half-spaces o r strips o r closed strips containing
G
and that, conversely, under some additional connectedness assump-
tions on
G
and on certain level sets of
h,
these support and
separation conditions are also necessary for o u r duality formulae to hold. or
JI E
The duality formulae of
[4] involved functionals JI E G S ,
where
-GS,
(i.e., support functionals
JI
of
G),
which have the advantage
that they can be easily computed for certain classes of sets (see
G
[4], formulae (1.19) and (1.21)). In [ 41, remark 1.2 c), we have observed that the duality for-
mulae of [4] present interest only in the case when the constraint set
G
is essential, i.e.,
when
inf h ( F ) < inf h(G);
(0.3)
clearly, this inequality is equivalent to the condition that there should exist an element
y‘E F
satisfying
h(y’) < inf h(G). The assumption (0.3)
(or (0.4))
(0.4) implies, obviously, that
In the present paper we shall study further the optimization problem ( O , l ) ,
under the assumption (0.3) (hence (0.5) will hold
throughout the paper).
We shall show that if
trary fixed element satisfying ( 0 . 4 ) , formulae of
y’E F
is an arbi-
one can replace in the duality
[4] the set GS by its subset { J I
E GS
I
sup JI(G)<Jr(y’))
385
OPTIMIZATION BY LEVEL SET METHODS
o r other similar subsets, and one can obtain new duality theorems,
which reduce the computation of the infima of
h
inf h(G)
to the computation o f
over some support hyperplanes of
G.
Thus, the
division of this paper into sections and subsections will be similar to that of
[h],
with two additional subsections on results of
weak and strong duality in terms of support hyperplanes of Let u s observe that a fixed element
G.
satisfying (0.4)
y’E F
arises quite naturally in some concrete optimization problems (0.1). F o r example, in the theory of best approximation, there are given
a normed linear space
y’E F ,
F,
a subset
G
of
F
and an element
and one wants to compute
(0.6)
dist (y’ ,G) = inf IIy’-gll, gEG
which is a problem of type ( O . l ) ,
= [ 0,+ m )
for the functional
h: F + R+ =
defined by
(0.7) clearly, we have
0 < dist(y’,G)
the balls with center
if and only if
(0.4) holds. Since
y‘, used in the theory of best approximation
to obtain duality formulae for ( 0 . 6 ) , are nothing else than the level sets of the functional
h
defined by ( 0 . 7 ) ,
it is natural to
attempt to extend the methods of the theory of best approximation to level set methods for the general problem ( O . l ) , assumptions ( 0 . 3 ) ,
(0.4).
In [2],
[ S ] , we have shown that this
leads to useful duality formulae f o r (0.1) when sets of
h
under the
G
and some level
are convex and satisfy certain topological assumptions.
In the present paper these convexity and topological assumptions are weakened s o as to become both necessary and sufficient for o u r duality formulae to hold.
IVAN SINGER
386
$1. THEOREMS OF WEAK DUALITY PRELIMINARIES
1.0
For convenience, let us recall and complement some of the main terminology and notations of
[4].
By flhyperplane"we shall always mean:
closed hyperplane.
In addition to the non-symmetric separation properties used in
[4],
we shall say that a hyperplane
if
#
J, E F*,
where
GIC
I
EY E F
c E R,
0,
$(Y) c c]
J, E F*\( O}
c E R
if there exists
ly separates
from
G1
I
G1
from
J,(y)> c];
strictly separates
G2
cor-
-
G1from
such that the hyperplane (1.1) strict-
G2.
G C F
Given a set
G2 c (y E F
and
respondingly, a functional G2,
strictly separates
and a functional
h: F
-t
g,
in the
sequel an important role will be played by the level sets
(0.4) mean that
the assumptions ( 0 . 3 ) and
Aa
f 8,
y'E Aa,
where
a = inf h(G). A s in
nected, if
[k]
, we
J,(G) C R
shall say that a subset
G
of
F
is connected -(i.e., an interval
is F*-=-
( a , @ ) , fi-
nite or infinite, closed or open from the right or from the left), for each
$ E F".
We shall denote by
Finally, for ( a , @ )= r y E R
u Eel,
fa,el
I
a <
a < Y <
= [ a , @ )u
@
in
@I, EB3.
R,
the closure of the set G .
as usual, = ( a , @ )u
rai,
( a , @ ] = ( a , @ )u
OPTIMIZATION BY LEVEL SET METHODS
387
RESULTS O F WEAK DUALITY I N TERMS O F CLOSED HALF-SPACES
1.1
CONTAINING
THEOFLEM 1.1. and
h: F
let
y'E
-I
Aa.
G
F
Let
b e a l o c a l l y convex s p a c e ,
a functional with
Aa
f
@,
where
G
a s u b s e t of F
a = i n f h ( G ) , and
The f o l l o w i n g s t a t e m e n t s a r e e q u i v a l e n t :
1'.
F o r each
c E
(h(y'),a)
there exists
JI,
E GS
satisfying
2O.
For each
c E [h(y'),a)
there exists
Q,
E GS
satisfying
SUP
3O.
There h o l d s
4O.
There h o l d s
1'.
For e a c h
c E
ecW
< JIc(Y)
(h(y'),a)
JlC(Y) < inf
F o r each
3'.
There h o l d s
sc).
there exists
QCW
c E [h(y'),a)
2'.
(Y E
6,
(1.5)
E -GS
(Y E Ac).
there exists
satisfying
(1.4')
6, E
-GS
satisfying
(1.69)
4'.
There h o l d s
388
IVAN SINGER
1 The equivalences '
PROOF.
e
2 O
3'
=.
Q
1'
2'
Q
t)
,
3 ' hold by [ !t]
Theorem 1.1 and Remark 2.10 c).
3O
Q
. ' 4
Since
y'E Aa,
we have
inf YE F
SUP
$EGS $(YOSSUP $(GI and hence, if 3'
$(Y)rsuP
4O. The implication 4O
holds, we obtain
Finally, considering
4'.
e
is
in (1.6)).
2
= -4,
$'
3'
3
we obtain the equivalence
This completes the proof of Theorem 1.1.
REMARK 1.1. C
(1.8)
$(G)
obvious (by the obvious inequality
k0
= inf h(G),
h(y)rh(y')
( $ E GS
a) Since
I
I
{$ E G '
sup $(G) < $(y')]
s u p $(GI
c
~ i ( y / ) )c
c GS, one can obviously add in
Theorem 1.1 the equivalent condition
5'.
There holds
However, in this paper we shall consider conditions of this type (i.e., involving
{ $ E GS
I
s u p $ ( G ) 5:
$(y')))
only when they
are not equivalent t o the corresponding conditions o f type (i.e., involving
( $ E GS
I
sup $ ( G )
< $(y')]);
(1.7)
see Theorems 1.3
and 2.3 below. b)
If
e > 0, one can add in Theorem 1.1 the following equi-
valent conditions, with the convention that if holds for any
6'. ing
GS
#
$,
Ac = 0 ,
then
(1.4)
0:
#
Q
and for each
c < a
#
Q
and for each
c E (a-e,a)
there exists
4,
E GS satisfy-
(1.4).
7'.
GS
satisfying
there exists
$, E GS
(1.4).
Indeed, the equivalences
Yo
c)
6'
o
3 O have been shown in
[4],
389
OPTIMIZATION BY LEVEL SET METHODS
Remark 2.10 c) and Theorem 1.1.
Similar remarks can be also made
for (1.5), (1.15), (1.16), (l.bf), etc. in the theorems of the present section, but we shall omit them.
ing
(h(y’),a)
instead of c E
fact that f o r each larly, for
(h(y’),a)
c E [h(y’),a)
the above convention for for
a-h(y’)
E ’ =
c)
we have
%
(-m,a)
@
(a-€,a)
we have
(or
(h(y’),a)
consists in the
y’E Ac f @
y‘E Sc f @ ) ,
we have Ac =
T h e advantage of consider-
Sc = Q);
(and, simi-
so we d o not need
o n the other hand,
= (a-E’,a).
B y the obvious inequalities 2 in
(1.6) and (1.7), one can
express 3 O and b0 of Theorem 1.1 in the following equivalent forms, re spec tively : go.
F o r each
c E
(h(y’),a)
there exists
6,
E GS
satisfying (1.10)
9O.
F o r each
c E
(h(y‘),a)
there exists
6,
E GS
satisfying
(1.10) and
Similar remarks can be also made f o r 3 1 , 41 and for the other results o f $1. Geometrically, formulae (1.9) and (1.7) mean that in
REMARK 1.2.
(1.6) it is enough to take the sup over all (respectively, which separate strictly)
G
E GS
from
which separate
y’.
In these
cases, the hyperplane
supports strictly)
G
and separates (respectively, has a translate separating
G
from
equivalent form
y‘.
One can also write (1.9) and
(1.7) in the
390
I V A N SINGER
i n f h(D
s UP
i n f h(G) =
JI
,G
)
(1.13)
and, r e s p e c t i v e l y , i n f h(G) =
sup
(1.14)
i n f h(D
$EGS
y'dU$ w h e r e , a s i n [ 41
,
Remark 1.1,
$,
space i n t h e d i r e c t i o n
D
JI
,G
i s the smallest closed half-
,G
containing
Since t h e geometric i n -
G.
t e r p r e t a t i o n s o f most of t h e o t h e r f o r m u l a e i n t h i s p a p e r c a n b e o b t a i n e d s i m i l a r l y from t h e c o r r e s p o n d i n g o n e s of [ 4 ] ,
we s h a l l ornit
them i n t h e s e q u e l ( w i t h t h e e x c e p t i o n of Remark 1 . 5 b e l o w ) .
I t w i l l be worth while t o s t a t e s e p a r a t e l y t h e f o l l o w i n g s u f f i c i e n t c o n d i t i o n i n o r d e r t o have ( 1 . 6 ) ,
(1.7),
(1.9),
(1.13),
(1.14): THEOREM 1 . 2 . and
h: F
-t
and t h e s e t s
Let
F
b e a l o c a l l y convex s p a c e ,
a functional with Sc
f
Aa
c E [h(y'),a)
with
on
i n i t i a l t o p o l o g y on
and i f e i t h e r
a r e compact f o r
whence a l s o ( 1 . 6 ) ,
Y'E
and l e t
If
Aa.
G
weaker t h a n o r e q u a l t o t h e
7
c E [h(y'),a)
a s u b s e t of F
a r e convex and c l o s e d f o r a
l o c a l l y convex t o p o l o g y F,
@,
G
F,
G
or the sets
t h e n we h a v e 2'
7 ,
Sc
with
o f Theorem 1.1,
(1.7), ( 1 . 9 ) , (1.13), (1.14).
The p r o o f i s s i m i l a r t o t h e s e c o n d p a r t of t h e p r o o f of [ 4 ] , Theorem 1 . 2 .
1.2
~ E S U L T S OF WEAK DUALITY
THEOREM 1.3. and
h:
F
-t
Let
F
IN
TERMS OF STRIPS CONTAINING
b e a l o c a l l y convex s p a c e ,
a functional with
s i d e r the following statements:
Aa
# a,
and l e t
G
G
a s u b s e t of F y'E
Aa.
Con-
391
O P T I M I Z A T I O N BY LEVEL SET METHODS
lo.
F o r each
c E
(h(y'),a)
there e x i s t s
6, E
GS
satisfying
F o r each
c E [h(y/),a)
there e x i s t s
6, E
GS
satisfying
F o r each
c E
there e x i s t s
6, E
GS
satisfying
(1.4). 2'.
(1.5). 3'. either
(h(y'),a)
( 1 . 4 ) or Jlc(g) < i n f q c ( A c )
4O. either
F o r each
c E [h(y'),a)
E G).
there e x i s t s
6, E
(1.15) GS
satisfying
( 1 . 5 ) or Q c ( g )< i n f
5O.
(g
$c(sc)
(g
5 GI.
(1.16)
There h o l d s
(1.17)
6O.
There h o l d s i n f h(G) =
7'.
inf
h(y).
(1.18)
There h o l d s
(1.19)
8'.
There h o l d s
(1.20)
IVAN SINGER
392
(1.18' )
i)
4'
3O, as well as '1 *
8'
and 2'
7 O , and the equivalence 5'
6O.
ii)
If
G
is F*-connected, then 5'
iii)
If
G
and the sets
then
3 '
iv)
v)
5'
...
and 6 O
=a
8 ' .
by [ 4 ] ,
6'
e
=a
3
lo
7O.
with
c E
Sc
with
c E [h(y'),a)
(h(y/),a)
are F*-connected,
7'.
(n = 1,
no e n'
are F*-connected,
...,8).
3
4'
a
are obvious (since
3O,
2O
Ac c Sc
lo
2
3
3O,
'8
3
7'
3
and since the ine-
in (1.19) and (1.17) are obvious).
Since
for any such
H
3
7'.
e
* 5O
2
3O
Ac
i) The implications 2'
qualities ' 1
e
and the sets
We have
PROOF.
6'
e
If G
then 3 '
3
5'
e
7'
5'
We have the implications '2
c
y'E Aa,
there exists
there exists
c
$, E GS
E (h(y/),a).
satisfying
Then, by lo,
(1.4) and hence,
Proposition 1.2 i) , we have (1.21)
Furthermore, by
(1.4) and
(l,ll), whence, by (1.21),
y'E Ac
we have, in particular,
OPTIMIZATION B Y LEVEL SET METHODS
=
b'
inf
h(y)
393
c.
2
Consequently,
and hence, since the opposite inequality is obvious, we obtain (1.20). The proof of the implication 3 O that for
c
5'
Since
3
6'.
Jic
and
y'E Aa, inf
YE F 4J(Y)E4J(G)
$(G)
Ji F F*
But, if
7 O .
Q E Gs)
sup ( - $ ) ( G )
5;
inequality
2
iii)
R,
< c < a,
-1 E
(whence
in (1.19), if 6'
G
$(y'),
2
G
Ji(y') $? $ ( G ) ,
and
we have either
inf $(G)
or
(-Ji)(y')
If
Ac
c E
with
(h(y'),a)
5O
G
6'
t)
v)
3
and the sets
3O
Replacing
equivalences no o n* Theorem 1.3.
3
5O
6,
9
by
6'
Sc
Q
-0,
...,8).
(n=1,
Q(y')
3
' 3
are and by i),
7O. with
c E [h(y'),a)
F*-connected, then, by [ 4 ] , Theorem 1.3, we have 5' above we have 4'
S
Hence, by the obvious
G').
ii) above we have 3'
Q
then, since
holds, we obtain 7 O .
and the sets
3
is F*-connected,
which is equivalent to
[4], Theorem 1.3, we have 5 O
If
(1.23)
sup $ ( G )
F*-connected, then, by
iv)
Then
holds, we obtain 6 O .
is an interval in
(whence
(h(y'),a).
By i), it is enough to show that if
ii) =)
c E
h(y) s h(y')
SUP
and hence, if 5'
we have
there exists
QEGS
@(Y' ) E Q ( G )
then 6 O
as in 3'
7 O is similar, observing
1
r)
are
4 O and by ii)
7'. and
Q
by
-$
we obtain the
This completes the proof of
394
IVAN SINGER
From Theorems 1.2 and 1.3 there follows THEOREM 1.4.
Under the assumptions of Theorem 1.2, we have (1.18)-
(1.20) and (1.18f)-(l.ZOf). REMARK 1.3.
Theorem 1.4 has been obtained, essentially, in
Corollary 2 . 3 and its proof
(see
also [3],
[S],
Remark 2.7 (a)).
RESULTS O F WEAK DUALITY IN TERMS O F CLOSED STRIPS
1.3
CONTAINING THEOREM 1.5. h: F +
and
G
Let
5
F
be a locally convex space,
a functional with
Aa
#
6,
G
and let
a subset of F y'E
Aa.
Con-
sider the following statements: .'1
For each
c E (h(y'),a)
there exists
Qc E
GS
satisfying
For each
c E [h(y'),a)
there exists
Q c E GS
satisfying
(1.4). 2O.
(1.5). 3O.
There holds (1.24)
. ' 4
There holds inf h(G) =
Q 5'.
inf h(y). QEGS YEF (Y' )iG-m Q (Y)Ern sup
(1.25)
There holds
11 -51
,
obtained from 1O-5'
procedure of Theorem 1.3.
similarly to the corresponding
395
OPTIMIZATION BY LEVEL SET METHODS
We have the implications 2O * lo
i)
valence 3O
is F*-connected, then ' 3
G
If G
iii)
and the sets
nected, then lo
If
iv)
e
3O
4O
0
o
Ac
We have no
o
Sc
I
$(y')
($ E GS
B'
3
'5
and 3O
on con-
c E (h(y'),a)
are
with
c E [h(y'),a)
are F -con-
*
-
$(G)]
I
sup $ ( G ) < $(yo}
C
c GS
=.
' 3
are obvious
and since the inequalities
2
in
The proofs of the implications
' 4 are the same as those of Theorem 1.3, implica-
1 =. 8 O and 5' tions '
(e
5O.
with
* lo and 5'
(1.25) and (1.24) are obvious). lo
=+
4O
C
( $ E GS
4O
a
(since
Sc,
o
(nzl, ...,5).
nt
The implications ' 2
C
and the equi-
5O.
PROOF. i) Ac
' 4
3
.'5
and the sets
G
nected, then lo o...e v)
'5
4O.
o
If
ii)
=r
=.
6 O , observing that
sup $ ( G ) = sup
O(G)
E F").
ii)
If
G
is F*-connected, the proof of 4O
=)
5O is similar to
that of Theorem 1.3 ii), implication 6' =. 7 O , observing that if JI E F" and $(y' ) $ (G), then either sup $ (G) < $(y' ) or inf
$(GI
> $(Y'
>.
iii) and iv) follow from [4], Theorem 1.5 and ii) above. the proof of v) is similar to that of Theorem 1.3 v).
Finally,
This com-
pletes the proof of Theorem 1.5. REMARK
1.4. From Theorems 1.2 and 1.5 there follows
milar to Theorem 1.4. in [ 3 ]
, Theorem
a result si-
Such a result has been proved, essentially,
2.3 and its proof.
