then the operator L is regular. Proof..
Let
x, / ~ L
~,
Lx=L Applying Fourier transforms to both sides of Eq.
(15) ...
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then the operator L is regular. Proof..
Let
x, / ~ L
~,
Lx=L Applying Fourier transforms to both sides of Eq.
(15) (15), we obtain
~(~) = ~'i'~ (~o) -- Ao)-* Aj~ (~ -- ~3) + ( ~ -- Ao) -~ ] (~) (fl is the Fourier transform of the function u). obtain the estimate
[I X I{L'<
Hence, using Plancherel's Theorem, we easily
~
tIX IlL' ~31)=1 sup l[ ( ~ -- A0)-1 Aj [{ + sup II (ia -- A0) -1 {{ Hf IIL~.
(16)
It follows clearly from estimates (14) and (16) t h a t o p e r a t o r L i s i n v e r t i b l e i n s p a c e L 2. From Theorem 1, o p e r a t o r L i s i n v e r t i b l e i n C. From t h e r e s u l t s i n [ 4 ] , s u c h an o p e r a t o r L, w i t h a l m o s t - p e r i o d i c coefficients, is regular. LITEPATURE CITED i. 2. 3. 4.
M . A . Krasnosel'skii, V. Sh. Burd, and Yu. S. Kolesov, lations [in Russian], Nauka, Moscow (1970). L. Dalecki and M. G. Krein, Stability of Solutions of Spaces, Amer. Math. Soc. (1974). J . L . Massera and J. J. Schaffer, Linear Differential Academic Press (1966). V . V . Zhikov, "The theory of admissibility of pairs of Nauk SSSR, 205 , No. 6, 1281-1283 (1972).
Nonlinear Almost-Periodic OscilDifferential Equations in Banach Equations and Function Spaces, functional spaces," Dokl. Akad.
2-ABOLUTELY SUMMABLE OEPPATOPs IN CERTAIN BANACH SPACES I. A. Komarchev
In this article we shall give necessary and sufficient conditions for the Orlich space (of sequences) to satisfy the equation ~2(co, X) = L(co, X). We first introduce some definitions and facts w h i c h w e shall need to prove our fundamental result. Let X and Y be Banach spaces. Denote by L(X, Y) the space of linear continuous operators from X to Y, and by X* and Y* the conjugate spaces of X and Y, respectively. Denote by co the space of all sequences which converge to zero. Let X be a Banach space with basis (ei)~<+=- Then we denote by X n the linear hull of the first n elements in the basis. The operator T ~ L (X, Y) is called p-absolutely summable (i ~ p ~ ~- oo) if there exists a number C > 0 such that for any natural number n and any x I, . .., xn E X we have the inequality i/p
de
sup
~*EX*,{Ix*{l=l
(*)
The l i n e a r s p a c e o f a l l p - a b s o i u t e l y summable o p e r a t o r s f r o m X t o Y f o r m s a Banach s p a c e , i f we i n t r o d u c e a n o r m ~ p ( T ) = i n f C, w h e r e i n f i s t a k e n o v e r t h e w h o l e o f C f o r w h i c h (*) holds. The s p a c e o f a l l p - a b s o . l u t e l y summable o p e r a t o r s f r o m X t o Y, w i t h t h e n o r m ~ p , ] i s d e n o t e d b y ~p(X, Y ) . The i d e a o f a p - a b s o l u t e l y summable o p e r a t o r f i r s t a p p e a r e d i n [ 1 . I f e a c h T ~ L (X, Y) i s a p - a b s o l u t e l y summable o p e r a t o r , t h e n we w r i t e Kp.(X, Y) = L(X, Y ) . _~ne proof of the following proposition, found in [2].
on which we shall rely substantially,
can be
Leningrad State University. Translated from Matematicheskie Zametki, Vol. 25, No. 4, pp. 591-602, April, 1979. Original article submitted January 24, 1977.
