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R^, then v{Q satisfies (2.4)-
156
CHAPTER 8
Proof. Let p < 0 be any number. Then the function Up{w) = U{w) - j j
\og\ho^{w,s)\ dv{s)
is harmonic in G~ and is of Mp by Lemma 1.3. By the same lemma, W-'^Upiw) = W-"" u{z) - 1 1
W-""log\bc,(w,s)\
dv{s).
is harmonic in G~. Writing the difference of two Poisson representations of : \w\ < RQ} and in G-{R) {R>Ro> i?*), W-'^Upiw), in G-{RQ) ^{weQwe arrive at the equality
r[
^-a^(^)^^P,i^o(C^)|^^|^ weG;{Ro), (3.7)
'^^ JdG-{Ro)
where
ipA^^s) = ^f
w--\ogK{C,s)\ ^^^'^^^-^Vci {R = Ro.R)
and QP^RQ, Qp^R are the Green functions of G~{Ro) and G~{R). We conclude that IP^R{W,S) — IP^RQ{W,S) > 0 when w G G~{RQ) and 5 G G~, since W~^\og\ha{C,^s)\ is subharmonic in G~ for any fixed s G G~. Now assume that i? and RQ {R> RQ > R^) are such that / \W-''U{Re-'^)\smddd<^oo Jo
{R =
RQ,R).
By Lemmas 2.1 and 2.2 of Ch. 7, the right-hand side of (3.7) remains bounded for a sequence p = pnt ^' Hence, by Fatou's lemma / / liminf [/p,ii(if;,5) J J^-^ p ^ - o
IP^RQ{W,S)]
dv{s) < M{w) < +oo,
where M{w) > 0 is a constant depending only on W^RQ and i?, while po < 0 has sufficiently small modulus. Letting here po -> —0, we obtain / /
liminf [/p,i?(if;,5) - Ip^R^{w,s)\ dv{s) < M{w) < +cx).
(3.8)
157
W E I G H T E D CLASSES O F SUBHARMONIC F U N C T I O N S
But for any R> 0 there exists
p->-0
C—w
^T^'^JdG-(R) fdG-{R)
C—w
W
W
dC
+
/"llm s\
-|
r ( l + a) J_\ims\ where (see (2.2) of Ch. 2) jR{t,w)
-f
27ri JdG-{R) C - (Re s + it) C ~w
C—w w
+
w
dC
Calculating this integral and returning to (3.8), we arrive at the estimate
+t
L(fi+*)' + (Re.)^ ML + t
dt > du{s) < M{-ivo)
\vo\
< 4-cx),
(S+*) +(^^«)'J where M{—ivo) > 0 depends only on VQ (—i?* < i;o < 0) and i?, i?o- Taking i? sufficiently large, we obtain that u{() satisfies (2.4). 3.3. P r o o f of T h e o r e m 3.1. Let U{w) G Mp be a function subharmonic in G~ and satisfying conditions (3.1), (3.2) for an a G (p — l,p], and let the associated measure u{Q of U{w) vanishes in a neighborhood of infinity. Then u{() satisfies (2.4) according to Lemma 3.1. Therefore the function U.{w) = U{w) - I I is harmonic in G'. W-''U.{w)
log Kiw, 01 di^iO
By Lemma 1.3, Ut{w) E Mp and = W-" u{z) - I I
W"' log \ho.(w, 01 dzy(0
is harmonic in G~. Besides, this function satisfies (3.1) and (3.2) since W~''lo.{w) = ' 1 1
W " " log K{w, 01 du{0
(3.9)
158
CHAPTER 8
satisfies (3.1) and (3.2). Moreover, the representation (1.12) implies that Km
^f
W-'^lJRe-'^)sm^di} = 0
(3.10)
and + 00
W-''Ia(u-\-iv)du = 0.
/
(3.11)
-oo
Hence, by Theorem 1.1 of Ch. 7 W--U4w) = - f ^ ,
'^^}^^ , ,
w = u^-iveG-,
(3.12)
where ii{t) is the measure determined from relations (3.4) and satisfying (3.5), (3.6). In addition, (3.3) is obviously true. Applying ^ V F - ^ ^ - " ) to both sides of (3.12) and using the results of Ch. 1, we obtain a representation of the form (2.3). Therefore, Theorem 1.1 leads to the conclusion that U{w) G SoL' Conversely, by Remark 2.1, for any U{w) G Sa the function W~^ u{z) is representable in the form (2.6). Now relations (3.3) and (3.6) follow from (3.10), (3.11) and Theorem 1.1 of Ch. 7. NOTES. The inversion z = w" ^ transforms our requirement on boundedness of the support of associated measures in classes Sa and Sa to the requirement that the supports of measures are disjoint from the origin. This is a natural requirement for the classical Blaschke product with factors of the form (1 -z/(!^)/ (1 - z/Q in G"*" and its generalizations.