396
IVAN SINGER
1.4
RESULTS OF WEAK DUALITY IN TERMS OF SUPPORT HYPERPLANES OF G
sup $(GI E
If we have (1.24), then, since
a(c),
there holds
also
Thus, it is natural to ask whether the opposite inequality holds (similarly to the obvious inequalities
2
in the preceding
formulae of weak duality), and whether one can replace in Theorem 1.5 the closed strips
B -$,G
the support hyperplanes
I
= EYE
H
$ ,G
$(Y)
(see
c41)
by
defined by (1.12) or by the support
hyperplanes
";I
,G =
Since the set hyperplanes (1.12), Indeed, even when
I
{Y E F
GS
$(Y) = inf
JI(G)I.
(1.121)
is too large and since for
$ E GS
the
(1.129) are too Ifthinff, the answer is negative. G
is a closed convex set and
h
is a finite
continuous convex functional on a finite-dimensional space
F
(so
(1.24) holds), the inequality in (1.27) may be strict, as shown by EXAMPLE 1.1.
so
Let
F = R2 ,
a = inf h(G) = 1.
the euclidean plane, and let
F o r each
Jlc(Y) = 'rll
c
with
0< c < a = 1
(Y = (Tl1,q2) E
let
(1.30)
F).
Then sup
qC(c)
= -1 < inf
qC(sc) =
inf
YE F
(-ql) = -c,
II Y1I so
(1.5) holds.
However, for
Q0
=
-6,
we have
sup
q0(G)
= 3
OPTIMIZATION BY LEVEL SET METHODS
q0 E
(so
397
and
Gs)
and hence the inequality in (1.27) is strict. Nevertheless, we shall show now that for the subset
( $ E GS
I
sup $(G) < $(y’)]
of
GS,
occurring in (1.26),
the si-
tuation is different, under certain additional assumptions.
To
this end, let us first give the following generalization of [2], Lemma 2.1: PROPOSITION 1.1. of IJJ
with
F
E GS,
GS
Let
f
@,
F
be a locally convex space,
fi
h: F +
such that either i)
Aa
a functional with
G
a subset
f 0,
and
is an interval (n=1,2,...),
$(A
a + :
$(s
1) is an interval (n=1,2,...) a+n $(Aa) is an interval, or iv) G fl Sa
or ii)
and
If there exists
interval.
y’E Aa
or iii)
#
0
and
Gn
f 0
$(Sa)
is an
such that
then a = inf h(G)
PROOF, i)
Take
E $(A by
l). a+n $(gn)
By
gn E G
2
inf h(Y). YE F 1 (Y)=suP $ (G)
such that
y’E Aa C A
h(gn) < a +
we have
sup $(G)
1 n,
$ (g,)
E
and (1.31), and since
1), whence, a+n $(A 1) is an interval, a+;; ‘I
we obtain SUP $(GI E t$(gn),$(Y’)1
Thus, for each such that
so
$(Y‘) E $(A
a+n C
(1.32)
n
there exists
$(yn) = sup $(G),
ii) is similar.
=
$(A
(1.33)
1)’ a+n
yn E F
with
which proves (1.32).
h(yn) < a
1
+ 5,
The proof of
398
IVAN SINGER
iv)
Take
we have
6 n sa,
yo E
$(Y' ) E $(Sa),
and (1.31), and since
SO
whence, by $(Sa)
Thus, there exists
n o(sa).
$(yo) s SUP
BY
Y'E
$(6) =
c
sa
SUP $(G)
is an interval, we obtain
ylE F
with
h(yl)
which proves (1.32).
= sup $(G),
$(yl)
$(yo) E a ( 6 )
5
a,
such that
The proof of iii) is si-
milar, which completes the proof of Proposition 1.1. We shall also need the following proposition and corollary, corresponding to
[&I,
Proposition 1.1 and Corollary 1.1 respective-
ly : PROPOSITION 1.2. of
F, h: F + i-it)
F
Let
r?
be a locally convex space,
a functional,
If either (1.4) or
c E R
with
Ac f
a subset
G
a,
$,
and
E
F".
1.41) holds, then
I inf
h(Y
).EF
oc(Y)=suP JlC(G) ii)
Conversely, if
y'E Ac
Q ~ ( A ~ )is an interval and there exists
such that SUP
s OC(Y'
1,
(1.36)
and if we have
then
(1.4) holds.
i i t ) If
qc(Ac)
is an interval and there exists
y'E
Ac
such
that
and if we have
(1.37')
OPTIMIZATION BY LEVEL SET METHODS
399
then (1.4f)holds. PROOF.
Similarly to
[&I,
proof of Proposition 1.1, it is immediate
that we have (1.37) if and only if
i-if) If (1.4) or (1.41) holds,then obviously, we have (1.38). Alternatively, i-if) follows also from and ii)
SUP Jlc(G), inf $,(GI
-
E ?Jc(G)
[4], Proposition 1.4 i). By
y'E
and (1.36), (l.38),
Ac
and since
Jlc(Ac)
is an in-
terval, we obtain (1.4) (whence, in particular, sup Qc(G) < Qc(y' Finally, iil) is equivalent to ii), considering
QL =
>.
-Jlc
This completes the proof of Proposition 1.2.
F, G
Even when
and
h
have "nice" properties, as in
Example 1.1, one cannot omit the assumptions (1.36) and (1.361) in Proposition 1.2 ii) and i i f ) respectively, as shown by EXAMPLE 1.2.
Let
F, G
and
h
0 < c < 1 = a = inf h(G)
be as in Example 1.1 and for each let
c
with
so
we have (1.35), but neither (1.4), nor (1.41).
The same example
motivates also the assumptions (1.36) and (1.369) in the following corollary of Proposition 1.2: COROLLARY 1.1. of
F,
h: F +
Let
F
be a locally convex space,
a functional,
c E R
with
Sc
#
0,
G
a subset and
$, E GS.
400
IVAN SINGER
If either (1.5) o r (1.51) holds, then we have (1.35).
i-it) ii) y'E
Conversely, if
$,(SC)
is an interval and there exists
satisfying (1.36) and if we have
Sc
(1.40)
then (1.5) holds.
If
ii')
is an interval and there exists
$,(SC)
y'E
sa-
Sc
tisfying ( 1 . 9 6 1 ) and if we have
then (1.5') holds. PROOF.
Parts i), i f ) follow from
Ac
C
Sc
and Proposition 1.2 i)
and it) respectively. The proof o f part ii) is similar to that o f Proposition 1.2 ii),
observing that if (1.40) holds, then
SUP fc(G)
f
(Y E s c ) .
$,(Y)
Finally, iil) is equivalent to ii), considering
Jr/c = -$,.
This completes the proof of Corollary 1.1. Now we are ready to prove THEOREM 1.6. h: F
and a)
be a locally convex space,
a functional with
-t
is F*-connected
A
a+n (n=1,2
,...) ,
d)
F
Let
G' fl
or
Sa f @
c)
and
G
and
a subset of F
6 , such that either
(n=l,2,...), o r b)
rl Aa f d,
Sa
Aa f
G
Aa
is 8'"-connected.
S a+n
is F*-connected
is F*-connected, Let
or
y'E
Aa
and con-
8, E
GS
satisfying
sider the following statements: lo.
(1.4).
F o r each
c E
(h(y'),a)
there exists
401
OPTIMIZATION B Y LEVEL SET METHODS
.'2
c E [h(y'),a)
For each
6 , E GS
there exists
satisfying
(1.5). . ' 3
There holds inf h(G) =
SUP inf $EGS YEF SUP 6 ( G k Q(Y' ) Q(Y)=suP
(1.41)
h(Y) '
6 (GI
11-3f, obtained from lo-3O similarly to the corresponding p r o cedure of Theorem 1.3. i)
We have the implications 2'
If the sets
ii) then
' 1
then lo
PROOF. a
2O
t)
Ac
with
c E
Sc
with
c E [h(y'),a)
nc
(n=1,2,3).
o
i) The implication '2
.'3
3O.
(h(y/),a)
are F*-connected,
are F*-connected,
3O.
We have no
iv)
' 1
t)
a
3O.
e
If the sets
iii)
a ' 1
By a), b),
sup Q(G) E
lo is obvious (since
Ac
C
Sc).
c) or d), y'E Aa, lo, Proposition 1.1 and
we have
inf h(G)
inf
;r
h(Y)
(1.42)
9
whence w e obtain 3 O (by Theorem 1.5). ii)
Assume that the sets
and that 3' Q c E GS
since
Ac
holds.
Ac
with
c E
(h(y'),a)
are F*-connected
Then, by (1.41), for any such c < a there exists
satisfying (1.11) (whence (1.36)) and (1.37).
Hence,
is F*-connected, by Proposition 1.2 ii) we obtain (1.4).
402
iii)
IVAN SINGER
If the sets
Sc
with
c E [h(y'),a)
are F*-connected and if
3' holds, then, similarly to the above proof of ii), using now corollary 1.1 ii), we obtain (1.5). Finally, the proof of iv) is similar to that of Theorem 1.3 v). This completes the proof of Theorem 1.6. REMARK 1.5.
Geometrically, formula (1.41) of Theorem 1.6 means
that inf h(G) =
sup
(1.44)
inf h(H),
HExG ,y'
where
#G,yl
denotes the collection of all hyperplanes
which support G Y'
H
in
and have a translate separating strictly G
F from
. Finally, let us make some complementary observations to
Proposition 1.1, collected in REMARK 1.6.
Under the assumptions of Proposition 1.1, but replacing
(1.31) by the stronger condition
(1.45)
Moreover, replacing (1.45) by the stronger condition
we have even
Indeed, if (1.45) holds, then f o r any
y'E Aa
and hence, by Proposition 1.1, we obtain (1.32). the inequality 2
we have (1.31)
Furthermore, if
in (1.43) is strict, then there exists
yo E F
OPTIMIZATION BY LEVEL SET METHODS
JI(G),
$(yo) E
with
403
such that
inf yEF $ (Y)=suP
h(Y)
’ h(Yo)*
(1.49)
Q (GI
But, by (1.32) and (1.49), we have
yo E Aa
and hence, by
(1.451,
SUP @(G)
inf @(Aa)
On the other hand, by 5
sup $(G),
5
whence, by (1.49), we obtain
contradiction with (1.50).
(1.50)
$(Yo).
$(yo) E $(G) we have
5
$(yo) < sup $(G),
This proves (1.46).
holds, then we have (1.45), whence also (1.46). by (1.47) and Proposition 1.2 i) (with
$(yo)
c=a)
in
(1.47)
Finally, if
On the other hand,
we have the opposite
inequality to (1.46) and hence the equality (1.48) (alternatively,
(1.48) also follows from (1.32), (1.43), (1.47) and [ 4 ] , Proposition
1.4 i) with
c=a).
Moreover, under some additional assumptions
(see Proposition 1.2 ii) and [4], Proposition
1.4 ii)), one can also
give results of converse type.
$2.
RESULTS OF STRONG DUALITY
2.1
RESULTS OF STRONG DUALITY IN TERMS OF CLOSED HALF-SPACES CONTAINING G
THEOREM 2.1. and
h: F -+
Let
<
F
be a locally convex space,
a functional with
Aa f 6,
following statements are equivalent: 1’.
There exists
2O.
There holds
Q o E GS
satisfying
G
and let
a subset of F y’E Aa.
The
404
IVAN SINGER
inf h(G) = max
inf
Q€GS
yEF
Q ( Y b SUP Q (G) 3O.
There holds inf h(G) =
1 1-3
t
,
(2.3)
obtained from 1O-3'
similarly to the corresponding pro-
cedure of Theorem 1.3. Q
2'
hold by [ 4 ] , Theorem 2.1,
and the proofs of the equivalences 2'
c)
3'
Q
3' arb similar to
those of Theorem 1.1, equivalences 3 O
Q
4'
3
41.
PROOF,
The equivalences ' 1
REMARK 2.1.
Q
Z0
By the inequalities and 3'
also express 2'
o 1'
2
in (2.2) and ( 2 . 3 ) ,
one can
of Theorem 2.1 in the following equivalent
forms , re spect ively : 4O.
Q o E GS
There exists
inf h(G) =
5O.
inf
Q o E GS
There exists
satisfying
SUP
h(Y)
(2.4)
*
satisfying (2.4) and
Q0W < Qo(Y/ 1.
(2.5)
Similar remarks can be also made for 2',
31 and for the
other results of 92. From Theorem 2.1 and [b], THEOREM 2.2.
set of
F
Let
and
and open, and let
be a locally convex space,
F
h: F
Theorem 2.2, there follows
+
y'E
a functional with Aa.
Aa
Then we have (2.3).
G
a convex sub-
non-empty, convex
OPTIMIZATION B Y LEVEL SET METHODS
RESULTS OF STRONG DUALITY I N TERMS OF STRIPS CONTAINING
2.2
We recall that the is the subset of
THEOREM 2.3.
h: F
and
-t
G
Let
of a set
G
in a linear space
G F
defined by
F
be a locally convex space,
a functional with
Aa f g j ,
a subset of F
G
and let
y'E Aa.
Con-
sider the following statements: .'1 2O.
There exists
q0 E
There exists
q0
inf h(G)
. ' 4
satisfying either (2.1) or
inf
h(y).
There holds max
inf
(2.9)
h(y).
There holds inf h(G) =
6'.
E GS
= max
inf h(G) =
5'.
satisfying (2.1).
GS
max
inf
h(y).
(2.10)
There holds
11-61, obtained from lo-6O similarly to the corresponding p r o cedure o f Theorem 1.3.
406
IVAN SINGER
We have the implications1°
i)
the equivalence ii)
If
and
Aa
are
' 3
y'E
2
6'.
The implications lo
By lo and
1
' 3
are
5Oad
o
5O.
.'5
o...o
then 2'
* 6'.
...,6).
obvious (by the obvious inequalities 3
6O
(n=1,
6O
lo
' 3 and ' 1
then 2'
core Aa,
ZO,
PROOF. i)
=,
' 4
o
* F -connected,
and
We have no o nl
v)
5O
a
o 4O.
Aa f Q
If core
iv)
' 2
is F*-connected, then
G
If G
iii)
3O
3
=,
5'
=)
=)
'3
and
' 4
3
in (1.19) and (1.17)).
[k] , Proposition 1.2 i)
(with c=a), we have (2.12)
Furthermore, by ' 1
and
y'E Aa
we have, in particular,
(2.5), whence, by the obvious inequality
in (1.20),
2
we obtain
(2.11).
The proof of the implication ' 2 that for
0,
-
is similar, observing
as in 2O we have
3 The proof of ' plication 5O
=,
=,
' 4 is similar to that of Theorem 1.3, im-
6'.
ii)
By i), it is enough to show that if
4O
.'5
3
5'
is F*-connected, then
G
The proof of this fact is similar to that of Theorem
1.3 ii). iii)
If
we have '2 iv)
G o
and
Aa
are F*-connected, then by
' 3 and by ii) above we have ' 3
w
4O
[4], Theorem 2.3, w
5O.
If we have (2.1) , then, by part i) , implication lo
holds 6'.
Assume now that we have core Aa f 0,
=,
6 O , there
y'E core A a
and
407
OPTIMIZATION BY LEVEL SET METHODS
that
Q0
E GS
satisfies (2.7).
so (2.5) holds.
[k], Lemma 2.1, we obtain
Then, by
But, by 2O and
[4], Proposition 1.2 i) (with c=a)
we have (2.12), whence by the obvious inequality obtain (2.11)
in (1.20), we
2
. $,
Finally, replacing
-$o
by
6
and
-4,
by
we obtain
v), which completes the proof of Theorem 2.3. REMARK 2.2,
Even when
G
is an open convex set and
h
F , with core Aa
vex functional on a finite-dimensional space
example (of
#
@,
y‘E core A
one cannot replace in Theorem 2.3 iv) the assumption by the weaker assumption
is a con-
a
as shown by the following simple
y’E Aa,
[4] , Remark 2.4 a)):
Let
F = R2 (the Euclidean plane),
let G =
h =
and let
Then
S0(Y) = ‘Il
xM,
I
m4lrl1+ll
h21)<
9
13,
the indicator functional o f the convex set
= a =
inf h(G)
(Y = (q1,Tl2) E F )
y’ = ( q i , q > ) E A ing
E F
C(rll’r12)
such that
a
sup $ ( G ) < $(y‘),
+-
(since
G
n
M
#
@ ) and for
we have (2.71, but for
q;
=
0 there is n o
I$
E GS
satisfy-
whence the right hand side of (2.11) is
-a,
so (2.11) does not hold. F r o m Theorem 2.3 and
THEOREM 2.4. set of
F
Let
with
F
GS
#
[4], Theorem 2.4, there follows
be a locally convex space, @
and
h: F
empty and convex, such that either y J E Aa
.
-t
<
G
G
a functional, with Aa nonor
Aa
is open, and let
Then we have (2.10) and, in the case when
y‘ E Int Aa,
we have (2.11).
a convex sub-
Int Aa
#
@
,
408
IVAN SINGER
2.3
RESULTS OF STRONG DUALITY I N TERMS OF CLOSED STRIPS CONTAINING G
THEOREM 2.5. h: F
and
-t
Let
fi
F
be a locally convex space,
a functional with
#
Aa
a subset of F
G
and let
r$,
y'E Aa.
Con-
sider the following statements: lo.
There exists
2O.
There holds
J,,
E GS
inf h(G) = max
.'3
satisfying (2.1).
inf
h(y).
(2.15)
There holds inf h(G)
.'4
=
max J,€ GS JI (Y' )dJI(G)
inf
(2.16)
h(y).
yEF
Jl(YNrn
There holds inf h(G) =
inf
max
11-41, obtained from 1O-4'
h(Y)
(2.17)
*
similarly to the corresponding pro-
cedure of Theorem 1.3. We have the implications lo
i) 2O
a
* 4 '
3
3 O and the equivalence
e
3O
3O.
ii)
If
G
is F*-connected, then
iii)
If
G
and
iv)
We have no
Aa CJ
'2
0
. ' 4
are F*-connected, then ' 1
nl
We omit the proof.
...,4).
(n=l,
*...a
4'.