306
0001-4346/79/2534- 0306507.50
9 1979 Plenum Publishing Corporation
P r o p o s i t i o n I. Let X be a Banach space with a normed unconditional basis (ei)~<+= and let (e~,)~<+= be a system biorthogonal to (ei)i<+= in X*. Then the following statements are equivalent :
1 ) II2 (Co, X) = L (co, X), 2) there exists a number K > 0 such that for any natural numbers m and n and any set of vectors (xi)~<m, z~ ~ X ~ we have the inequality
(,~?=z[I x, [[~)'/'~ K II ,,~?=1(~'~=1[e~ (x,)[~)'l'ej I[. W e give the definition of an Orlich space of sequences. Everywhere we shall consider real spaces. Let M be a function which is c o n v e x and increasing on [0, +~) such that M(0) = 0 and H(t) > 0 if t > 0. Then M is called an Orlich function. If M satisfies the condition ]im M (2z)/M ( x ) < q-co, then M is said to satisfy the h=-condition at 0. We note that if an x-+0
Orlich function M satisfies the A=-condition
sup M ( 2 x ) / M ( z ) < + o o
at zero, then
for any
0 < x-~K
number K < +~. Definition. Let M be an Orlich function. Consider ZM, the space of all real numerical sequences x = (xi)i<+=, for which the following n o r m is finite:
II x II = inf {t > 0 : ~ ' ~ 1 M (I x, l/t) < i}. Then lM w i t h t h i s n o r m i s c a l l e d c a l l t h i s an O r l i c h s p a c e ) .
a space
of Orlich sequences
( h e n c e f o r w a r d we s h a l l
simply
O r l i c h s p a c e s a r e Banach s p a c e s . M o r e o v e r , i f an O r l i c h f u n c t i o n M s a t i s f i e s t h e h=c o n d i t i o n , t h e n t h e s e q u e n c e o f v e c t o r s (e~)~<+=, w h e r e e~ =(8ii)1<+= and 6 i j i s t h e K r o n e c k e r s y m b o l , f o r m s an u n c o n d i t i o n a l b a s i s i n lM. This b a s i s i s u s u a l l y c a l l e d s t a n d a r d . Hencef o r w a r d we s h a l l assume t h a t a l l t h e O r l i c h f u n c t i o n s u n d e r c o n s i d e r a t i o n s a t i s f y t h e h a condition. We now b e g i n definition.
to present
the
fundamental result
of this
article.
We f i r s t
introduce
a
Definition. Let T be a real function defined on the interval [b, a]. We say that T is quasiconcave if there exists a number C > 0 such that for any natural number n and any x I, .... zn ~ [b, a] , we have the inequality
, ((:s;'_, In the definition of quasiconcavity, we may replace the closed interval by an open or semiopen interval. THEOREM. Let M be an Orlich function, t Consider the function M,, defined on [0, I] and defined by the equation 1}It(x) = I]I (~Ix) (x ~ [0, I]). The following statements are equivalent :
1) II2(c o, lM) = L (Co, lM), 2) Mt i s We f i r s t sup M ( x ) = i .
quasiconcave. make a r e m a r k . It is sufficient t o p r o v e t h e t h e o r e m o n l y i n t h e c a s e when In fact, if sup M ( x ) = r , t h e n c o n s i d e r N(x) = M ( x ) / r ( t h e f u n c t i o n N i s
well defined, since r > 0). It is easily seen t h a t the mapping T: ZN -> IM, given by the equations T.fi = el (here (f~)~<+~ and (el)~<+= are standard bases in ZN and ZM, respectively), is an isomorphism from the space ZN into ZM. But if the Banach spaces X and Y are isomorphic, it is easily seen from the definition of a p-absolutely summable operator that the equation ~p(Z, X) = L(Z, X) holds if and only if Hp(Z, Y) = L(Z, Y) (here Z is an arbitrary Banach space and p is an arbitrary real number in [i, + ~ ) . Therefore, in the proof we shall assume that sup M (x) ----'1: o~<~t #We recall that we are only considering O~lich functions which satisfy the A=-condition at zero. (For brevity, we sometimes write simply the A=-condition, instead of the h2-condition at zero. )
307
LEMMA i.