CHAPTER 9 FUNCTIONS OF a-BOUNDED TYPE IN SPECTRAL THEORY OF NON-WEAK CONTRACTIONS
1. FACTORIZATION OF REGULARIZED DETERMINANTS 1.1. For any p > 1 we denote by Cp the class of continuously invertible contractions T in a separable Hilbert space 53 for which the operator Dj^=I — T*T belongs to the Neuman-Schatten ideal 6p. The set Ci coincides with the class of all invertible weak contractions [104]. We define the characteristic function WT of the operator T as in [12]: WT{Z)WT{0)
WT{0)
= [/ - DT{I
- ZT)-'^DT]
= (r*T)i/2 I S T ,
I ST,
VT = DTS).
It is easy to verify that the operator-function Wj^{'z) differs from the characteristic function @T{Z) of B.Sz.-Nagy and C.Foias [104] by a constant isometric factor. Let us recall from [104, 12] that WT{Z) is holomorphic in \z\ < 1, where its values are two-sided contractions in 2 ) T , i.e. W>J^{Z)WT{Z) < I and WT{z)Wf{z) < / in \z\ < 1. Since / - W^\0)
= W^\0)
[(r*T)i/2 - /
\^i
= W f i ( 0 ) [ / + ( T * T ) i / 2 ] ~ \ r * T - / ) |S)T and
I - WT{Z)
= I-
W^\0)
+ DT{I
- zT)-^
DTW^\0)
operator / — WT{Z) belongs to &p for any z ^ a{T~^). the regularized determinant driz) = detpWriz)
= Y[Xk{z)expl k
^
, DT
-:[1 - Xk{z)y
^ 3=1 ^
e 62p,
the
Hence for each p > 1
(1.1) J
is holomorphic wherever the operator-function WT is holomorphic [37, Ch. IV]. In (1.1), {Xk{z)} is the set of eigenvalues of the operator WT{Z). Further, the functions VT = detpWT{z)Wf{z) (1.2)
160
CHAPTER 9
will play an important role, and one can state that formulas (1.1) and (1.2) give a correspondence between the operators T e Cp (where p is a natural number) and the functions (IT and 7>T which are holomorphic in |z| < 1. 1.2. The main result of this section states that (IT and J>T both belong to M.M.Djrbashian's class Na [19, Ch. IX]. As it is well known, the functions (IT and J>T are bounded in |2:| < 1 if p = 1. For the general case we have T h e o r e m 1.1. If p > 2 is an integer and T G Cp, then the holomorphic functions (IT and T>T belong to Np-i^^ for any e > 0. For proving this theorem we need L e m m a 1.1, If p > 2 is an integer, then for 0 < r < 1 i- / ^^
\\DT{I-zT)-'DTrp\dz\
< (l-r)-(^-i)p|.||^,
(1.3)
J\z\=r
where \\ • ||p is the norm in &p. Proof. First, suppose p = 2^ {k > 1). By the elementary properties of eigenvalues (A^) and singular values {sj) of compact operators [37, Ch. II] \\DT{I - ZT)-'DT\\1
= Y. s f '
{DT{I - ZT)-'DUI
-
ZT*)-'DT)
3
= J2 ^T i^Til - ZT)-'DUI - ZT*)-^DT) 3
= Y, Af' {Dl{I - zT)-^Dl{I
-
zT*)-')
3
< Y ^T i^Tii - zTr'DUi - zT*r') 3
3
<{l-rrP/'\\DUl-zT)-'Dm/;^
{\z\=r).
Using these relations k — 1 times we get \DT{I
- ZT)-'DT\\1
< ( 1 - r ) - ^ n ; . ' 2"' \\Dl'-' (/ -
=
fc-1
zXy^D^ 2
il-r)-^P-^)\\Q{I-zTr'Q\\l, 2
•
||2
2
FUNCTIONS O F Q-BOUNDED T Y P E IN ...