OPTIMIZATION BY LEVEL SET METHODS
2.4
409
R E S U L T S OF STRONG DUALITY IN TERMS OF SUPPORT HYPERPLANES OF G
As shown by Example 1.2, we may have inf h(G)
=
inf
h(Y)
<
and yet strict inequality
Q c E Gs
=
inf
n
and
-GS h(Y)
in (1.27) (see Example 1.1).
9
However,
we shall prove now THEOREM 2.6.
and
Let
h: F +
be a locally convex space,
F
a functional with
Aa f 0 ,
b), c) or d) of Theorem 1.6, and let
y'E
G
a subset of F
satisfying one of a), Aa.
Consider the follow-
ing statements:
lo.
There exists
2O.
There holds
b 0 E GS
satisfying (2.1).
inf h(G) =
11-21, obtained from l0-Zo similarly to the corresponding pro-
cedure of Theorem 1.3. i) ii) iii) PROOF. ii)
We have the implication lo = 2'.
If
Aa
*
is F -connected, then '1
We have no e n f
t)
2'.
(n=1,2).
Part i ) follows from the second part of Remark 1.6.
Assume that
Aa
(2.18), there exists
is F*-connected and that
Qo E
holds.
Then, by
satisfying ( 2 . 5 ) and
GS
a = inf h(G)
2O
=
inf
h(Y)
(2.19)
410
IVAN SINGER
Hence, since
Aa
is F*-connected, by Proposition 1.2 ii)
we obtain (2.1). Finally, the proof of iii) is similar to that of Theorem 2.3 v).
This completes the proof of Theorem 2.6. From Theorem 2.6, using the separation theorem, combined
with
[4], Lemmas 2.2 i), 2,3 i) and [ 41, Theorem 2.6, there
follows THEOREM
2.7.
set of
F
y’E Aa.
Let
and
F
be a locally convex space,
h: F +
If the sets
(n=1,2,...) o r a+n
convex and if either convex and
G
Aa f Q ,
a functional with A i)
Aa
is open, with
G
a convex suband let
(n=1,2, ...) are
S
a+n is convex and open o r a = inf h(G) <
+a,
ii)
h
is
then we have
(2.18). REMARK 2.3.
Theorem 2.7 has been proved, essentially, in [2]
Theorem 2.1 and Remark 2.2 (d), and their proofs.
,
OPTIMIZATION BY LEVEL SET METHODS
411
REFERENCES
1.
V.
KLEE,
Separation and suport properties of convex sets-a
In: Control theory and the calculus of variations,
survey.
Academic Press, New York, 1969.
235-303. 2.
I. SINGER,
Generalizations of methods of best approximation
to convex optimization in locally convex spaces. 11: Hyperplane theorems.
3.
I. SINGER,
J. Math. Anal. Appl. 69 (1979), 571-584.
Duality theorems f o r linear systems and convex
J. Math. Anal. Appl. 76 (1980), 339-368.
systems.
4. I. SINGER, Optimization by level set methods.1: Duality f o r mulae.
In:
Optimisation: Th6orie et algorithmes.
Proc.
Internat. Confer. held at Confolant, March, 1981. Lecture Notes in Pure and Appl. Math., Marcel Dekker, New Y o r k (to appear).
INCRE ST Department of Mathematics Bd. Pgcii 220, 79622 Bucharest and Institute of Mathematics, Str. Academiei 14 70109 Bucharest, Romania.
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Functional ,4nalysis, Holowtorpliy uird Approximution Theory 11, G.I. Zaputu (ed,j @ Ekevier Science Publislrers n. I-. (North-Hollulidj, 1984
SPACES FORMED BY SPECIAL ATOMS I1
Geraldo Soares de Souza (Dedicated to Patricia, Lesley and Geraldo Jr., my children)
1. INTRODUCTION
We define the space
2<
for
p <
where
m
T
CE = (f: T
f(t) =
C cnbn(t)} finite is the perimeter of the unit disk i n the -I R;
I)fl/
= Inf
C lcnlP fir&& where the infimum is taken over all possible representations of f. complex plane.
Cop
is endowed with the "norm"
cPo
Each
bn
is a special p-atom, that is, a real-valued function
defined on
T, which is either
xL(t),
where
I
b(t)
-1o r 2rr
E
b(t)
=
T, L
is an interval on
-
b,
1 p x X R ( t ) +
is the left half
+*
of and
I
and
xE
space
R
is the right half.
denotes the length of
111
the characteristic function of Cp
as being the completion of
f E Cp
E.
CE
Then we define the
under LO
i f there is a sequence
(b,)
and a sequence (cn) of numbers such that m f(t) = C cnbn(t). The "normtfin C p is n=1
C
We may say
I
of special p-atoms
rn
cnIP <
and
m
n=1
m
= Inf
C n=1
cnlP
where the infimum is taken over all possible representations of
In this paper we are interested in those For
p = 1
p
in the range
we may refer the interested reader to
The spaces
Cp
9
f.
21 c p < l .
b].
need some further comments since they are
defined in the abstract sense. under the metric sarily complete. metric
d.
Observe that Cop is a metric space 1 for p < p < 1, but not necesd(f ,g) = /(f-gll CE Next consider the completion of C g under the
This completion is denoted by
Cp,
since
f E Cp implies
414
GERALD0 SOARES DE SOUZA
f
that ric
is an equivalence class of Cauchy sequences under the met-
d,
f = (fn)nzl
say
the “norm11 of
f
in
Cp
as
-+ 0
n,m
= lim CP n-ko is not a genuine norm.
is defined by
11.11
notation t ~ n o r mmeans ~~
IIfn-fmll
where
-+
then
m,
.
))frill
The
cpo
CP In this paper we study some properties of the spaces
Cp and
the computation of its dual spaces, which is the key result of this paper. T o make the presentation reasonably self-contained, we shall include a resume of pertinent results and definitions,
2. PRELIMINARIES DEFINITION 2.1 = {g: T
-+
R,
The Lip a
The Lipschitz space continuous,
g(x+h)
-
Lip a
Lip a =
is defined by
g(x) = O ( h a ) )
for
0
< a < 1.
norm is given by
X
DEFINITION 2.2 A
a
The generalized Lipschitz space
= [G: T -+ IR,
0 < a < 2.
The
continuous,
Aa
+ G(x-h)
G(x+h)
-
is defined by 2G(x) = O(ha)]
for
norm is given by
-
= sup IG(x+h) + G(x-h)
mo
“GIIAa
2G(x)I
.
ha
X
We would like to point out that Aa
=. [G: T + R,
1 < a < 2,
A
a
can be defined as
absolutely continuous and
G
0 < a < 1,
and for
*a
G‘E Lip(a-l)]
is the same as
Lip a ,
for for
these two claims we state the following theorem. THEOREM 2.3 ( A . Zygmund). where
1< a < 2
and
absolute constants
N
r
g E Lip(a-1)
G(x)
=
and
M
g(t)dt.
such that
if and only if
,
G E Aa
Moreover there are two NIlgllLip(a-l)
4llGll
Aa
4
415
SPACES FORMED BY SPECIAL ATOMS I1
5
g E A,,
0<
If
Mllgl/Lip(a-l).
a < 1
g E Lip(a-1)
then
if and only if
moreover the norms are equivalent. R.R. Coifman [l] observed that a distribution
real part of a boundary function
=
E C ; I z I < 11)
where
D
(a,),
of p-atoms and a sequence
{ z
(c,),
of numbers, such that
. .
C cnan(t). ( A real valued function n=1 is called a p-atom whenever a is supported on an
and
I C T,
interval
for
if and onlylif there is a sequence
m
c lcnlP < m n=1 defined on T
(F E Hp(D)
F E Hp(D).
~~
m
is the
f
f(t) =
la(t)l
5
III-l/p and
c
= 0.) Moreover,
a(t)dt
m
letting
X(f)
equal the infimum of
lcnlP
over all such repre-
n=1 sentations of
f,
h(f) as
ReHP
for
and
0 < p
2;
there exists absolute constants
M
and
N
such
< NIIFll~p, we shall denote the set of such
f
II fll
= X(f). We point out this result is true R~HP 1 / 2 , however we have to slightly modify the defini-
tion of p-atoms.
We are not going to deal with
p
in this range.
The interested reader is directed to [l]. Notice for
p1 <
p < 1,
Hp
the boundary values o f
must be
taken in the sense of distributions, because, as the example F(z) =
2n
1+2 1-2
shows, one cannot uniquely recover
F
from the
pointwise boundary value of its real part.
3.
SOME PROPERTIES OF
Cp
In this section we state and prove some properties of the space
cP.
LEMMA
3.1.
Cp
is an embedding in
ReHP
for
1
2 < p
2;
1,
that is,
the inclusion mapping is a bounded linear operator. PROOF.
II fll
ReHP
Obvious from the definition of
ReHP
and
Cp,
that
416
GERALD0 S O A R E S DE SOUZA
LEMMA 3.2.
If XI
I C [0,2n],
and
is the characteristic function of an interval I
xI
then
E Cp,
moreover
IlxIll
s
cp A
P
depends only on
PROOF.
A 11 p
Ip
where
p.
One can easily observe that it suffices to prove this
x]
I = [ O , 2l-l where N is a fixed non-negative integer. 2 In The idea is to expand xI in Haar-Fourier series on [0,2n]. theorem f o r
fact, we recall that the Haar system on
is defined by
[0,2rr]
elsewhere. 2n
m
Consequently, the expansion of
XI
is
=
/-
If we split geometry of
Ink
Ink and
as in the definition of
[0,1-2n
shows that
2
and
a&
= 0
n=O k=l
an l
otherwise, thus the expansion of
#
xI
(qnk) 0
ank$nk(t),
then the
for 0 s n < N in Haar-Fourier
series becomes
(3.3)
.
2n 1/2 by computing the coefficients anl we get anl = (E) s 2n where s =-. Substituting these values into (3.3) we have ,N .c so
(3.4) 2-p
=
2n 2P
(F)
qn,(t)
Now observe that
bn(t)
n = O,l,...,N-1,
then we may write (3.4) in the form
(3.5)
are special p-atoms for
SPACES FORMED BY SPECIAL ATOMS I1
xT
therefore
417
E Cp; moreover by definition of “norm” in
we
Cp
I
llxIll
have from (3.5) that
CP
N-1 C
=
A
P
1-p
Zrr
(-)
we have
s [ C (-) n=O 2n
llxIll
n=O
S
is an arbitrary number in
expansion of
=
AplIIP.
then taking
Thus the theorem is
[O,
(0,2rr],
2n -1.
Now if I = [ O , s ] where 2n we can write the dyadic
and apply the above argument.
s
X C O , ~ ]- X C O , ~ ]
that the operator boundedly into
Cp,
xI
and s o
a Taf = f
E
where
in fact
Finaily if
0 < a < @ s 2n,
I = (ale],
any interval, say
XI
Isp,
CP
proved for intervals of the form s
Zrr 1 - p
N-1
I
is
then
On the other hand, observe
Cp.
fa(x) = f(x-a),
\lTafll r:
/IfllCP
,
so
maps
Cp
if we take
CP
I I X ( ~ , ~ ] I I ~ IIx(o,B-al P
11 cP
Ap(B-a)’.
Thus the theorem is proved.
The next result is some sort of Hblderfs inequality between Cp
and
1 P
Lip(-
-
1)
which will be very crucial in determining the
linear functionals on THEOREM 3 . 6 . then
PROOF.
I
i,
(Hblderfs Type Inequality). llgIlLip a
f(t)g(t)dtl
Let
cP.
(IIfll p)l/p C
If
f E Cp
where
a =
and 1
P
g E Lip a
-
1.
f(t) that is,
f
is finite linear combination of
Then
special p-atoms.
f
k
/
I
’T
where
tn E In
,
and thus using the fact that
g E Lip a
we get
418
GERALD0 SOARES DE SOUZA
Then it fol
the theorem is proved for any we have
:C
is dense in
Cp
in
f so
CE.
Now by definition of f E Cp
the extension f o r any
Cp is
just routine, hence the theorem is proved. The next results give us a different way to define a norm in the Lipschitz space COROLLARY 3.7.
only on
Lip a
If f E Cp
and
for
a = P
g E Lip a
-
1. for
1 a = P
-
1
then
p. I
combining these two inequalities involving Lip a-norm we get the desired result.
SPACES FORMED BY SPECIAL ATOMS I1
4 . DUALITY
=
r,
a = Cp.
:$,
Consider the mapping f(t)g(t)dt,
1 P
1.
g
where
Cp
+
R
is a fixed function in
One can easily see that
Cp
is obvious that by the usual argument
is equivalent to the continuity of
g E Lip a,
qg
Cp.
for
,
h
I\glILipa
qg
that is,
is a bounded
Consequently, we have that for each
is a bounded linear functional on
point, a natural question is: on
Lip a
is not a normed space it
I$,(f)l
$,
=
$,(f)
is a linear functional on
JI,
Moreover, despite the fact that
linear functional on
defined by
Cp.
At this
Are these all the linear functionals
We anticipate that the answer is yes; in order to formulate
Cp?
the theorem which leads to this answer, we need some notation. X*
Throughout this paper
X,
will denote the dual space of
Q
that is, the space of bounded linear functionals
X
on
with
the norm
We recall our definition of bounded linear functional just
THEOREM 4.1 (Duality theorem). 1 unique g E Lip a , a = 1 that is, $(f) = A
P
on
[
Q(f) =
p.
~ ( g =)
PROOF.
then
Ap/lgjlLip a
If
1
F<
then there is a
p < 1
such that
f E Cp.
$ E
( C ' ) " .
II$I)
s )IgllLip a
$ = $,
,
Conversely i f
Moreover there is a constant
C+Y: Lip
where
a + (Cp)*
AP
depends only
defined by
is a Banach space isomorphism. $(f)
= { T
f(t)g(t)dt,
Theorem 3 . 6 implies that is,
S
$ E (Cp)*
for all
Therefore the mapping Qg
and
f(t)g(t)dt
JT f(t)g(t)dt
'T such that
-
P
(
If
JI E (Cp)*,
so
JI
then we already have seen that
is a bounded linear functional, that
it remains to prove the other direction. In fact,
GERALD0 SOARES DE SOUZA
420
$ E (Cp)*
let
Observe t h a t
and d e f i n e
A
P
- G(s)
G(s+h)
J,
t h e b o u n d e d n e s s of
G(s)
=
Jl(x[,,,])
= J ~ ( X ( , , , + ~) ~ and t h u s Lemma 3 . 2 and
t e l l s us t h a t
o t h e r h a n d , u s i n g t h e d e f i n i t i o n of
since
i n (4.2)
so that
IlGII,,
<
s
A hp
O n the
we g e t
1
x
[s,s+h]
//bllCP s 1.
where
P
i s continuous.
1 = 7 x t) -
b(t)
a s p e c i a l p-atom we h a v e J,
G
G
(2h)'
n e s s of
- G(s)l
lG(s+h)
i s an a b s o l u t e c o n s t a n t , t h e r e f o r e
Consequently,
s E [O,Zrr].
for
is
t) [s-h,~]
T h e r e f o r e u s i n g t h e bounded-
we g e t and t h e r e f o r e
0 ,
G E
Aa
a = l/p.
for
So b y
a t h e r e m a r k s made r i g h t a f t e r D e f i n i t i o n 2.2 we h a v e t h a t
c
absolutely continuous, therefore there e x i s t s a function G(s) =
such t h a t
1 g E Lip(p
I,
-
g(t)dt.
~ ( t ) g ( t ) d t and t h u s i f
=
Jl(x,)
-1 b(t) = 7 X & 1111 $(b) =
I
)
+
b(t)g(t)dt
AP
constant
II gII Lip
T
$(X[:o,sl)
=
.III~/P X,(t) 1
a
i s a s p e c i a l p - a t om t h e n
and t h e r e f o r e f o r any
By C o r o l l a r y
d e p e n d i n g o n l y on
for
a
Consequently
1 = P
-
1
p
f E Cp
such t h a t
ApllgllLip
d e f i n e d by
bnfs
a
IlJlll
4
4
and t h u s t h e t h e o r e m i s p r o v e d .
m
< p <
we h a v e
3.7 we h a v e t h a t t h e r e i s a
I n [ 3 ] t h e author a l s o introduced t h e spaces
where
on
i s any i n t e r v a l
= fT x , ( t ) g ( t ) d t .
$ ( t )= fT f ( t ) g ( t ) d t .
2
g
[O,Sl
have t h a t
4
is
Thus b y Theorem 2 . 3 we h a v e
O n t h e o t h e r hand t h i s i m p l i e s t h a t
1).
G
Bp = { f : T + R ; f ( t ) =
a r e s p e c i a l p-atoms
Bp
for m
C
c n b n ( t ) ; C Icnl<m) n =1 n=1
defined i n the introduction.
We
SPACES FORMED B Y SPECIAL ATOMS I1
421
m
IIf(IBp
= Inf
C lcnI where the infimum n=1 is taken over all possible representations of f. One of the main endow
Bp
with the norm
results about
Bp
is the following.
JI
TIIEOREM 4 . 3 (Duality Theorem). on
Bp
for
a = -l - l
1 7 < p < 1 if and only if there is a unique
such that
P
is a bounded linear functional
rp:
over the mapping
$(f) = Lip a
I,
f(t)g(t)dt
(BP)"
-t
for all p(g) =
defined
f
JI
g E Lip a,
E Bp.
More-
is an iso-
metric isomorphism. The proof of this theorem follows basically the same line as Cp, that
however we point out that a basic difference between them is Cp
is a Frbchet space while
Bp
is a Banach space.
Certain-
ly the fact that
Bp
is a Banach space makes it easier to work
with this space.
F o r example by using the duality theorem and the
Hahn-Banach theorem one can prove without much difficulty that can be identified with the space of analytic functions disk
I D = ( z E C,
'0
IzI
< 1)
10
Bp
together these results about
Bp
1
? < p < 1.
the famous duality theorem for tells us that
Cp c Hp
1
a = P
-
1.
Hp
Cp
Hp;
is shows
again we
Now if we put
we have as a consequence
in [ 2 ] .
In fact, Corollary 3.1
continuously, on the other hand one easily
can see from definition of Therefore we have
and
Bp
We point out that [ 2 ]
is the smallest Banach space containing
emphasize that we are working with
for
Romberg and A.L.