Let M be an Orlich
If there exists that
L > 0 such that
~ , ~9 Z ~~= 1 M ( x ~ ) < 2 ,
function
satisfying
the h=- condition, and
for any natural number n and any numbers
we have the inequality
M(]/(~jy=lX])/n)~>.L/n,
M(x)-~ ~.
sup
0 < xi <
J such
then M is quasiconcave
Suppose that the lemma is not true. Then for any ~ > 0 there exists Proof. number m and real numbers x:, . ., x m in [0, !] (not all zero) such that
a natural
(I) Consider two cases. I)
Let
~=~ :~l (x0 > We may assume that X =
(xi)i~<mis
(2)
I.
a nonincreasing
sequence,
Consider the natural number n and the finite sequence fying the conditions
k0 = 0 ,
of such an n and
of natural numbers
(ki)i
kn ~ m, k i < k ~ + l ,
t~i i ~ Az=~_~+l M (x;) ~ 2 The existence
pu
for
(ki)i~<~ follows
i ~ n and ~ k ~ 1+1 ~I (xi) ~ 2.
(3)
from (2) and from the fact that
r---- sup
o~l
M(x)=
i. Since M satisfies the Au-condition at zero, we can assume that n is sufficiently large. We shall assume that n > 8. We may also assume that 2 = m/n is an integer. In fact, if this is not so, then in place of the sequence (xi)i-<_m, we consider the sequence (Y~)~<([l]+1)~ (here [2] denotes the integral part of l), where Yi = xi for i-~ m and Yi = 0 for i > m. Then inequality (I) becomes
[-!/[%,([1]+1)r~2~
[1 /m/.'l([/]'{-1)r~ 2~
f~-~([/]~-,)n ~ [ (Yi)) / /T$-~
2~ (~2,(m+~~)'=~ ~I (yO) / if[l] + l) n). Thus, w i t h o u t l o s s o f g e n e r a l i t y , n o t a t i o n we r e w r i t e t h e e l e m e n t s
we may assume t h a t 2 i s an i n t e g e r . For convenience of o f t h e s e q u e n c e (xi)~<m, v i z . , b y c o n s i d e r i n g P i = ki+~ --
ki(i = O, ., n I), and s e t t i n g xij = x r if r = k i + j ( i ~ ] ~ qi (i = 0 . . . . , n -- i) defined hy the equations qi = min (~, Pi), = ! ! ~--1 n--1
ki+~--ki). Consider and consider the sets
Y~ {xi):]~q~}, Z~ = { x q : ] ~ l } . S e t Y = w ~ = 0 Y ~ , Z = [ J i ~ g d t h e n Z N Y = r and Z ~ Y = X. We p r o v e t h a t f o r any y ~ Z , M ( g ) ~ 2/I. I n f a c t , l e t M ( y ) ~ 2/I. By t h e d e f i n i t i o n of Z, t h e r e e x i s t j and i s u c h t h a t y = x i j and j > 2. S i n c e x i ~ > x i i f o r i . ~ k ~ ] ((x])/<m i s a nonincreasing sequence), then
~'=~ M (x~r) ~ ~=~ M (x~) > ~=~ M (x~.0 > lUll = 2. This.contradicts inequality (3). We now r e c o n s t r u c t t h e s e t s Yo, - - , Yn-~. Take Yo. I f qo < l , c h o o s e (2 -- qo) e l e m e n t s i n Z. D e n o t e them by yi . . . . . y~_~o. Set Yo = Yo [J {Yl. . . . .
y~_qo}. We
note that these always exist,
since
Z contains
~_~(l--q~)
elements,
and
q~
1
for
any i ~ n--i. If qo = l, we do not change the set Yo but set Y" = Yo and go straight to Y~. y' Consider the set Zo=Z~{yi ..... y~-qo}if ~ 0 ~ l , and Zo = Z if qo = ~- We construct z in exactly the same way as we constructed Yo, where Z is replaced by Zo, etc. Thus we obtain sets Yo ..... Yi~-i such that n--I
! and each Yi contains exactly I elements. It is easily seen that for any f ~ n - - I
We rewrite the elements we have the inequality
-~'~=iM (~) < ~. In fact,
308
let Y~ = {zll..... Xlqi, zl..... s(~_qi~), where
zj ~ Z.
! of Yi as uij
(] ~ l).
(4) Then
M(z) ~ 2/l
( h e r e we a r e u s i n g t h e i n e q u a l i t y (1) l o o k s l i k e t h i s :
for
,-o - , = . From (4) and (5) we see immediately
<
z ~ ~.