With this identification one can say that
a real characterization of such spaces. that
on the
f
satisfying
These spaces were introduced by P.L. Duren, B.W. Shields in [ 2 ] .
Bp
Hp
Cp C Hp C Bp
and
Bp
that
Hp C Bp continuously.
which implies
Thus we get the duality of
Hp
Lip a C (Hp)* C L i p a for
$<
p < 1.
422
GERALD0 SOARES DE SOUZA
(Hp)* = Lip(p1
That is,
-
1).
One natural question to ask is: and
N
dependent only on
for
3< p <
p
such that
In other words are
1 ?
as Frgchet space?
Do there exist constants M
I f]lHP
MllfllCP
Cp
Hp
and
I;
NIlfllCP
the same space
The answer to this question is negative, and
will come out shortly in a joint paper with Gary Sampson [ 7 ] . 5 . INTERPOLATION THEOREM In this section we present an elementary theorem on the interpolation of operators acting on
Cp
In order to state
spaces.
it we need some definitions.
Y >
0
Let
f
be a real valued measurable function on
let
m(f,u)
I{.
= m(lfl ,Y) =
E T, If(X)
called distribution function of
T. m(f,y)
on
I*[
f,
I
>
~1
I.
T.
For
m(f,~)
is
means the Lebesgue measure
is non-negative, non-increasing and continuous from
the right. A
sublinear operator
T
is a mapping from linear space of
measurable functions defined on a measure space into measurable functions defined on another measure space satisfying i) ii)
IT(f+g)(x)l
ITf(x)l
5
ITafl = la( ITfl, whenever
A sublinear operator
IlTll = supEIITflly:
I,
IIfllX
A function If(t) IPdt)l’P
I;
is a scalar and
TI X
-t
Y
is in
is said to be bounded if
f
<
defined on m ,
f
T
is said to be in belongs to
I(f/lp =
for which
L(p,-),
LP
.
We shall say that
usually called weak
space, if there exists a positive number A A P (7) In both definitions p lies in (0,m).
.
f
13 <
a measurable function
L P
a
almost everywhere.
T.
domain of
= (
+ ITg(x)l
such that
m(f,y)
The least constant
SPACES FORMED BY SPECIAL ATOMS I1
L P
in the definition of weak
A
is considered as "norm" in L(p,m).
IIflli
We recall that one can easily show that / -
ape' rn(f,a)da
II fl
that is,
THEOREM 5.1.
Let
T: L P1
1 2 < po < p
L(pl,m)
-t
=
is continuously embedded in
(+)".
m(f,y)
linear operator mapping and
L P
and that
0
L(p,=),
423
1 < p1 <
I;
PO
T: C
-I
with norm
-.
T
is a sub-
boundedly with n o r m
L(po,m) M1,
Suppose
MO
that is
respectively. t 1-t L with llTfllp I KMOMl (Ilfll p)l'p, P C an absolute constant depending only on po, p1 and
T: C p
Then
,
+-
-t
where p,
is
K
- --
t +
p
Po
0 < t < 1.
p1
PROOF.
f
Let
+w 1
(5.3)
xL(t).
be a special p-atom, that is, Therefore P -Po
(IIfll
P -Po
and
1 1 1 7
4
C We now evaluate
-1 f(t) = 7 - X R ( t ) 1111
(Ilfl ,
IP1
1
III
in terms llTflIp using the definition of LP-llnorml'
of the distribution function,
We have, m
11lTfIl; P aP"m(Tf
=
,U)da
EP'lm(Tf
+
aP-'m(Tf
,a)da =
,U)dU,
Then (5.2) implies that
Using (5.3) and the hypothesis on
p f s we obtain
u >
0.
+
424
GERALD0 S O A R E S DE SOUZA
PO 1
- I(Tfl1; P
-I P-Po
1
p
MI
u
,
- P'P1 (7
B
where
is a
We get PO
(5.4)
III
PI-P
u = B 1I-F
is arbitrary we may take
constant to be determined.
Since
P-PI
P1
- P-Po
1
u
As
P-Po
MO
1
Mo
P IlTflIpP
P-Po
P1 M1
=P-P0
+ - P1-P B
B > 0 is arbitrary, replace B
P'P1
by the value that makes the
expression minimal, namely take
Substitutin in (5.4)
. p1-p
Po
5.P-Po
P1-Po Mo
Po
Observe that P 1-t =
and
PI
P1-P P1 .+ -.-P1-Po P
P-Po
.-
-
P
P1-Po'
( 5 . 5 ) becomes
IITfllp
k
h(t) =
if
Z ' cnbn(t) n=1
p-atoms, that is,
bnls
then
P-Po P1'PO -= p
- 1.
- + - lmt Po
p1
. M1
Taking
,
P1-P .-P1-Po
Po t =-.-
P
0 < t < 1,
.
(p-p,
.
Pl'PO
and thus
Consequently,
is a finite linear combination of special are equal p-atoms of type
f, we have
and thus
(5.6)
IIThllp
t 1-t m O M l ( ~ ~ h ~ l C p ) l where ~P
p
K = (p )P-Po , + P1 P
That is, the theorem is proved for a dense subspace of Cg
,
so
it can be extended to all
( 5 . 6 ) in
Cp.
Then,if
f
E Cp,
Cp,
U P
Cp,
'
namely
preserving the inequality
( 5 . 6 ) implies that
SPACES FORMED B Y SPECIAL ATOMS I1
IITfllp
l-t(l fl
KMOMl
. ' / ' )
425
The p r o o f is completed.
CP
The interested reader can find similar types of interpolation theorem in [ 81 and 191
.
REFERENCES 1.
R.R. COIFMAN,
A real variable characterization of
Hp,
Studia Math., 51 (1974) 269-274. 2.
P.L. DUREN, B.W. ROMBERG and A.L. SHIELDS, Linear functionals on Hp with 0 < p-1, J. Reine Angin Math., 238 (1969) 32-60.
3.
GERALD0 SOARES DE SOULA,
Spaced formed by special atoms,
Ph.D. dissertation, SUNY at Albany, 1980.
4.
.......................
5.
-----------------------, Spaces
and RICHARD O"EIL, Spaces formed with special atoms, Proceedings Conference on Harmonic Anal y s i s , 1980, Italy. Rendiconti del Circolo Matematico di Palermo, 139-144, Serie 11, #l. 1981. formed by special atoms I,
to appear, Rocky Mountain Journal o f Mathematics.
6.
.......................
and GARY SAMPSON,
A real character-
ization of the pre-dual of the Bloch functions, to appear, London Journal o f Mathematics.
7.
....................... acterization o f
and
------------, An
cP, In preparation.
8.
-----------------------, Two theorems
9.
-----------------------, An interpolation
10
11.
analytic char-
on interpolation of operators, Journal of Functional Analysis, 46, 149-157, (1982).
o f operators, to appear, Anais la Academia Brasileira de Cigncias 54, # 3 (1982).
-----------------------, The dyadic
special atom space, Proceeding Conference on Harmonic Analysis, Minneapolis, April 1981, Lectures Notes in Mathematics, #908, Springer-Verlag, 1981. A. ZYGMUND, Trigonometric Series, 2nd red. ed., Vols. I,II, Cambridge University Press, New York, 1959.
Department of Mathematics Syracuse University Syracuse, New York 13210
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Fui~ctionul~4rralysis, Holornorphy and Approximation Theory 11, C.I. Zapata (ed.) @ Ekevier Sciencc Publishers R. V. (North-Holland), 1984
427
A HOLOMORPHIC CHARACTERIZATION OF c*-ALGEBRAS
Harald Upmeier
$1. INTRODUCTION
The theory of operator algebras (i.e. C*-algebras and von Neumann algebras on complex Hilbert spaces) is of increasing importance to many branches of mathematics, e.g.
integration theory,
operator theory, algebraic topology and in particular mathematical physics and quantum mechanics.
Since C*-algebras provide a natural
framework for the foundations of quantum mechanics and quantum field theory it is an important problem to characterize the class of C*algebras by certain properties, for instance motivated by physical experiments.
So far two characterizations of operator algebras in
different categories have been obtained.
The first is A. Connest
characterization of von Neumann algebras in terms of self-dual homogeneous Hilbert cones [ 81,
the second is the work of Alfsen and
Shultz [2,1] characterizing the state spaces of C*-algebras using the geometry of compact convex sets and their affine function spaces. Although the methods of these papers are quite different, approaches have a common feature:
both
the characterization of C*-al&ras
in the respective category can be divided into two steps, the first being the characterization of the larger class of Jordan C*-algebras (JB*-algebras) and the second being the characterization of (associative) C*-algebras among all (non-associative) Jordan C*-algebras. ( F o r the case of self-dual homogeneous Hilbert cones this point of
view has been adopted by Bellissard and Iochum
[S]).
The second step
428
HARALD UPMEIER
i n v o l v e s some concept of " o r i e n t a t i o n " of The g e o m e t r i c
2
Torden o p e r a t o r a l g e b r a .
o b j e c t s a s s o c i a t e d w i t h an o p e r a t o r a l g e b r a A
i n t h e p a p e r s mentioned above a r e t h e H i l b e r t cone a s s o c i a t e d w i t h A
v i a Tomita-Takesaki t h e o r y ( i f
s p a c e of ometry. A
A,
A
h a s a p r e d u a l ) and t h e s t a t e
r e s p e c t i v e l y , b o t h endowed w i t h a n a t u r a l a f f i n e ge-
On t h e o t h e r hand,
D
t h e open u n i t b a l l
h a s an i n t e r e s t i n g holomorphic s t r u c t u r e :
D
of a C*-algebra
i s homogeneous with
r e s p e c t t o holomorphic automorphisms and i s t h e r e f o r e a bounded symm e t r i c domain.
The aim of t h i s p a p e r i s t o g i v e a holomorphic
c h a r a c t e r i z a t i o n of C*-algebras
i n terms of t h e holomorphic s t r u c -
t u r e of t h e a s s o c i a t e d open u n i t b a l l . f o r J o r d a n C*-algebras
The c o r r e s p o n d i n g r e s u l t
i s t h e main theorem i n [ 6 ] .
a c h a r a c t e r i z a t i o n of C*-algebras
Here we o b t a i n
among a l l J o r d a n C*-algebras
u s e s t h e s t r u c t u r e of t h e L i e a l g e b r a of a l l d e r i v a t i o n s .
which
A similar
i d e a i s u n d e r l y i n g i n [ B ] ; t h e c o n c e p t of H i l b e r t cones however i s o n l y a p p r o p r i a t e f o r o p e r a t o r a l g e b r a s h a v i n g a Banach p r e d u a l . I n $ 2 we g i v e a s h o r t i n t r o d u c t i o n i n t o t h e t h e o r y of J o r d a n C*-alg e b r a s and t h e i r holomorphic c h a r a c t e r i z a t i o n i n c l u d i n g t h e cons t r u c t i o n of t h e J o r d a n t r i p l e p r o d u c t f o r bounded symmetric domains i n complex Banach s p a c e s .
In
$3 some p r o p e r t i e s o f t h e L i e a l g e b r a
of a l l d e r i v a t i o n s of a J o r d a n C*-algebra needed i n t h e s e q u e l .
The c o n c e p t of " o r i e n t a t i o n " o f a J o r d a n C*-
algebra i s introduced i n result,
a r e s t u d i e d which w i l l be
$4,
and
$ 5 c o n t a i n s t h e p r o o f of t h e main
s a y i n g t h a t o r i e n t a b l e J o r d a n C*-algebras
$ 2 . JORDAN c*-ALGEBRAS
AND THEIR
HOLOMORPHIC
a r e C*-algebras.
CHARACTERIZATION
J o r d a n a l g e b r a s made t h e i r f i r s t a p p e a r a n c e i n m a t h e m a t i c a l p h y s i c s and quantum t h e o r y ,
( P . J o r d a n 1932):
Let
H
s t a r t i n g from t h e f o l l o w i n g o b s e r v a t i o n be a complex H i l b e r t s p a c e .
Then t h e
A HOLOMORPHIC CHARACTERIZATION
Banach space
#(H)
OF c*-ALGEBRAS
429
H (which
of all bounded hermitian operators on
can be interpreted as (bounded) observables of a quantum mechanical system) is not closed under the associative operator product
xy,
but with respect to the anti-commutator product
#(H)
becomes a non-associative algebra.
As a consequence the anti-
commutator product of two observables has a physical interpretation whereas, i n general, the operator product does not.
The product
(2.1) satisfies the following identities: (J1)
xay = yox
(52)
x o(x0y) = xo(x
(Commutativity)
2
2.2 DEFINITION.
Let
A
2
(Jordan-Identity).
oy)
be a (not necessarily associative) algebra
over the real o r complex numbers with product denoted by all
x,y E A.
Then
A
xoy
for
is called a Jordan algebra i f the identities
(Jl) and (J2) are satisfied. Since the algebras appearing i n quantum mechanics are in general infinite-dimensional, it is desirable to consider Banach Jordan algebras.
A particularly important class of Banach Jordan
algebras are the so-called Jordan C*-algebras (i.e. JB*-algebras and JB-algebras) which have been introduced and thoroughly studied by Alfsen, Shultz and St$rmer
2 . 3 DEFINITION. and unit element
Let e.
X
131.
be a real Jordan algebra with product
Then
X
is called a JB-algebra i f
Banach space with respect to a norm x,y 6 X
X
is a
x ~ 1 x 1 such that f o r all
the following properties hold:
xoy
430
HARALD UPMEIER
I*I
I n t h i s c a s e t h e JB-norm
on
i s u n i q u e l y d e t e r m i n e d by ( i )
X
and ( i i ) . Note t h a t J B - a l g e b r a s a r e " f o r m a l l y r e a l " , i.e. 2 2 x = o f o r a l l x1 xn E X . x1 +...+ xn = o i m p l i e s x1 =...= n
,...,
The complex a n a l o g u e of J B - a l g e b r a s
2.4 DEFINITION. ZOW,
Let
u n i t element
a JB*-algebra
u,v E Z
such t h a t f o r a l l luovl
(i) (ii)
and i n v o l u t i o n
Then
Z I - Z * .
i s a Banach s p a c e w i t h r e s p e c t
Z
if
be a complex J o r d a n a l g e b r a w i t h p r o d u c t
Z
e
a r e t h e s o - c a l l e d JB*-algebras:
i s called
Z
t o a norm z - l z l
t h e following p r o p e r t i e s hold:
IUI*IVI
I;
=
IEUU*U31
IUI
3
9
where
(2.5)
{Uv*w)
:=
uo
-
(v*ow)
v*o
denotes t h e Jordan t r i p l e product o f Wright and Youngson [29,30] i n t r o d u c e d above a r e e q u i v a l e n t : Z := X'
plexification
+
(w.u)
wo
(uav*)
u,v,w E 2 .
have shown t h a t t h e c o n c e p t s Given a J B - a l g e b r a
X,
becomes a JB*-algebra
= X @ i X
t h e com-
with respect
t o a unique ftJB*-normff; c o n v e r s e l y t h e s e l f - a d j o i n t p a r t
x of a JB*-algebra JB*-algebras
:= [ x E Z : x* = x)
i s a J B - a l g e b r a under t h e r e s t r i c t e d norm.
Z
and J B - a l g e b r a s a r e o f t e n c a l l e d " J o r d a n C*-algebras"
b e c a u s e o f t h e f o l l o w i n g example.
2 . 6 EXAMPLE.
Let
be a u n i t a l C*-algebra,
Z
i.e.
Z
i s a complex
a s s o c i a t i v e Banach * - a l g e b r a w i t h u n i t s u c h t h a t product, i n v o l u t i o n and norm a r e r e l a t e d by t h e c o n d i t i o n
for all
z E 2.
Then
Z
becomes a JB*-algebra
commutator p r o d u c t ( 2 . 1 ) and t h e JB*-norm
under t h e a n t i -
c o i n c i d e s w i t h t h e C*-norm.
To s e e t h i s , n o t e t h a t f o r a s s o c i a t i v e * - a l g e b r a s product (2.5)
1 (uv*w + [ uv*w} = 2 I n particular,
t h e Jordan t r i p l e
reduces t o
[zz*z)
= zz*z.
wv*u)
.
This implies, v i a the s p e c t r a l
theorem for h e r m i t i a n o p e r a t o r s and t h e C*-condition:
A HOLOMORPHIC CHARACTERIZATION
OF c*-ALGEBRAS
431
In a fundamental paper [12] Jordan, von Neumann and Wigner classified all formally-real (simple) Jordan algebras of finite dimension.
A natural extension of this classification is the follow-
ing list of all JB-factors of type 1 [ 2 2 , 3 ] :
2.7 EXAMPLE. (i)
Let
M
denote the field
R
of real numbers, the field
of complex numbers o r the skew-field respectively. H(E)
Let
E
IH
be a Hilbert space over
C
of quaternions,
M.
Then the set
of all bounded K-linear self-adjoint operators on
E
is a
JB-algebra under the product (2.1) and the operator norm. (ii)
Let
Y
be a real Hilbert space of dimension
scalar product
(xly).
Then the direct sum
r2
with
X := R
@
Y
becomes a JB-algebra called spin factor under the product
(a1
Y,)
0
(a2 CB Y,)
:=
(ala2
+ (y11y2))
CB (aly2 +
a2y1).
The name "spin-factor" stems from the fact, that in quantum mechanics spin systems obeying the canonical anti-commutation relations are in close connection with Jordan algebra representations of a suitable spin factor as defined above. (iii)
The set
Z3(CD)
of all self-adjoint
9x3 octonion ma-
trices is a JB-algebra under the product (2.1) which cannot be embedded into an associative algebra and is therefore called the exceptional Jordan algebra.
The octonion matrices of higher
rank do not form a Jordan algebra, since
0
is not associative
[ 7 ; Ch. VIII, Lemma 8 . 2 1 . Jordan algebras and JB-algebras in particular have found several applications to quantum mechanics, for instance using the affine geometry of the state space of a JB-algebra or the projective geometry associated with the exceptional Jordan algebra.
How-
ever, even more promising seems to be the relationship between J o r d a n C*-algebras and complex analysis, more precisely the Jordan algebraic
432
HARALD UPMEIER
description of bounded symmetric domains i n complex Banach spaces. F o r the sake of completeness we give a short survey about this relationship. a mapping
Z
Given a complex Banach space f: D
a E D
each point
is said to be holomorphic if locally around
Z
-+
D C Z,
and a domain
there exists a convergent expansion m
Pm: Z -+ Z.
into a series of m-homogeneous continuous polynomials The polynomials
where
f(m)(a)
Pm
are uniquely determined by
is the m-th derivative of
ed as a multilinear mapping [18; $51.
in
f
and
f
a E D,
all biholomorphic automorphisms of a domain
D
matter of notation, denote by
Z(E)
namely
consider-
A bijective mapping which is
holomorphic in both directions is called biholomorphic.
led the autornorphism group of
a,
D
The set of
forms a group, cal-
and denoted by
Aut(D).
As a
the algebra of all bounded
linear operators on a real or complex Banach space
E.
Generalizing a well-known theorem of H. Cartan for domains in
Cn,
the author [ 2 4 ]
and independently J.P. Vigu6 [ 2 8 ] have s h m
that for a bounded domain automorphism group
D
in a complex Banach space
G = Aut(D)
The essential idea behind G
this result is the construction of the Lie algebra of set of all complete holomorphic vector fields on
5
on a domain
DC
2
6 = h(z) where
h: D -+ Z
cates that
6
the
carries in a natural way the
structure of a real Banach Lie group.
vector field
Z,
D.
as the
A holomorphic
can be written as
a , az
is a holomorphic mapping and the symbol
a az
indi-
is viewed as a holomorphic differential operator,
associating to each holomorphic mapping
f: D -+ Z
the holomorphic
A HOLOMORPHIC CHARACTERIZATION OF c*-ALGEBRAS
mapping
5f: D
-I
Z
f’(z) E X ( Z )
where
433
defined by
is the derivative of
f
in
z
E D.
Given
another holomorphic vector field
the commutator vector field
has the characteristic property
for all holomorphic mappings vector space
3(D)
f: D
-t
2.
It follows that the complex
of all holomorphic vector fields on
a Lie algebra under the commutator product.
D
Regarded as an ordina-
ry differential equation, each holomorphic vector field on nerates a local flow on
D.
A
becomes
ge-
D
vector field is called complete if
D.
this local flow can be enlarged to a global flow on
An equiva-
lent formulation is given in
2.8 DEFINITION.
A holomorphic vector field
5
= h(z)
aZ
on
D
is
called complete if there exists an analytic real 1-parameter group tl-gt
E Aut(D)
for all
z E D.
satisfying the differential equation
In this case the transformations
determined for all
since locally
gt
t E R
aut(D).
are uniquely
and are denoted by
is given by an exponential series.
The set of
D
is denoted
all complete holomorphic vector fields on a domain by
gt
For domains i n general,
aut(D)
is not closed under
434
HARALD UPMEIER
In case
formation of sums and commutators. aut(D)
Q:=
D
is bounded, however,
Satz 2 . 6 1 .
9
Moreover
nential mapping
r24;
3(D)
turns out to be a real Lie subalgebra of
is a Banach Lie algebra and via the expo-
5-exp(la5),
this Banach Lie algebra induces on
the structure of a real Banach Lie group with Lie algebra j.
Aut(D)
As a consequence of Liouvillels theorem, we have
9 n is, =
0.
A crucial property of bounded domains is the following result.
2 . 9 CARTANIS UNIQUENESS THEOREM. g E Aut(D)
Suppose some
PROOF.
a E D.
f
of
g
a =
g f idD.
Assume,
0.
o
o
of order
for
Choose
k
2
5
there is an expansion
h
By induction, the n-th iterate
>k.
gn
has the expansion
= z
gn(z) where
g’(a) = idZ
is a k-homogeneous continuous polynomial and
0
vanishes in
and
be a bounded domain.
g = idD ’
Then
We may assume
pk
g(a) = a
satisfies
minimal, such that around
where
Dc Z
Let
>k.
h’ has order
bounded domain
D
+
+
nPk(z)
h‘(z),
Since the transformations
gn
leave the
invariant, it follows from Cauchyfs inequalities
[18; 5 6 , Prop, 31 that geneous polynomials.
{nPk : n
2
This implies
O}
is a bounded set of k-homo-
Pk =
0,
a contradiction. Q.E.D.
By differentiation, Theorem 2 . 9 implies that each vector field
5 =
f o r some
h(z)
aE a Z
a E D
aut(D)
satisfying
h(a)
= o
and
h’(a)
= o
must vanish identically.
The possibility of describing a bounded domain ically (e.g. in terms of
Aut(D)
or
aut(D))
D
algebra-
can only be expected
HOLOMORPHIC CHARACTERIZATION OF c*-ALGEBRAS
A
if
D
435
admits sufficiently many holomorphic automorphisms. The class
of bounded symmetric domains defined below fulfills this requirement in an ideal manner. 2.10 DEFINITION.
D
A bounded domain
a E D
is called symmetric if for each point s E Aut(D) a
~'(a) = -idZ ' a
By Theorem 2.9 the automorphism a,
is uniquely determined.
bounded symmetric domain
D,
transitive on
D
s a , called the symmetry
Moreover,
D
is homogeneous, i.e.
and
a
is an
Aut(D)
is
and is biholomorphically equivalent to a bounded
satisfying
o E D
and
e
it
D = D
mains of this kind are called circular).
D
homogeneous domain
o E D
2 a = idD
s
s a . It can be shown [28,14] that each
isolated fixed point of
domain
there is a mapping
with the following properties: s (a) = a, a
in
Z
in a complex Banach space
for all
t E R
(do-
Conversely, every circular
is symmetric, since the symmetry
can be transported to any other point in
D.
zu-z
at
In particular,
is a bounded symmetric domain. It turns out that there is a natural way of associating with each bounded symmetric domain a Jordan triple product generalizing the triple product ( 2 . 5 ) .
The following Lemmas are the crucial
steps towards this algebraic construction. 2.11 LEMMA.
D
Let
field
5 = h(z)
h = h
+ hl + h 2 ,
polynomials.
a=
E
be a bounded circular domain.
9
:= aut(D)
where
Then each vector
is polynomial of degree
hk: Z + Z
4 2 , i.e.
are k-homogeneous continuous
Moreover, the vector fields
436
HARALD UPMEIER
belong to
3
and there is a Cartan decomposition
y = 4 e p
(2.12) where
4
PROOF.
[s,
:=
Since
:
D
be the expansion of vector fields
]
,b :=
and
5,.
since
:
5 E y].
a E 9 az
.
Therefore m
is a Lie algebra.
$4
5 =
Let
5,
C m= o
3
5 E
(5-,
6 := iz
is circular,
:= ad(6) E C( J ) ,
8
y
5 E
I
around
into m-homogeneous polynomial
o
By Euler's relation
This implies for any polynomial
p E El[@],
m
2 p(8) = 8 ( e +l).
Choose
~(8)s = o for all
Then
p(i(m-1))
= o
as a consequence of Theorem 2.9.
m > 2, which implies
as asserted.
Further,
5,
=
2
5-,
= -0 5 E
0.
y
for
m
But
Hence
2.13 LEMMA.
Let
D
Ad(s),
f o
5 = 5 , + 5, + g 2
5,
and hence
where
hence
p(i(m-1))
I t is clear that (2.12) is a Cartan decomposition of the involutive automorphism
2,
5;
:=
s(z)
- z-,
= 5
ZXZXZ + Z
denoted by
E f.
relative to
j
Q.E.D.
-z.
be a bounded symmetric circular domain.
there is a unique mapping
,
Then
( u , a , v ) c {ua*v]
having the following properties: (i)
{ua*v]
is complex bilinear symmetric in the outer
variables and conjugate linear in the inner variable a E Z . (ii)
The subspace
p
p =
c j [(U
defined in 2.11 has the form
-
{ZU".])
Moreover, defining the operator I=
u
a v*
: u E z].
on
Z
by
(u
V*)Z
{ u v " ~ ] , the following Jordan triple identity is valid:
:=
A HOLOMORPHIC CHARACTERIZATION OF c*-ALGEBRAS
u,v,x,y E Z ,
for all
[x,~]
where
-
:= ? q ~ p?,
437
denotes the com-
mutator of linear operators. PROOF.
D
Since
is homogeneous, it follows from (2.12) that the
evaluation mapping I ) -+ Z , ive.
defined by
h(z)
By definition, the vector fields in
(u-qu(z))
=, a
where
Z + Z
9 , :
u
u
?
c p rp,p11
c
(since (since
Then (i) and (ii) are satisfied. sequence of the fact that
/J
is surject-
have the form
is a 2-homogeneous polynomial,
which is uniquely determined by depends conjugate linearly on
a. a -h(o),
D
3 n
is bounded) and iJ =
Define
0).
Property (iii) is a direct con-
is a Lie triple system, i.e.,
p.
A Banach space
Z
with a composition
{ua*v}
satisfying
properties (i) and (iii) is called a Banach Jordan triple system. Using Jordan triple systems, W. Kaup [ l ' r ] has obtained an algebraic
In particular,
characterization of all symmetric Banach manifolds.
the Banach Jordan triple system associated to each bounded symmetric domain
D
D
via 2.13 characterizes
uniquely.
A particularly
important class of Jordan triple systems are the JB*-algebras with Jordan triple product ( 2 . 5 ) .
Therefore the problem arises which
bounded symmetric domains correspond to JB*-algebras.
It turns out
that the appropriate holomorphic condition relates to the notion of tube domain. 2.14 DEFINITION. tion
Z := X
6:
.
Let Let
X
be a real Banach space with complexifica-
R C X D~
:=
be an open convex cone.
iz
E
z
:
z-z*
2i E n ?
is called the tube domain with the base the conjugation of
Z
Then the domain
with respect to
R. X.
Here
z
HZ*
denotes
HARALD UPMEIER
438
X := R
I n case
n
and
:= ( x E R : x > o } ,
i s t h e f a m i l i a r upper h a l f - p l a n e
in
( x 2 : x E X}
n.
terior
of
,
the
i s a convex cone h a v i n g non-empty
in-
i s c a l l e d t h e upper
Dn
The holomorbhic c h a r a c t e r i z a t i o n
Z.
i s g i v e n i n t h e f o l l o w i n g theorem, t h e main r e s u l t
which g e n e r a l i z e s t h e p i o n e e r i n g work by M .
2.15 THEOmM.
A bounded symmetric domain
a l g e b r a i f and only i f domain.
D
More p r e c i s e l y ,
o f a JB*-algebra
n
X,
o f t h e JB*-algebra
JB*-algebras
o f [ 61
For any JB-algebra
The a s s o c i a t e d t u b e domain
half-plane
D
T h e r e f o r e t u b e domains a r e
C.
o f t e n c a l l e d ‘!generalized h a l f - p l a n e s “ . s e t of s q u a r e s
t h e domain
D C Z
Koecher [ 171
.
b e l o n g s t o a JB*-
i s biholomorphically equivalent t o a t u b D :=
t h e open u n i t b a l l
( z E Z
: IzI
<
l}
i s a bounded symmetric domain which i s biholomor-
p h i c a l l y e q u i v a l e n t t o t h e upper h a l f - p l a n e
of
Dn
Z.
Conversely,
each bounded symmetric domain which i s b i h o l o m o r p h i c a l l y e q u i v a l e n t t o a t u b e domain can be r e a l i z e d a s t h e open u n i t b a l l o f a JB*algebra.
F o r t h e p r o o f of Theorem 2.15 which u s e s t h e Gelfand-Naimark embedding theorem f o r J B - a l g e b r a s [ 3 ]
and a d e t a i l e d a n a l y s i s o f
t h e L i e a l g e b r a of a l l complete holomorphic v e c t o r f i e l d s on t u b e domains, t h e r e a d e r i s r e f e r r e d t o [ 1 6 , 6 ] .
I t should be noted t h a t
t h e biholomorphic e q u i v a l e n c e between t h e open u n i t b a l l JB*-algebra
Z
and i t s upper h a l f - p l a n e
t h e s o - c a l l e d Cayley t r a n s f o r m
Dn
u: D -+ D
R
D
of a
i s g i v e n by means of
which i s c o m p l e t e l y de-
f i n e d i n J o r d a n a l g e b r a i c t e r m s , namely ~ ( z =) ( z t i e )
where
e
i s t h e u n i t element of
Z
taken i n t h e Jordan t h e o r e t i c sense.
0
(etiz)”
,
und p r o d u c t and i n v e r s e a r e
A HOLOMORPHIC CHARACTERIZATION OF c*-ALGEBRAS
439
$ 3 . DERIVATIONS OF JORDAN c*-ALGEBRAS The Jordan algebraic characterization of bounded symmetric domains
and symmetric tube domains developed in $2 was based on
D
a detailed analysis of the Lie algebra D.
holomorphic vector fields on
D
9
= aut(D)
of all complete
It is shown in $4 that for domains
equivalent to a tube domain there is a natural notion of "orien-
tation" associated with
$
characterizing the class of all C*-al-
This concept is similar to the one introduced in [ 8 ] , but
gebras.
has to be modified in order to apply to JB*-algebras in general which need not be Banach dual spaces.
The appropriate modifications
are motivated by some results on derivations of Jordan C*-algebras [25,26,271. Let space
Z.
circular.
D
be a bounded symmetric domain in a complex Banach
D
Without loss of generality we may assume that Let
9 =k@P
9
:= aut(D).
Lie algebra
is
be the Cartan decomposition (2.12) of the By definition of the Jordan triple
product, the vector fields in
p ,
which are sometimes called
"infinitesimal transvections", have the form (u
-
{ZU*Z])
a az
u E Z
for
and are therefore explicitly given in terms of the
Jordan triple product associated with
D.
Our first aim is to give
a Jordan triple characterization of the Lie algebra
3.1 LEMMA. if
?, E X ( Z )
for all
t E
& a
belongs to
h
if and only
is a derivation of the Jordan triple system
irUV*Wi = r(Au)v*wi
PROOF.
?,z
:
Z,
i.e.
u,v,w E Z:
(3.2)
all
A linear vector field
4
Suppose, !R,
?, E X ( Z )
+ E ~ ( x ~ )+* E ~ ~I ~ * ( I ~ ) ~ . satisfies the identity (3.2).
the invertible transformation
Then for
gt := exp(t?,) E G L ( Z ) is
440
HARALD UPMEIER
The domain
D
has several characterizations in terms of the Jordan
triple product (cf. [14; $31) which imply that
t E
By definition o f
R.
k, 5
Conversely, suppose since
5
is linear.
z,u E 2 .
:=
a 12 7 a
E
h
.
xz
a
Then
a.
E
[s, p ]
c f
,
This implies
+ (z(xu)*z}
= 2[(Xz)u"z}
X[ZU*Z}
for all
it follows that
for all
gt E Aut(D)
By polarization, ( 3 . 2 ) follows.
Q.E.D.
The Lie algebra of all derivations of a Jordan triple system 2
is denoted by
By Lemma 3.1,
aut(Z).
has also a direct interpretation in terms of the Jordan triple
D.
system associated with
Using the commutator notation for
linear operators, the defining identity ( 3 . 2 ) of a derivation can b e reformulated as follows: cx,u
u,v E Z.
for all := {uv*z).
0
= (xu)
V*]
Here
u
0
0
v*
v* E X(Z)
Using the fact that
[ p ,?]
ing Jordan triple identity 2.13.iii u
whenever
u,v E Z.
v*
+ u
-v
u*
(xv)*
0
is defined as
c
h
(u
0
V*)Z
:=
or checking the defin-
one can easily show that
E aut(z)
The linear subspace generated by these oper-
ators is an ideal of the Lie algebra all inner derivations of
Z
aut(Z),
and denoted by
called the ideal of int(Z).
The inner
derivations will play an essential role in the sequel. Suppose now that
D
C Z
is a bounded symmetric domain of
"tube type", i.e. having a realization as a tube domain.
By
A HOLOMORPHIC CHARACTERIZATION
Theorem 2.15 we may assume that
Z.
algebra structure on
is the open unit ball of a JB*-
D
D
Further,
Z
valent to the upper half plane of
Dn := where
defined by
z-z*
denotes the involution of
z -z*
or of the convex cone
(x2 : x E X)
X := (x E Z : x* = x)
gebra
is biholomorphically equi-
E Z : 2i E
( Z
441
OF c*-ALGEBRAS
n1, Z
and
n
is the interi-
of all squares i n the JB-al-
associated with
Z.
Using a complex-
ified version of Cartants uniqueness theorem for vector fields [15] it has been shown i n [15,16] that the Lie algebra
A:=
aut(D
n
)
of
all complete holomorphic vector fields on the (unbounded) tube domain
Dn
consists of polynomial vector fields of degree
and
I;2
has a direct sum decomposition
h
(3.3) where
hj
components in
Z
@
Lo
X,
and
namely =
{
U
aS
: u E
az
\ :x
R, = ~ h az 0
aut(n)
which are homoge-
(3.3) can be interpreted in terms of the Jordan al-
-1 = ([zu*z) 1
x),
: u
E X)
and
E aut(n)),
denotes the Lie algebra of all infinitesimal trans-
formations ("derivations") meter group
R, ,
Analogous to the case of bounded domains the
j+l.
L
where
3
consists of all vector fields in
neous of degree
gebras
h-,
=
gt = exp(th)
1
of the cone
generating a l-para-
of linear automorphisms of
aim is to show, that the Lie algebras
aut(Z)
and
0.
aut(n)
O u r next
can be
regarded as llduals' Lie algebras i n the sense of symmetric space theory.
3.4 DEFINITION. satisfying
Let
X
be a JB-algebra.
S(xoy) = (6x)oy + xo(6y)
A linear map
for all
x,y E X
6 : X
-+ x
is called a
442
HARALD UPMEIER
derivation of
X.
Denote by
aut(X)
the Lie algebra of all deri-
vations of X. Modifying the proof for C*-algebras given in [ 1 9 ; Lemma 4.1.31
it
is easy to show that (everywhere defined) derivations of 33-algebras are norm-continuous.
3 . 5 DEFINITION.