I n t h e new n o t a t i o n ,
8X=0 g=,
M
inequality
(5)
that
Our p r o b l e m is now to estimate the quantity
from b e l o w , and u s i n g t h e a r b i t r a r i n e s s o f ~ > 0, t o o b t a i n a c o n t r a d i c t i o n . t h e s e t o f a l l t h o s e i ~ n - - i f o r which
Denote by A
~zi=1 i (uij) > t/i0.
(7)
Then {A{> n/8 ( h e r e {A l d e n o t e s t h e c a r d i n a l i t y o f t h e s e t A). n/8. We e s t i m a t e t h e f o l l o w i n g sums from above:
Suppose t h e o p p o s i t e :
{A{ <
]l Using inequality
(4), we obtain
~.~ 2j=, M (u~j)<
~
4 < 4~18 = ~/2.
(s)
From the definition of the set A, for the second sum we. have the estimate
~ \~' M (uu) <~ ~'~A t/t0 ~ /--Ji~A ~Jj=i Combining inequalities
.(9)
(8) and (9) and bearing in mind that n >11 8, we obtain
~,?-1?z
-~i--~ -~;=~ M (uij) = ~ _ A ~j=~ M (u~)) + ~ a We now recall that the sequence
obtain the inequality
n/tO.
Ej=~ M (u~i) ~ n -- i.
(ui/)~<~-,,i<~ is precisely the sequence
i=o ~i=z M(u~j)>n--i,
which c o n t r a d i c t s
(xl)~~,
(10).
(10) and use (3) to
Without loss of gen-
e r a l i t y , we may assume t h a t (7) h o l d s f o r ~ ~ [n/8]--t. Since M satisfies the A~-condition, i t f o l l o w s from t h e i n e q u a l i t y 15 [n/8] > n t h a t t h e r e e x i s t s a number P > 0 such t h a t
Here P depends only on the function M.
We now take i0 ~
21=1U~:] = min {~i=l ui~: Since M is monotonic,
[n18] -- i such that
i -~< [ n / 8 ] - i}.
it follows that
M(V~':o /'.
2j:lu~i)/([nlS]l)) >
/-~[n/8]--I
.
M(I//(j=lU~])/I) 2'
~
(12)
S i n c e t/iO.~ti=lM(Uw)<~4 ( t h i s f o l l o w s from (4) and ( 7 ) ) , i t f o l l o w s from t h e f a c t t h a t the function M is convex, is strictly i n c r e a s i n g , and s a t i s f i e s the A~-condition, that there e x i s t numbers 0 < t~ < Ca ( d e p e n d i n g o n l y on M), and a number c, ti~
h~-condition and c ~
that~,r
--
t
....
,.
t. From t h i s .
.
-
-
~ .
.
.
.
e q u a t i o n a n d t h e s t a t e m e n t o f t h e lemma, .
t~, we see there exists a constant P, ~
M(V ~j=iu~,i/ i)>/P,fl.
0 (depending only on M) such that
(13)
309
Comparing
(Ii), (12), and (13), we obtain n--1
Now let
~,m ~=lM(X0 ~. t.
Hence it follows that if m > 0 is suffi-
~,~ ~=~ M(x,) = t. Since t > 0, we can choose a natural
Set
number N s u c h t h a t N t ~ l , (Nq-l) w h e r e u i j = x i f o r any ] ~ N q - i .
2
Pl/P < ~e.
and bearing in mind (6), we can write ciently small, case I) does not occur.
II)
l
t > I.
Thus,
I < (N q- I) t < 2. Now take a family (u~l)i<m,~
Then
~'~ ~j=l ~.~+~ u ~" (:~=~ ~ !(m(N + t)))
M(y
M(V
:
m
2
(14)
and
From (i) and Eqs.
M(~/
(14) and (15) we obtain
- ~ ~N+I X~r~ ~3N+I (~'=l#Jj=l u~j)l(m(N + 1))) ~ e ( ~ . = l ~ = 1 M (ucj))/(m(N q- t)) ~ 2e/(m(N q- l)),
w h i c h , i n view o f t h e a r b i t r a r i n e s s inequalities
of r > 0, contradicts ~
the statement
o f t h e lemma and t h e
N+I
Thus Lemma i is proved. LEMMA 2. l)
Let M be an Orlich function.
The following statements
are equivalent:
lie (c O, lM) = L (c o, lM),
2) t h e r e e x i s t s a r e a l number C > 0 s u c h t h a t numbers %ij (i ~ m, ] ~ n), s a t i s f y i n g t h e f o l l o w i n g 0<X~/~t
f o r any n a t u r a l relations
numbers m and n and r e a l
(i~m,i~n),
(16)
i . ~ j = I M ( X ~ j ) < 2,
(17)
we have the inequality
Proof. We first prove that condition 2) in Proposition i is equivalent to the following condition 2') : 2') There exists a number K > 0 such that for any natural numbers m and n and for any normed vectors x~ ~ X n (i = I ..... m) , we have the inequality tr-~
9
In the proof we clearly only need the implication trary vectors
x~. . . . . Find the natural number i, . ., m) such that
Xm ~ X '~, xi 4 = 0
i o ~< m . f o r w h i c h
a =
kia' < II x~ IP < 2k~a2 and set ko = 0.
Consider the sequence
(19)
. ~ l~J , (xOp)v, ~ II. IIY,j=, (~=,
2') ~
2). Let 2') hold.
(i =
r a i n ti x i
t .....
-Consider the arbi-
rn).
II = II z,, II, t a k e n a t u r a l
(i = t,2 . . . . .
numbers k i
(i =
m),
y~ ~ X ~ (i = I . . . . . J), J = ~i0ki, defined as follows :
yj = xil(~f~a), if I + ~:----'0k, < ] < 2~=0 k,. Then I < IIuy, < ~ 2 (2 = I ..... ]). J). Hence it follows easily that
310
Inequality
(19) holds for the vectors
yyl II y,
I] (] =
i .....
I t is easily shown that the latter inequality can be rewritten as follows :
which completes the proof
(2,~_.~_,II x, ii,)v, < V~ g II Z}=, ~=~ I~* (~)l~)'/'~J II, 9 of the implication 2') ~ 2).
We now pass to the real proof of Lemma 2. We see that 2) ~ i). We show that 2') holds, and thus that the equation l]~ (co, I M ) = L (co, IM)follows from Proposition i and the equivalence of 2') to the second part of Proposition I. Consider arbitrary natural numbers m and n and normed vectors xl ~ X ~ (i = I .... , m). prove that
We
VF < c~ IIZj=~ ~=~ l e* (x0p)'~,e~II, where C = I/C~ is the number in Lemma 2.
This is equivalent
)Jjl M (C~ ]/(~.'lll e* (z,)l,)l,,) >
to the inequality
(20)
~..
If we prove that (21) then using the convexity of M, we immediately obtain then for any i ~ m
(20).
Since [] x~ ][ = i (i----I..... m),
~ i i (Ie~ (x01) = I. Then to prove
(21), it is sufficient
Conversely,
to use statement 2) of Lemma 2.
let [Is (co, IM) = L (co, IM). Then to prove 2) of Lemma 2, it remains to use
Proposition I, taking
x i = ~'Y=1%r
= i .... , m).
Lemma 2 is proved.
Proof of the Theorem. i) Let II~ (Co, IM) = L (co, IM). We prove that the function M x (x) = M (V~ is quasiconcave, suppose not. Then from Lemma I, for any e > 0 there exists a natural
number
m and
0~
xt ~ I (i ----1 ..... ra),satisfying the inequality
i~.ilJ}l(x0~2,
for
which M ( V ( ~ . : i x ~ ) / m ) < elm. Consider now the matrix of real numbers (%ij)~<m.j<m, setting Xil = xj, and whose remaining rows are c y c l i c permutations of the first row. Then
i -.~ ~.~IM(~0)..< 2
(i = 1 , . . . . m)
and
which contradicts
Lemma 2, since ~ > 0 was arbitrary.
2) Now let 7~11(x) = M ( ~ f - ~ we use Le=~na 2.
be quasiconcave.