Let
X
be a JB-algebra. Mxy :=
the multiplication operator on
Denote by
XOY
x E X.
X induced by
(Note that left and right multiplications cannot be distinguished since
x
is commutative).
I n terms of the Jordan triple product (2.5) on complexified JB*-algebra
2 := ' X
X
or on the
the multiplication operators
have the form
M~ = x where
e
is the unit element of
derivations,
aut(X)
consists
e",
X.
Similarly as for Jordan triple
of all operators
6 E X(X)
satisfy-
ing the commutator identity
r6 x E
whenever 3 . 6 LEMMA.
,M~I=
M~~
X.
Let
X
complexification
be a JB-algebra with open positive cone Z.
Put
MX := EMx
:
x E XI.
R
and
Then there exist
direct sum decompositions aut(Z) = aut(X) @ iMX and aut(n) = aut(x) e M ~ . (Note that
aut(2)
the JB*-algebra PROOF.
Z
Obviously,
consists of all Jordan triple derivations of satisfying ( 3 . 2 ) ) . aut(X)
is contained in
aut(Z)
and in
aut(Q),
A HOLOMORPHIC CHARACTERIZATION
since each Jordan algebra automorphism of
Z
automorphism of invariant.
OF c*-ALGEBRAS
X
443
is a Jordan triple
(by complexification) and leaves the cone
Moreover every
x E X
satisfies
x
e* = e
0
x*
0
n by
(2.13.iii), whence iMX = ix
ex
1 = 2 (ix
-
e*
0
e
0
(ix)")
E aut(Z).
n
u E
the fundamental formula for Jordan algebras 2 [ 7 ; Ch. 111, Satz 1.51 implies that Pu := 2 M U M E G L ( Z ) leaves
Further, for any
-
U
R
F o r every
invariant.
x E X
exp(x) E R
we have
and
exP(2Mx) = pexp(x) by [ 7 ; Ch.XI, Satz 2 . 2 1 .
I t follows that
X
the converse inclusions, suppose
Hence
X e = ix
satisfies
for some
be = 0.
x E X.
MX
6 := X
-
6 E aut(X).
This implies
iMx.
x := Xe E X.
Similarly, for each
Therefore
By C16; Prop. 5.4.viiI
6 := l,
- Mx E
X
aut(2)
E aut(0)
aut(n)
Then 6 E aut(Z)
X = 6 + iMx,
Since
we have obtained the desired decomposition for obviously direct.
T o prove
By (3.2),
E aut(Z).
Let
aut(0).
C
which is
we have
satisfies
6e = 0.
6 E aut(X).
this implies
Q.E.D.
An important consequence of Lemma 3 . 6 is the fact, that the Lie algebra a JB-algebra
aut(n) X
associated with the open positive cone
R
is not only a real Banach Lie algebra under some
appropriate norm, but carries also an involution
X-h*
satisfy-
ing the properties
(I")* = [x,ll]*= In fact, if with
X E aut(n)
6 E aut(X)
of
and
x
rll*,X*]
*
is uniquely decomposed as x E X,
A"
define := -6
+ Mx
I. = 6 + M x
444
HARALD UPMEIER
The geometric meaning of this involution on
aut(R)
is clarified
by the following examples.
3.7
EXAMPLE.
X := B ( E )
Let
E
introduced in 2 . 7 .
n
Then
positive definite a(-linear operators on
is the cone of all
E.
Denote by
real W*-algebra of all K-linear operators on tion the Jordan product on induced from
S.(E),
X
a = a* E X.
+ xa*
-
la E aut(n)
C(E)
3 . 9 EXAMPLE. X
-b
aut(n)
of
X
has the form
+ xa*
xa = ax
Associate to each a E 1 ( E )
defined by
lax := ax
Then Lemma 3.6 implies that
part
6
a = -a* E S. (E).
where
(3.8)
phism
+ xa".
a-la
yields a surjective homomor-
of real. involutive Lie algebras.
Z
Suppose
is a (unital) C*-algebra with self-adjoint
and open positive cone
n.
Then it is well known
[25; C o r . 2.121) that each Jordan derivation 6 rivation of algebra
Z
Der(2)
Since by defini-
On the other hand, it is well known (cf. c25;
6 x = ax
the derivation
the
it follows that
Lemma 2.61) that each derivation
x E X,
E.
X(E)
is the anti-commutator product
2~ ax = ax + xa = ax
for all
Let
(!R,C,H).
be the JB-algebra of all hermitian K-linear (bounded)
operators on
for
K E
be a Hilbert space over
E
E aut(X)
in the usual (associative) sense. of all derivations of
Z
(cf.
is a de-
The complex Lie
has an involution D-D*,
defined by D*z := -(DZ*)
for all of
2
z E Z.
Let
ad(a)z
associated with
:= az
a E Z.
-
za
* be the ftinnertt derivation
Then
aut(X) = [ S E Der(Z)
: 6"
=
-61
A HOLOMORPHI c CHARACTERI zATI ON OF
and
= ad(a*)
ad(a)*
for all
+
a E 2.
-ALGEBRAS
445
It follows that
+ 1 ad(a)
(1.10)
6
defines a homomorphism
aut(n) + Der(Z)
6
Ma-
C*
of real involutive Lie
algebras. Now assume in addition, that
is a (unital) C*-algebra
Z
having only inner derivations, that is Der(Z)
= (ad(a)
: a E Z}.
For example all W*-algebras and all simple unital C*-algebras have this property [ 1 9 ; Th. 4.1.6 and Th. 4.1.111.
Then
fined as in (3.8) yields a surjective homomorphism
a-ha
de-
Z + aut(n)
of
real involutive Lie algebras.
Z
For C*-algebras
in general, a representation of 1 E aut(n)
in the form (3.8) is still possible, but the "implementing operatorf' a
has to be chosen from the second dual space
Z
since derivations of
Ztt (a W*-algebra)
are in general not inner.
characterizing those operators
tt a E Z
The problem of
occuring in (3.8) is quite
difficult and this difficulty is responsible for the rather complicated notion of "orientation" introduced in
$4.
In $4 orientations will be defined in terms of the involutive Lie algebra
aut(n)
of the cone
n
belonging to a JB-algebra
X.
Of course, using Lemma 3.6 an equivalent condition could be imposed on the "dual" Lie algebra (2.12) of
j
= aut(D)
aut(2).
In view of the decompositions
h
and (3.3) of
:= aut(Dn)
the notion of
orientation has also a holomorphic interpretation in terms of the Lie algebras and
9
and
h.
of complete holomorphic vector fields on
Dn , respectively. T o define the concept of orientation we must have a closer
look at inner derivations.
By Lemma 3.6 the Lie algebra
of all derivations of a JB-algebra
X
aut(X)
can be viewed as the "non-
D
446
HARALD UPMEIER
t r i v i a l " p a r t of aut(X)
aut(Z)
and
aut(f2).
Moreover, t h e subspace of
spanned by a l l commutators
= [x
[Mx,My]
of m u l t i p l i c a t i o n operators f o r which i s denoted by
int(X).
0
e",
e"]
Y
x,y E X
forms an i d e a l i n
The elements of
i n n e r d e r i v a t i o n s of t h e JB-algebra
X.
int(X)
aut(X)
are called
S i m i l a r a s i n Lemma 3.6
t h e r e e x i s t s a decomposition
(3.11)
i n t ( Z ) = i n t ( X ) €9 i M X
Obviously t h e i d e a l
i n t ( f 2 ) := i n t ( X ) €9 MX
. of
aut(f2)
i s in-
v a r i a n t under t h e i n v o l u t i o n . L e t u s c o n s i d e r some examples o f i n n e r d e r i v a t i o n s :
3.12 EXAMPLE. ( i ) Let
be a C*-algebra w i t h s e l f - a d j o i n t p a r t
Z
a,b E X ,
Given
X.
an elementary c a l c u l a t i o n shows
I t f o l l o w s t h a t t h e concept of J o r d a n i n n e r d e r i v a t i o n i s more r e s t r i c t i v e t h a n t h e u s u a l n o t i o n of i n n e r d e r i v a t i o n , s i n c e t h e lfimplementing o p e r a t o r " i s r e q u i r e d t o be a f i n i t e s u m o f commutators in
Z.
*
For W - a l g e b r a s ,
t h i s condition i s (up t o c e n t r a l
e l e m e n t s ) always f u l f i l l e d , hence i n t h i s c a s e b o t h n o t i o n s c o i n c i d e ( c f . t h e more g e n e r a l s t a t e m e n t i n Theorem 3.15). ( i i ) The JB-algebras
over
LK E
W(E)
a s s o c i a t e d w i t h a H i l b e r t space
and t h e e x c e p t i o n a l Jordan a l g e b r a sf
{R,C,H}
E
3
(8)
have only i n n e r d e r i v a t i o n s , c f . Theorem 3.15. ( i i i ) Let
X = R CB Y
H i l b e r t space
be t h e s p i n f a c t o r d e f i n e d by some r e a l Y.
Then t h e L i e a l g e b r a
i d e n t i f i e d with the L i e algebra o p e r a t o r s on
Y.
The i d e a l
O(Y)
int(X)
aut(X)
can be
of a l l skew a d j o i n t bounded corresponds t o t h e s e t of a l l
A HOLOMORPHIC CHARACTERIZATION OF c*-ALGEBRAS
operators in
o(Y)
447
having finite rank.
Before stating the fundament a1
approximation the orem" for
JB-algebra derivations, let us give a short survey about the structure theory for JB-algebras as developed in [ 2 2 , 2 3 , 3 , 2 0 ] . JB-algebra
the second dual space
X,
Xtt
Given a
of the real Banach space
is again a JB-algebra with respect to the (commutative) Arens product [ 20,111.
JB-algebras which are Banach dual spaces are called
JBW-algebras (in analogy to associative W*-algebras). gebra
X
Any JBW-al-
has an (essentially unique) decomposition into three or-
thogonal weakly closed ideals
x = xrev
(3.13)
CB 'spin
such that the "reversible" part adjoint part of a 'spin
(resp.
real W*-algebra
Xexc)
'
Xrev
'exc
*
can be realized as the self-
(on a complex Hilbert space) and
is a JBW-algebra having only factor represen-
tations of spin type (resp. exceptional type). By Example 3.12.iii. infinite dimensional spin factors have a lot of outer derivations.
On the other hand, the remaining types
of JBW-algebras behave rather nicely with respect to inner derivations.
This is expressed in the following theorem [ 2 5 ; Th. 3.51:
3.14 THEOREM.
Let
X
be a JBW-algebra.
Then
aut(X) = int(X)
and only if the dimension of all spin factor representations of remains bounded.
if X
In particular, reversible and purely exceptional
JBW-algebras have only inner derivations. The proof uses the extension theorem for derivations of reversible JC-algebras [ 2 5 ; Th. 2.51,
Sakails theorem on the innerness of de-
rivations of von Neumann algebras [l9, Th. 4.1.61 and results about commutators in von Neumann algebras. Given an arbitrary JB-algebra
X
and a derivation 6 E aut(X),
448
HARALD UPMEIER
we may extend
6
a canonical way.
to a derivation
btt
of the second dual
Xtt
in
Since derivations vanish on the center of a JB-
algebra, the direct sum decomposition (3.13) of the JBW-algebra X t t is invariant under the extended derivation Theorem 3.14 to this situation.
6 tt.
Hence we may apply
Modulo some technical arguments one
obtains the so-called approximation theorem for derivations on an arbitrary JB-algebra
3.15 THEOREM.
Let
gebra
of
aut(X)
X X
X
[ 2 5 ; Th. 4.21 :
be a JB-algebra.
Then the derivation al-
is the closure of the ideal
int(X)
o f all
inner derivations with respect to the strong operator topology on S(X),
i.e. the topology of pointwise norm-convergence.
Simple examples show that in general aut(X)
int(X)
is not dense in
with respect to the topology of uniform norm-convergence.
B y Lemma 3.6 and (3.11), the approximation theorem can also be formulated in terms of the Lie algebra derivations of the JB*-algebra gebra
X.
Z = 'X
aut(Z)
of all Jordan triple
associated with the JB-al-
Equivalently, in holomorphic terms, Theorem 3.15 says
that each "infinitesimal rotation"
5 E h can be pointwise apprsx-
imated by linear combinations o f commutators of infinitesimal transvections in
?.
After having clarified the main properties of derivation algebras of Jordan C*-algebras, we will define in $ 4 the notion of orientation of a JB-algebra
X
to be a complex structure on aut(Q)
(modulo center) given by a closed operator which is densely defined with respect to the strong operator topology.
In $ 5 it will be
shown that a JB*-algebra has an orientation in this sense if and
only if it is a C*-algebra.
A HOLOMORPHIC CHARACTERIZATION
OF c*-ALGEBRAS
449
$4. ORIENTATIONS OF JORDAN c*-ALGEBRAS In order to define "orientations" on a JB*-algebra in terms of the Lie algebra
aut(n)
we have to clarify one technical ques-
tion, namely the structure of the center of
aut(n).
Recall the
notion of center for Jordan algebras [ 7 ; Ch. I, $51.
4.1 DEFINITION.
Let
X
be a JB-algebra.
is the set of all elements Y E
x E X,
Then the center of
such that
[Mx,My]
X
= 0 whenever
x.
4.2 LEMMA. Let
Let
X
be a JB-algebra with open positive cone
aut(n) = aut(X) 69 MX
(cf. Lemma 3 . 6 ) .
Then the center of
tiplication operators PROOF.
= May
X
Suppose
= 6
+
[Mx,My].
6 y = 0.
Since
Mx
aut(n)
aut(n)
consists of all mul-
x E center()().
with
+ Mx E center(aut(n)).
F o r each
y E X,
= [ 6 ,MY3 + [Mx,My] = Evaluation at the unit element e E X gives
My E aut(f2)
we have
be the Cartan decomposition of
n.
and therefore
is arbitrary,
y
Conversely, suppose
0 = [X,My]
6 = 0
x E center()().
x E center()[).
and
A := center()[)
Since
is the
self-adjoint part of an abelian C*-algebra, it follows that 6(A) = 0
for all
6 E aut(X).
and also with all operators
Therefore M
Y
for
Mx
y E X
commutes with
6
by Definition 4.1. Q.E.D.
Since . A := center()()
-X
:= X/A
is a closed subalgebra of
X,
is a Banach space under the natural norm
Similarly,
aut(n)
Banach Lie algebra
is a real Banach Lie algebra and the quotient
&(n)
= aut(X) E? &
with the multiplication operator
Mx.
by identifying
x E X
By Lemma 4.2, the centers of
450
HARALD UPMEIER
aut(n)
and
int(n)
coincide, hence
int(n) The involution of
:= int(n)/center
aut(n)
f~
X.
leaves the center invariant and there-
fore induces an involution of
s(n)
such that
=(n)
is a
Modulo centers, the homomorphisms (3.10) and
*-invariant ideal. (3.8)
= int(X)
turn out to be isomorphisms:
4.3 EXAMPLE. rivations.
Let
Z
n
Let
be a unital C*-algebra having only inner debe the open positive cone of
Z.
Then the ho-
momorphism (3.10) induces an isomorphism
aut(n)
(4.4)
Similarly, the homomorphism
-+ Der(Z).
Z + aut(n)
defined by (3.8) yields
an isomorphism Z/center + ~ ( 0 ) . Modulo the isomorphism
ad: Z/center
-t
Der(Z)
the above isomor-
phisms are inverse to each other. We are now in a position to define "orientations" on Jordan C*-algebras.
4.5 DEFINITION. Denote by
Let
e ( R )
of the JB*-algebra
be a JB-algebra with open positive cone 0.
the involutive Lie algebra of all infinitesimal R
automorphisms of
X
modulo its center. Z = 'X
An orientation of
associated with
X)
X
(or
is given by the
following : (i)
A (not necessarily closed) ideal
int(n) (ii)
G ( R )
containing
and invariant under the involution.
A complex structure
Lie algebra such that
X
C
J:a-ta making J:
1~
+
g(n)
a
a complex involutive is a closed operator.
is called orientable, if there exists an orientation on
X.
A HOLOMORPHIC CHARACTERIZATION OF c*-ALGEBRAS
451
4.6 REMARK. By the approximation Theorem 3.15, the ideal
(i)
in
e ( n )
under pointwise convergence, s o that
"densely defined" with respect to this topology.
However,
is &
The technical difficulty that the complex structure only be defined on a proper ideal in
present if
aut (n)
J
=
X
can-
s(n).
not always be chosen to be uniformly dense in (ii)
is dense
&(n)
J
may
is not
is a Banach dual space, since in this case
=(R).
O u r first aim is to give an equivalent notion of orientation which
is technically easier to handle.
4.7 THEOREM. X
Let
X
be a JB-algebra.
in the sense of 4.5 are in
Then the orientations on
correspondence with continuous
1-1
IR-linear mappings
J:
x + aut(x)
satisfying the property
(4.8)
Cbia,bib]
for all 6(a,b) PROOF.
a,b E X.
have
Cb,d
= 6 i 6 (a,b)
-
6ia := Ja E aut(X)
Here we define
and
:= diab E X. Let
J:O+
int(X).
be an orientation.
a
follows that u, = ing
=
@
Since
J h * = -(Jh)*.
restriction
JIX
operator and
&
This implies by
x c u,
J
again.
is an ideal in
c
U.
aut(X)
it contain-
JX
C
Since
c?. C aut(X). J: a +
Denote the
e ( n )
is a closed
is a Banach space, it follows that
2
is a closed IR-linear operator.
X.
=(R)
is a complex involutive Lie algebra, we
J:
continuous on
b
2 , where
Since
+ aut(x)
By the closed graph theorem
Further, since
CL
J
is
is a complex Lie algebra under
HARALD UPMEIER
452
we have for all
J,
' 6 ia
and 6i 6
Hence the property
(4.8) is satisfied. J:
Conversely, suppose that
If +
aut(X)
R-linear mapping having the property (4.8).
a := {Gia Given
a,b E X ,
E center(aut(0))
It follows that
5
aut(X)
-b
suppose
n
aut(X)
= 0.
CB ~ f ~c
( n ) .