To prove the equation ll~(c0, I M ) =
L (co, IM)
Consider m, n and the set of numbers (%U)~<m. j 0 be the constant in the definition of quasiconcavity of M~. Then
which completes the proof of the theorem. COROLLARY i. Let M be a differentiable Orlich function with the A2-condition, N(x) = M'(x)/x be a decreasing function on (0, i ] . Then ll~(co, i M ) = L ( c o , IM).
and let
Proof. C o n s i d e r M~ (x) = M ( V x ) (x E (0,tl). Then t h e f u n c t i o n M~(x) = M ' (Fx)/(2Vx) is monotonically decreasing. Hence it follows that the function MI is concave on (0, i], and therefore also quasiconcave on this interval, a n d it only remains now to apply the theorem. To prove the following corollaries we need some definitions and results. Definition.
Let
p > 2
and X be a Banach space.
p r o p e r t y if for any unconditionally
convergent series
X is said to satisfy the Orlich p~,:=ix~ in X, the numerical series 311
~=i 11x~ li p
also converges.
Definition. Let p ~ 2. The Banach space X is called a space of cotype p if there exists a number C > 0 such that for any natural number n and any x j ~ X ( ] = i ..... n) we have the inequality
(i~l]E';=l rj(t)xj
~t) '/'>C( Y?j=l
Here r] (t) = sign sin (2i-1 2=t) i s the Rademakher f u n c t i o n . thus :
\lip
IIxj IIp)
.
Inequality
(22) (22) can be rewritten
I t i s w e l l k n o ~ (see, e . g . , [ 2 ] ) t h a t i f the e q u a t i o n E=(c=, X) = L ( c o , X) h o l d s f o r the Banach space X, then X s a t i s f i e s the O r l i c h 2 - p r o p e r t y . The f o l l o w i n g s t a t e m e n t a l s o h o l d s (see [3] ). Proposition. Let X be a Banach space with an unconditional basis. K2(co, X) = L(co, X) holds if and only if X is a space of cotype 2.
Then the equation
Using these fffcts, we obtain the following corollaries of the theorem. COROLLARY 2. Let M be an Orlich function satisfying the A2-condition, ~/i (x)= 7~f (~). If M~ is quasiconcave, then the space 1 M satisfies the Orlich 2-property. COROLLARY 3. Let M be an Orlich function with the A=-condition. of cotype 2 if and only if the function M~ is quasiconcave.
Then 1M is a space
The author thanks B. M. Makarov for his help with this article. LITERATURE CITED i. 2. 3.
A. Pietsch, "Absolute p-summierende Abbildungen in normierten Raumen," Stud. Math., 28, 333-353 (1967). E. Dubinsky, A. Pelczyncki, and H. P. Rosenthal, "On Banach spaces X for which I]2(L~, X) = B (L~, XJ," Stud. Math., 44, '617-648 (1972). B. Maurey, "Theorems de factorisation pour les operateurs lineaires a valeurs dans les espaces ~ , " Asterisque, ii, 1-194 (1974).
PROPERTIES OF SYSTEMS OF n-DII~NSIONAL CONVEX SETS IN FINITEDIMENSIONAL LINEAR SPACES A. G. Netrebin
In this article we consider combinatorial properties of systems of convex sets in a linear space. In Sec. 1 we prove Theorems 1-3, which are analogs of theorems of De Santis [i], GrHnbaum and Katchalski [2, 3], and we introduce an analog of theorems of Hadwiger and Debrunner [4]. The theorems of these authors establish several properties of systems of convex sets in the space R n. Theorems I, 2, and 4 state that these properties (with certain changes) hold for systems of n-dimensional convex sets in a finite-dimensional space. We also prove (Theorem 5) a generalization of Boltyanskii's and Soltan's theorem [5]. In Sec. ~ 2 we use several statements from Sec. 1 to study the nerves of systems of n-dimensional convex sets in the space R d. Standard terms and notation will be used. i. To prove Theorems 1-4 we need the following two theorems. The first of these (A) was proved by De Santis [i], and the second (Theorem B) was proved partly by GrHnbaum [2] and finally by Katchalski [3]. Institute of Mathematics and Mechanics, Ural Science Center, Academy of Sciences of the USSR. Translated from Matematicheskie Zametki, Vol. 25, No. 4, pp. 603-618, April, 1979. Original article submitted November 22, 1977.
312
0001-4346/79/2534- 0312507.50 9 1979 Plenum Publishing Corporation