(4.8) that
+ [6ia,6ib] E
[Ma,Mb]
Therefore
6ia = 0 implies is injective.
a E X.
observation, a
it follows from
Define
tMalMb1 = -[bia,6 ib] = -6 i 6(a,b)
(4.9)
J:
x]
: a E
is a continuous
a 6 center()[).
To show that
= 6(b,a),
-6(a,b)
6 i 6(a,a) = Cbia,6. la] = 0 . By the above E center()[). Since b i a is a derivation of
Then
6 (a,.)
commutative C*-algebra containing
Hence
Hence
6(a,a) = 0
a,
apply [19; Lemma 4.1.21.
and skew-symmetry follows by polarization.
next assert that for every
6 E aut(X)
b E
and
X:
In fact, by the approximation Theorem 3.15, given any is a net -+
b j E int(X)
x E X
there
with the following properties, the symbol
denoting norm convergence in 6jx + 6 x ,
We
8 .b + 6 b J
and
This implies by norm-continuity of f6 j y 6 i b I ~= 6 J.(dibx)
X:
6j(6ibx)
8 ib
-+
6(bibx).
(cf. $ 3 ) :
- gib(6 j ~ ) C6 ,6 +
i b ] x'
c* -ALGEBRA s
A HOLOMORPHI c CHAR ACTERI zATI ON OF
On the other hand, since 6 j E int(X) from
(4.9), it follows
(4.8) that = big J,bX = -6ix(6 jb) + -bix(6b) = 6 i(6 b)x
r6 j,6ib]X
Since
aut(n) & +
Comparing both limits gives (4.10).
by definition, (4.10) implies that
&
& C
.
6
using the skew-symmetry of
J:
J& by
C
453
=(n)
containing
by
(4.9). The injective mapping
aut(X)
has a well-defined extension
:= -2 E
5
J(bia)
a E X.
for every
,
hence
(
L&
,J)
satisfying
J : U +
Obviously, -5'
elementary verification shows that
h , E~ Lt
is an ideal in
U
= id,.
= [X,JP]
J[X,b]
for all
is a complex involutive Lie algebra.
Finally, the continuity of
2
on
J
implies that
closed operator, since the decomposition
J:&
-t
= aut(X)
%(Q)
is a
&
@
5
is
Q.E.D.
topologically direct.
4.7
An immediate corollary of Theorem
4.11 COROLLARY.
An
is
*
Z
Every (unital) C -algebra
has a canonical
orientation. PROOF.
Let
be the self-adjoint part of
X
A := center(X) a
E X
Z.
is the self-adjoint part of the center of
Z.
For
define i i 6 iax := 5 [ a,x] = 2 (ax-xa)
Obviously the derivation class
Then
2 E
2
modulo
which implies
lbial
A.
6 ia E aut(X)
depends only on the residue
Further, f o r all
121
b
.
x E X
for the operator norm on
X.
It fol-
lows that the well-defined mapping
J: sending
to
J a :=
(4.8) is satisfied.
'
ia
2
4
aut(x)
is continuous.
By definition, property Q.E.D.
454
HARALD UPMEIER
F o r C*-algebras there is a simple criterion for the con-
tinuity of the canonical orientation: 4.12 PROPOSITION. a C*-algebra J
(i) (ii)
Z.
J:&+
Let
be the canonical orientation of
U.
Then the following statements are equivalent:
is continuous.
The ideal
Inn(X)
of all inner *-derivations on
the associative sense) is closed in (iii)
is closed in
PROOF.
(i)
* (ii).
2
is a homeomorphism.
-t
(ii)
(iii).
=)
(iii)
=)
(i)
2
Since
d
is continuous.
Then the mapping
is a Banach space,
Inn(X)
is
aut(X).
Obvious from
.
J
{6ia : a E X] = Inn(X)
Since
complete, hence closed in
aut(X).
s(n).
Suppose J:
L&
= Inn(X) @
5.
is supposed to be a Banach space and
by definition a closed operator,
J
J
Q.E.D.
By Proposition 4.12 the canonical orientation
a unital C*-algebra is known to be continuous for
J
C*-algebras [ 211 , f o r AW*-algebras and n-homogeneous C*-algebras If
Z
is a W*-algebra, it is even true that J:
is an isometry, if aut(X)
tf
x
-t
aut(x)
is equipped with the quotient norm and
carries the operator norm with respect to
However, for C*-algebras in general,
J
recent example of an AF C*-algebra with found in f 41
.
X
[13,11].
is not continuous. J
of
several classes
of C*-algebras, namely for simple and, more generally, primitive
[4].
is
is continuous according to the
closed graph theorem.
4.13 REMARK.
(in
Z
A
discontinuous can be
A
OF c*-ALGEBRAS
HOLOMORPHIC CHARACTERIZATION
§5. ORIENTABLE JB*-ALGEBRAS
C*-AJJXBRAS
ARE
In this section we prove the converse of 4.11: gebra
with an orientation is a C*-algebra.
Z
sible C*-algebra structures on the orientations of several steps. algebra
Z
Z.
455
Z
are in
Every JB*-al-
Moreover, the poscorrespondence with
1-1
The proof of this result is carried out in
Denote by
X
the JB-algebra associated with a JB*-
n.
with open positive cone
According to Theorem 4.7,
orientations are considered as continuous maps
&
J:
+ aut(X)
sa-
tisfying property (4.8).
5.1 LEMMA. A spin factor
(i) (ii)
(i) Suppose,
Hilbert dimension aut(X) = W
Since
is orientable if and only if
n,
n
3.
2
Then
n = 3,
for
=(n)
Since
rank f6,,b2]
2
5:
c CL,
for all
(ii) and
Let
X =
a,(@).
dim & = 26.
morphism
J:
Let
X
2 +
n z 4.
it follows that [Ma,Mb]
has rank
B y defini-
r2
whenever
it follows from (4.9) that
4, X
This implies
n
d
Any orientation
J
of
X
7:
Q.E.D.
X +
2
be a JB-representa-
*
tion into a JBW-algebra and
n"
X
which is weak* dense.
the open positive cones of
second dual space
Xtt
of
X
X
and
2,
Denote by
respectively.
is a JBW-algebra and
unique extension to a JEW-representation
= 52
would induce an iso-
a contradiction.
be a JB-algebra and let
4.
is orientable by 4.11.
= a,(C)
Then it is well known that dim(aut(X))
aut(X),
of
is not abelian by (4.9).
aut(X)
81,b2 E int(X).
n =
On the other hand, for
x
is an orientation of a spin factor
J
tion of the spin factor product, a,b E X.
4.
dim X =
is not orientable.
H3(0)
PROOF.
X
n : Xtt
+ ?.
TT
n The
has a The kernel
456
of
HARALD UPMEIER
TT
in
Xtt
is a weak* closed ideal, thus of the form ker(n) = Pl-cXtt
where
c E Xtt
, 2 Px := 2Mx
is a central projection and
-
M
deX
notes the quadratic representation. Lemma 3.6, the second transpose
X E
Now suppose
Xtt
E X(Xtt)
By
commutes with
pc such that the
. + + ( I ) E aut(n")
Therefore there is a unique element
aut(n).
following diagram commutes -. x-x
tt
TT
-.
x -
Xtt i7
Obviously,
TT*: aut(n)
aut(n")
-t
a E X,
Lie algebras and for all
Since
TT
aut(n)
-t
preserves centers, there is an induced homomorphism
&(n"),
5 . 3 LEMMA.
Let
JB-algebra
X
PROOF.
is a homomorphism of involutive
TT:
-t
the skew symmetry of
T
-t
-X
aut(X)
a E X.
for
Since
X
T
+ E~
-.
X.
Then
,.
X
is orientable.
be the orientation of
Suppose
and put
X
n(a) E center(%).
6 (a,b) := Giab
aut(%) ~ ~
.
TT+
be a representation of an orientable
onto a JBW-algebra
J: &
Let
6ia := J p
again denoted by
Given
b E X
implies
vanishes at
ive, evaluation at the unit element of
2
"(a)
and
yields
TT
is surject-
n+bia =
0.
Hence
CHARACTERIZATION OF c*-ALGEBRAS
A HOLOMORPHIC
TT
(a) c n.6
4 57
induces a well-defined mapping
ia
..,-
J:
-J
T o see that
x+
aut(2).
is continuous, assume that
J
However, since the left hand side depends only on In*(Sia)n(x)l
n*
and
N
6
In(.)\
and
which proves that
X
#(H)
is not orientable by Lemma 5.1, it follows
from Lemma 5.3 that
X
has no factor representations onto
the exceptional ideal of
X
X
... x
+ xn
5.5. COROLLARY.
E X
X c #(H)
whenever
xl,.
is called reversible if
..,xn E
Every orientable JC-algebra
x.
X c #(H)
is re-
versible. PROOF.
Let
n: X
-b
?
be a dense JB-representation into a spin .-.
d.
factor X.
Then
weak* dense. ..,
is
Q.E.D.
Recall that a JC-algebra
... x
Hg(a)).
vanishes, hence
a JC-algebra.
x1
for some
H.
a3(0)
By [3; Th. 9.51
is
with an orientation is a JC-
algebra, i.e. a norm-closed unital subalgebra of
Since
.-.
J
Q.E.D.
Each JI3-algebra
complex Hilbert space
~(x),
= (i-~*b~,)(~b)
is a homomorphism.
5.4 COROLLARY.
PROOF.
na
(4.8) follows from the fact that rr(biab)
continuous.
and
a,x E X
observe that, for all
we have
N
has norm
r(X)
Since
.-
X
is a norm closed subspace of
X
is a Hilbert space,
n(X) =
X
.-. X.
which is By Lemma
5.3,
X
Lemma
4.1, Lemma 4.4 and Lemma 4.51 it follows that the 12-part
is orientable, hence
of the weak closure of
X
dim
=
4 by Lemma 5.1.
is reversible.
is reversible [I.; Th. 4.61.
By [l;
This implies that
X
Q.E.D.
458
HARALD UPMEIER
5.6 LEMMA. X C H(H).
cp
and every state
A :=
Let
PROOF.
be a derivation of a reversible JC-algebra
Then
x E X
for all
6
Let
C*(X)
of
X.
be the C*-algebra generated by
X.
extension theorem f o r reversible JC-algebras [25; Th. 2.51 exists a *-derivation
on
D
satisfying
A
DIX = 6
By [31; Th. 21 there exists a self-adjoint element closure
B
of
A
By the there
and ]Dl L 2161. a
in the weak
such that 6 x = i[a,x]
x E X
for all
extending
cp
and
21al = ID1
b
216
I.
(cf. [ 9 ; Lemma 2.10.11 ) .
$
Let
be any state of B
Then, by the Cauchy-Schwarz
inequality,
X
Given a JB-algebra
with orientation
in constructing a C*-structure on
the crucial step
is the extension of the
X'
orientation to the second dual space this extension it is essential that operator.
J,
Xtt, J
In order to construct
is assumed to be a closed
We have already seen (Cor. 4.11) that the canonical
orientation of a C*-algebra has this property. Suppose
[3; p.231
p
p : XxX + X
is a continuous bilinear mapping.
has a unique bilinear extension ;:
XttXXtt
-t
Xtt
having the same norm and satisfying the following properties:
(5.7)
ac-p(a,b)
is weak* continuous f o r
a,b E X
(5.8)
b t-p(a,b)
is weak* continuous for
a E X,
tt
,
b E Xtt,
By
CHARACTERIZATION OF c*-ALGEBRAS
A HOLOMORPHIC
If p
is the Jordan product in
the Arens product in
Xtt
X,
making
the extension
Xtt
459
is called
into a JBW-algebra.
We
will apply the "Arens process" to the skew symmetric bilinear mapb
ping
associated with an orientation.
5.9 THEOREM.
Let
be a JB-algebra with orientation
X
the second dual space a E X
For
PROOF.
Xtt
J.
Then
is orientable.
bia
define
:= J a E aut(X)
and let
8: XxX
-I
X
be the continuous skew symmetric bilinear mapping given by 6(a,b)
of
-
:= Gia(b).
6 ,
Let
written as
a':
XttxXtt
diab := 8(a,b)
Xtt for
be the Arens extension a,b E Xtt.
hence
Fia
13;
Arens product
ria E a E Xtt,
since
X
bounded.
(5.11)
Suppose
and to the commutative
(aj)
and
aut(xtt)
- -
is weak* dense in
strong and weak convergence in ively.
Xtt
(bj)
by
B
j
u
b
j
B
b
are nets in
implies
by (5.10) and
versible by Cor. 5.5.
6(aj,bj)
Xtt
-
Let us denote
a with
,
respect(a,) norm J
C(a,b).
U
The second summand converges to aut(Xtt)
Xtt.
and
Then a
a E X,
Cor. 3.41 it follows that
(5.10) whenever
a'
Applying (5.7) to
E aut(Xtt).
If
b i a E aut(X),
is the second transpose of the derivation
bia
cj E
-t
Xtt
o
by (5.7).
Further,
is reversible, since
X
is re-
Hence we may apply Lemma 5.6 to obtain
By definition of strong convergence, this term converges to
o
460
HARALD UPMEIER
a'
sj?ce
is bounded and
(aj)
is norm bounded.
Hence (5.11)
follows. As a consequence of (5.11),
s
is skew symmetric since
is skew symmetric and the unit ball of the unit ball of
Xtt
by [ 3 ;
6
is strongly dense in
X
Prop. 9.91.
It follows that for all
x E Xtt Biax = -bixa = o whenever
a E center(Xtt),
Therefore -t
aut(Xtt)
a-Bia
since derivations vanish on the center.
induces a well defined mapping
which is continuous since
property (5.11) for
a'
is.
Xtt/center
and the strong continuity of the Arens
straightforward to show that property By Theorem 4.7,
3
-t
Using [3; Prop.3.91,
[ 3 ; Prop. 3.71 as well as skew symmetry of
product
6.
s'
3:
(4.8) for
is an orientation of
6
E,
it is
carries over to
Xtt.
Q.E.D.
Theorem 5.9 is the essential tool for constructing the C*-structure on an orientable JB*-algebra
Z.
The rest of the
proof uses some results of the structure theory for JB-algebra state spaces developed in [2,l]. tiplication operators
Mx
Let
X
be a JB-algebra with mul-
and quadratic representation
Px := 2 Mx2
-M
2 . X
Then
Pxy = {xy*x]
projection.
Then
(x,y) rr [xp*y]
for all X
x,y E X,
cf. (2.5).
Let
p E X
be a
:= P X
is a JB-algebra with multiplication P and unit element p. F o r a E X we have by P
P
C3; (2.35)l Pl,pa
BY
I:3;
Lemma 2.111
= P1-P PP a =
0.
,
(5.12) Hence the restriction
[Ma,Mp] = o = [M a'ppl pP(Ma)
:= MalXp
is well defined and coin-
*
A HOLOMORPHIC CHARACTERIZ AT1 ON OF C -ALGEBRAS
cides with the multiplication by
5.13 LEMMA.
If
X
every projection For
PROOF.
a
in
X
is orientable, then
X
P
461
.
P
is orientable for
p E X.
a E Xp
put
By (4.8) and (5.12),
b := 6 iap.
6ib = [6ia,6ip] = -[M Therefore, for every
P
,M ] = b
0.
x E X
This implies = -6 iap = -b E
6
.
center(X)
As in the proof of Theorem 4.7, it follows that since
ip
is a derivation.
b = dipa =
0,
bia
The derivation property of
imp1ie s [Gia,MP1 = o = Hence
:= bialXp
pp(bia)
is a well defined derivation of
a E center()(
particular, for
1 6 ia’Ppl*
P
)
xP
and
Giab = -6. iba = Hence ping
=
pp(Gia)
0.
L EM M A .
Let
X
P
.
In
’
0.
A n elementary calculation shows that the mapinduces a well-defined orientation on
pp(Ma)l-)pp(6ia)
5.14
X
X
P Q.E.D.
be an orientable JB-algebra.
Then
X
is of
and denote by
F
the
.
complex type (cf. [I; $31). PROOF.
Let
S
be the state space of
face generated by two pure states
cp
X
and
$.
not equivalent in the sense of [1;$2], then generated by
cp
and
J,
[l; Prop. 2.31.
valent, then by [l; Prop. 2.31 k
2
2.
F
If cp F
If cp
and
$
are
is the line segment and
$
are equi-
is a Hilbert ball of dimension
B y the proof of [ l ; Th. 3.111, there exists a projection
462
HARALD UPMEIER
Since
V := Xtt is a spin factor with state space F. P is orientable, Xtt is orientable by Theorem 5.9, hence
such that
p E Xtt X
is orientable by Lemma 5.13.
V
k = 3.
Hence
This means that
[I; C o r . 3.31,
X
Lemma 5.1 implies
k + 1 = dim V
=4.
has the 3-ball property and by
S
Q.E.D.
is of complex type.
We are now in a position to define the C*-algebra structure associated with an orientation.
I I
JB*-norm
and involution
X := {x E Z : x* = x} x E X
for on
X.
Let
Z
z +z * .
be a JB*-algebra with Suppose the JB-algebra
has an orientation
and denote by
Mx
J.
Jx
:=
biX
Define
the multiplication operator by
x
x,y E X define
For
xy := MxY
(5.15)
+
1
6 ixy E
z
and extend this product by bilinearity to a product on
Z.
We
first list some elementary properties of (5.15).
5.16
(i)
xy
PROOF.
the following properties hold:
p xy + yx = MxY
(ii) (iii)
x,y E X
For
LEMMA.
-
2
yx = 1 6 ixy i
(xY)" = YX. These properties follow from the fact, that
metric and
is skew symmetric in
diXy
Mxy
is sym-
(x,y).
Q.E.D.
It follows from 5.16.iii, that the complexification Z := X" of a JB-algebra
X
with orientation becomes an involutive algebra
under the (extended) product (5.15) and the JB*-involution.
5.16.1, the anti-commutator product in original Jordan product.
Let
n:
X
-I
?
Z
By
coincides with the be a weak" dense represen-
tation into a JB-factor of type I (called type I factor representation).
Since
? = x:~
for some central projection
c E
xtt,
it
463
A HOLOMORPHIC CHARACTERIZATION OF c*-ALGEBRAS
follows from Theorem 5.9 and Lemma 5.13, that each orientation on X
X
induces a unique orientation of
having the property
(5.17) for all
x E X.
5.18 LEMMA.
Let
JB-algebra
X
induced by
IT.
n: X
-+
2
be a type I factor representation of a
with orientation.
-
Endow
with the orientation
X
Then the complexification ll:
z
z"
-+
is a homomorphism of involutive algebras.
PROOF.
Let
x,y E X.
Then by (5.2) and (5.17),
We can now prove the converse of 4.11:
5.19 THEOREM.
Let
complexification
X
be a JB-algebra with orientation.
Z := X'
Then the
becomes a C*-algebra under the product
(5.15), the JB*-involution and the JB*-norm.
The underlying Jordan
algebra of this C*-algebra is the original JB*-algebra structure on Z
and the canonical orientation coincides with the given one. F o r each type I factor representation
PROOF.
2
= #(H),
exists a complex Hilbert space
H
with
of complex type by Lemma 5.14.
By
(4.4) and
Z
:= (ab)c
relative to the product (5.15). rr[abc]
hence
Z
since
X
-
Z
:=
2'
is
".
X
becomes a
Consider the associator [abc]
in
there
[ 8 ; Prop. 4.121,
has only two orientations under both of which C*-algebra.
r: X + ?,
-
a(bc) B y Lemma 5.18,
= [na,rrb,rrc] =
0,
is associative, since the type I factor representations
464
KARALD UPMEIER
form a faithful family on z E Z
[ 7 ; Cor. 5.71.
X
Similarly, for any
we have by Lemma 5.18,
In(z*z)I
= I(TrZ)*(.Z)I
= InzI 2
,
hence
The inequality similar way.
lzwl
Hence
S
2
IzI*IwI
z,w E Z
for all
is a C*-algebra.
is shown in a
The remaining statements
follow from Lemma 5.16.
Q.E .D.
It follows from Cor. 4.11 and Theorem 5.19 that a JB*-algehra
Z
is orientable if and only if it is a C*-algebra; moreover the
possible C*-algebra structures on with the orientations on
Z.
Z
are in 1-1 correspondence
Together with Theorem 2.15, we obtain
the desired holomorphic characterization of C*-algebras:
5.20 MAIN THEOREM.
Let
space
is biholomorphically equivalent to the open
Then
Z.
D
D
be a bounded domain in a complex Banach
unit ball of a (unital) C*-algebra if and only if the following conditions are satisfied: (i)
D
is a bounded symmetric domain which is biholomor-
phically equivalent to a tube domain, (ii)
The JB*-algebra structure on
Z
associated with
D
is
orientable. Moreover the possible C*-algebra structures on correspondence to the orientations on
Z.
2
are in
1-1
A HOLOMORPHIC
CHARACTERIZATION
OF c*-ALGEBRAS
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277-290. d e r math.
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von NEUMANN,
J . , WIGNER, E . ,
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KADISON,
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LANCE, E . C . ,
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J.R.,
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29-64. D e r i v a t i o n s and
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Banach manifolds.
Math. Ann. ;128 (1977), 39-64.
15. KAUP, W., UPMEIER,H., uniqueness theorem.
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An infinitesimal version of Cartants Manuscripta math. 22 (l.977), 381-401. Jordan algebras and symmetric Siege1
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KOECHER, M.,
Math. Z. lJJ (1977), 179-200.
An elementary approach to bounded symmetric Houstonr Rice University 1969.
domains.
18. NACHBIN, L., Topology on Spaces of Holomorphic Mappings. Erg. d. Math. 5. Berlin-Heidelberg-New York: Springer
1969 19.
SAKAI, S.,
C*-algebras and W*-algebras.
Erg. d. Math.
60.
Berlin-Heidelberg-New Yorkt Springer 1971. 20.
SHULTZ, F.W.,
On normed Jordan algebras which are Banach
dual spaces. 21.
STAMPFLI, J.,
J. Funct. Anal.
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The norm of a derivation.
Pac. J. Math.
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(1970) 737-748. 22.
Jordan algebras of type I.
STP(RMER, E.,
Acta Math.
115
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S T P ( N R , E.,
Irreducible Jordan algebras of self-adjoint
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Automorphism groups of Jordan C*-algebras.
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DEPARTMENT O F MATHEMATICS U N I V E R S I T Y O F PENNSYLVANIA PHILADELPHIA,
SOC.
84 ( 1 9 7 8 ) ,
263-272.
The norm o f a d e r i v a t i o n i n a W*-algebra.
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This Page Intentionally Left Blank
Punctionul Analysis, Holoniorphy und ,4pproxiniation Theory 11, C.1. Zupata (ed.) 0ELcutrr Scierice Pirblislicrs B. 1'. (Abortli-Holland), 1984
A PROPERTY OF FFU~CIIET SPACES Manuel Valdivia
SUMMARY
F
Let
E.
be a dense subspace of a Fr6chet space
If F
is not barreled there is an infinite dimensional closed subspace G of
E
such that
F
G = {O} ,
Some consequences of this result
are given. The linear spaces we shall use are defined over the field K
of the real or complex numbers. Given a dual pair and
p(L,M)
ively.
If
(L,M)
are the weak and the Mackey topology on A
L,
the linear hull of
topology derived from the gauge of
A
(u,x)
L
The word "space1'is used
A.
E' is its topological dual.
we shall write sometimes
respect-
endowed with the normed
to denote a Hausdorff locally convex topological space. a space,
u (L,M)
absolutely convex subset of
is a U(L,M)-bounded
LA
we denote by
,
x E E
If
instead of
If E
and if
u E E'
~ ( x ) . A space
a Baire-like space if given an increasing sequence
(An)
is
E
is
of closed
m
absolutely convex sets of
E
E
there is a natural number
u An n=1 such that An
such that
no
is absorbing in is a neighbourhood 0
of the origin.
N
denotes the set of the natural numbers.
I shall need f o r the proof of Theorem 1 the following result: a)
If E
then
is an infinite dimensional metrizable separable space,
E
has the property of quasi-complementation.
This result was obtained by Mackey in
121
for normed space but the
470
MANUEL VALDIVIA
proof given there remains valid for metrizable spaces.
F
F
Let
THEOFLEM 1.
be a dense subspace of a Fr6chet space
is not barrelled there is a subspace
E
of
G
If
E.
such that the
following conditions are satisfied: is closed;
1.
G
2.
the dimension of
3.
~n
PROOF.
is infinite;
G
F = [o}. T
Let
be a barrel in
the origin and let
F
which is not a neighbourhood of
E'.
be its polar set in
To
If
(Un)
is a
fundamental system of absolutely convex neighbourhood of the origin in
F
we take x1 $? T, u1
22 x1 E U1,
E
with
To
Iul(xl)l
> 1.
Supposing that we have obtained
such that Iuj(xj)l
let
H
and
L
*,...,un}
(ul,u
> 1,
n
in
E' TO
in
E'
and
E'
F
[x1,x2,
..., n}
and
x
,...,un]
respectively.
Since [u1,u2
which is a topological complement of
X
these is a
X >
1
such that
n
H)
2 TO.
TO X(X
On the other hand,
F
i,j = 1,2,...,n,
ifj,
be the orthogonal subspaces of
generate a linear space
H
= 0,
uj(xi)
and therefore
L
+
TO
(1)
is a finite codimensional closed subspace of
T flL
is not a neighbourhood of the origin in
L
and thus there is 1
Yn+1
~22(n+l) Ln
B y using (1) we can find
u n+ 1 E To
Un+l
9
n H
and
T*
X
E K,
j = 1,2,.
..,n,
A PROPERTY OF F ~ C H E TSPACES
so that
By setting
XY,+~
= x
~
+ it ~ follows that
1
Xn+l E
~22(n+l)un+l
9
Un+l E
Xn+l @
U ~ ( X ~ +=~ U) ~ + ~ ( X=~ 0),
TO
9
j=1,2,...,ny
n
Then, sequences
c
AjUj
+
(up)
in
F
= IX(
Iun+l(xn+l)l (x,)
and
> 1.
Un+l)(Yn+l)l
j=1 and
E' can be select
such that x € - 1 U P 22P P
For every positive integer
,
xp@T,
n
we find a sequence
u € T o , P
...,k(n), ...
l(n),2(n),
of even positive integer numbers which are pairwise differente such that if
k(n)
#
h, k, m
h(m).
and
Let
n
V
m f n
then
be the linear hull of
EU2n-1
and let
are positive integers with
+
1 m\(n) n,k = 1,2,..*1 :
be its closure in
E'[u(E',F)].
The closure of
V
in
contains
E' TO
b1'U3,
* * *
lu 2n-19
*.I
and therefore 2n-1 E
U
Let
A
7,
n = 1,2,...
.
be the closed absolutely convex hull in
{ 22xl, 24x2,. , Since the sequence
(22nxn)
E
of
.,22nxn,. . .} .
converges to the origin in
E
the set
472
A
MANUEL VALDIVIA
is compact.
P
Let
be the linear hull : n = 1,2
{U2n-1
P
We take a non-zero element of
Then there is a positive integer
p,
Given a positive number
2e
of
v E V
with
E
,...] .
1
%
p
n,
6
< lapl,
so
that
a
P
f 0.
suppose the existence
such that : x E A] < E .
supfI(v-u)(x)l We can write m
c c j=1 kEP
v = with and in
m
;?
n, P j
@ j,k E K ,
u-v
1 @
j,k ('2j-1
+
"k)
j
a finite subset of even positive integer numbers
k E
,
P
.
j = 1,2,. .,m.
The coefficient of
u
2p-1
coincides with
a
-
C kEPp P p 9 k
e
+
and therefore
from where we get
lapl In u-v,
%,
the coefficient of
take an element
ek
of ' k
we have that, since
I kEP c
'p,kI '
k E P p , is
*
If we
of modulus one such that
K
k E Pp
Pp,k = l@p,kl'
Pi n P j
c kEP
e > sup(I(u-v)(x)l
1 k Bp,k
= 0,
ifj,
i,j=l,2,,..,m,
and
e k zk xk E A,
: x E A) 2
I(u-v)(
c kEP
k ek2 xk)l
2
A PROPERTY OF FRESCHET
SPACES
473
tion.
< 2 e , according to ( 2 ) , which is a contradiclap We can conc ude that P f l M = (01 if M is the closure o f
V
E’[y(E‘ , E ) ] .
and therefore
in
M
be the orthogonal subspace of
M
gonal subspace of
in
5
it follows that
E
7,
is U(E’ ,F)-dense in
M
closed in
and of infinite codimension in that space.
u(E’,E)
for
Then
F.
in
S
If
E
and let
S
be an algebraic complement of
in
denotes the closure of
R.
R
be the ortho-
R.
is o f infinite codimension in
S
Let
?
in
S
Y
Let
We take linearly inde-
pendent vectors
in Y.
denotes the closed linear hull of ( 3 ) in
Z
If
5 n
is separable and
of infinite codimension in
Z
to a) we obtain a quasicomplement subspace Then
=
G
is closed in
G
n ( 5 n z)
=
E,
G
Z.
5 n
of
E
Z
it has infinite dimension and
(03.
Z
According in G
n
Z.
F =
q.e.d. G
In order to prove the following lemma let subsets of a space
then
E
be a family of
which are bounded closed absolutely convex
and satisfying the following conditions: a.
If
is a finite part of
(2
that
A
3
If
A1,A2 E G.
C.
If
A
We denote by
G
there is
and if
E’[-i]
A3 E G.
with
then
LA E G .
A > 0
the space
Let
di tions :
F
such
be a subspace of
A3
3
A1 U A2.
endowed with the topology of the
E’
uniform convergence on the elements of LEMMA.
A E G
$.
b.
E
there is an
E
E
G.
satisfying the following con-
474
1. 2.
Then,
MANUEL VALDIVIA
n A EA n F
F
F
is of finite codimension in
E‘[T]
is closed if Let
PROOF.
x
{xi : i E I]
A E G
is closed,
E A , A E Ci.
is complete.
be a vector in
E
which is not in
a family of vectors in
E
such that
is a Hamel basis of an algebraic complement of
I.
be the family of all finite subsets of A E Ci
we find a continuous linear form on
x
and
We give an order relation
s
in
p1,p2 E N
then
it takes the value
only if
and
1
A1,A2 E G
fl C f2
,
p1 L p2
and
If
F
if
(x,xi : i E I) in
E.
Let
f E 3, p E N
u(f,p,A)
(3,N,G):
Let
F.
on
E
C
and
such that
fl,f2 E 3 ;
(f19~1,A1) s (f2,p2,A2) A1
3
if and
A2.
We consider the net
We take that
2
m<
C
> 0 and C .
Since
is a set
D E Ci,
contains
B.
If
B E G. EB
n
a part
z
E B
F
We find a positive integer
m
is of finite codimension in
g E 3
and an integer
we have that
q E N
such
EB
there
such that
OF F&CHET
A PROPERTY
SPACES
475
from where we get
-
I(U(fl’P1,A1)
implying that (4) is Cauchy in in
E’[T].
z
C ,
e,
be the limit of (4)
u
Let
= 1.
u(x)
On the other hand, if
> 0 there is a positive integer r
r
2
<;
5
Since
E’rT].
it follows that C
I
U(f2,P2,A2),4
E A.
Then if
B
rA,
3
f
A E G
and
E 3
and
E
z
F,
given
such that
p E N
it follows
that 1 * r <e ,
I(u(f,p,~),z)l and thus of
F
zXo,
be a barrelled space such that
E
is a subspace of
F
then
is closed.
F
E.
EA
is of codimension less than
AnF
,E)]
is
E
less
is barrelled.
F
We suppose first that
E
E’[y(E’
with codimension in
E
compact absolutely convex subset of that
does not belong to the closure q.e.d.
Let
If
complete.
PROOF.
x
E.
in
THEOREM 2.
than
= 0. Accordingly
U(Z)
2’O
Let
be a weakly
A
is a Banach space such in
EA.
Every infi-
nite dimensional Banach space has dimension larger or equal that 2‘O,
[ 13, and therefore
EmF
is of finite codimension in
canonical surjection, space i n
E/F
orthogonal to
EA.
If cp: E
F’
is the subspace of
the topology induced by
weak topology and therefore
-b
E/F
is the
generates a finite dimensional sub-
v(A)
and thus if F
is finite dimensional and thus
Ea/EAnF
F‘
y(E‘
,E)
E ’ f l (E’ , E ) ]
on
FL is the
is topologically isomorphic to a
product of lines and thus it has a topological complement E‘[k(E’
,E)]
therefore
.
Then
F
is barreled.
F
has a topological complement in F‘[y(F’,F)]
E
is isomorphic to
L
in
and
L
and
476
MANUEL VALDIVIA
F' f p (F' , F ) ]
therefore
is complete.
F
We suppose now that in
F,
T.
hull of of
E
.?. be its closure in
let
B
Let
and let
Banach space.
dimensional subspace
less than
?
absorbs
B
n
EBnH
of
then a barrel in
M
E
H
B
n
H
EBnH
EBnH
fl F
E
accord-
had codimension
is barreled.
is closed in
E.
is closed and therefore
and, since
T =
of the origin and
which is a
EBnH fl G = { O j ,
such that
We have now that
and thus
lemma to conclude that
,
EB
were not barreled there is an infinite
G
M.
H
be the linear
H
in
But this contradicts that
in
2'
be a barrel
be a weakly compact absolutely convex subset
If "BnH
ing to Theorem 1.
T
Let
and let
E
be the closure of
M
E.
is dense in
Then
We apply the
-
H = E.
T
is
is barrelled, a neighbourhood
is a neighbourhood of the origin in
F. The general case reduces to the former ones. q.e.d. COROLLARY 1.2. a subspace of
Let
E
E
be an ultrabornological space. XO
of codimension less than
2
then
If F
F
is
is
barrelled.
If G subspace of
[4]
is a barrelled space every countable codimensional is barrelled, [ 3 ] .
G
Using this fact we proved in
that every countable codimensional subspace of an ultraborno-
logical space is bornological.
Using Corollary 1.2 a slight modi-
fication of the proof given in [ k ] provides with the following THEOREM
.
subspace of
Let
E
be an ultrabornological space.
of codimension less than
E
2'
then
If F F
is a
is borno-
logical. THEOREM
4. Let
is complete.
E
If F
be a Baire-like space such that is a subspace of
E
E'[i(E'
,E)]
of codimension less than
A PROPERTY OF FR$CFET
2Xo
F
then
plement in let
(T,)
477
is Baire-like.
If F
PROOF'.
SPACES
is closed in
E
F
and thus
E
then
F
has a topological com-
is Baire-like.
F
If
is dense in
E
be an incrasing sequence of closed absolutely convex m
sets of
F
u Tn n=1
such that
Tn
the closure of
u Tn. -
in
is absorbing in
n = 1,2,...,
E,
and
Let
F.
H
Tn
be
the linear hull
m
of
n=1 subset of
If
E,
B
denotes a weakly compact absolutely convex
EBnH is barrelled (see the proof of Theorem 2)
and therefore a Baire-like space, [3], and thus there is a positive integer
n0
origin in in
E.
n
such that
B
n
€I
is a neighbourhood of the
TnO
EBnH.
absorbs
'no
Then
B TI H
and
B
n H
is closed
The conclusion follows as in Theorem 2. q.e.d.
COROLLARY 1.4.
Let
be a subspace of
be an ultrabornological space and let
E
E
of codimension less than
Baire-like space then
F
XO
2
.
If
E
F
is a
is a Baire-like space.
REFERENCES 1.
LdWIG, H.,
-5 ,
2.
-
Studia Math.
18-23 (1934)
MACKEY, G., SOC.
3.
h e r die Dimension linearer Rtiume.
Note on a Theorem of Murray.
Bull. Amer. Math.
2 , 322-325 (1946).
VALDIVIA, M.,
Absolutely convex sets in barrelled spaces.
Ann. Inst. Fourier, Grenoble, 2 l , 2, 3-13 (1971).
4. VALDIVIA, M.,
On final topologies.
193-199 (1971)
Facultad de Matemhticas
Dr. Moliner 50 Burjasot
- Valencia -
Spain
J. Reine angew. Math.
251,
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