161
where Q = DZ! . Further,
/
\\DTiI-zT)-'DTt\dz\
J\z\=r
P
oo
= 27r(l - r)-(^-2) ^Sp(QT^'Q2(T*)^Q)r2^' 3=0
since
|2 = Sp(0(7 - zT)-'Q\l ||Q(J - zT)-iQ||2'
-
zT*r'Q)
= J2 Sp{QT^Q^{TyQ)z^z''. j,k>0
Recalling that Q = D^' , we get
Hence (1.3) holds for p = 2^ {k > 1). Note that the above argument remains vahd if we replace DT by any normal operator G with the spectral decomposition i
i
We shall use this later. Returning to the proof of (1.3), for any p > 2 we shall use the well known Hadamard theorem on three lines and some techniques similar to one worked out in the proof of the Riesz-Thorin theorem on interpolation of operators [7]. First note that if the values of a function F{ip) are in &p, then TV
sup G
/
\
(
f^
1 ^^^
^ sp{F{,f)G{^)}dcp]^ = |y_^ m^w^d^j
,
where the supremum is taken over all ©^-valued functions G{(p) for which
\\G{^)\\ld^ = l,
(1.4)
1 + 1 = '^-
J —t
Of course, we presupposed that F and G satisfy the standard requirements providing the existence of the integrals in (1.4). It is easy to verify that (1.4) is an immediate consequence of the precise estimate |Sp{FG}| < ||i^||p||G||q (see [37]) and Holder's inequality.
162
CHAPTER 9
For interpolating the inequality (1.3) (already proved for p = 2^~^, 2^) for all p G [2^~^,2'^], denote po = 2^"^, pi = 2^ and similar to the proof of the Riesz-Thorin theorem set 1 1— z z 1 1— z z 1 1 . / . ^ . x P{z) Po P i ' q{z) qo Qi' Pj Qj and for any n > 1 consider the operator-function ri
_p_
_ ^
n
k=l
j=l
where l/p-\- 1/q = 1 and {ek}, {%(v^)}> {'^i(^)} ^^^ some weak measurable orthonormal sets of vectors, 5^ > 0 and aj{(p), [djiy^)]"^ are nonnegative, bounded, measurable functions. In addition, we suppose that
X:sf = l
and
rJ2a%v)d^
fc=i
=l
•'-'^j=i
for any fixed n > 1. Now introduce the entire function fr{z)
= ^
f
Sp{^niz){I-re"'T)-^^n{z)Gn{
k,m,j=l
wheie Pk,m,j{^) = {ek,Vj{(f)){uj{(p),em){{I - re^'^T) ^6^,6^). The inequality (1.3) is true for p = po and for the normal operator $n(^)- Consequently 1/PO
\Miy)\ <
(27^)^-'!^^Ij\^n{iy)il-re'^T)-'^niiy)\Zld^'^ r
\
p-K
l/qo
<(27r)-i||$^(i2/)||^^(l_^)-i+i/Po|^j|Gr„(^,iy)||^orf^ 2p
=(27r)-i|X:
Po >j 1/po
go
(l-r)-
af^\^) 1 1/90
^27r)-^^.f
(1-r)-^
i/go
/
0
>j 1/go
y
_ ^
5 ] a ((/:?)d(yi? ^ = [27r(l - r)] ^o.
Takings = pi we similarly come to the inequality \fr{l-{-iy)\ < [27r(l—r)] ^/^^. Consequently, \frW\
< [27r(l-r)]""^~^,
0 < ? ? < 1.
163
FUNCTIONS O F Q:-BOUNDED T Y P E IN ...
Now let p e (2^ "^,2^) be arbitrary and let the corresponding T? G (0,1) be chosen from 1
1-1?
p
po
- =
^
+-, Pi
1
1-7?
q
go
- =
7?
/I
1
Qi
\p
q
+-
\
- +- =1 . J
Then obviously n
^nW
= Y^Sk{'
n
,ek)ek,
Gn(^,^) = ^ a j M ( . ,Vj{cp))uj{ip)
k=i
(1.6)
j=i
and therefore ^)]d^ < [27r(l-r)]
•^ j ^ Sp{$n(7?)(/ - re'^Tr^^nmOni^,
-i/q
for any n>\ and any functions 0n('^), Gn{^,'&) satisfying (1.5). Using this inequality and (1.4) instead of the first condition of (1.5), we get ^
fj^nWil
~ r e ^ ^ T ) - ^ ^ 4 ^ ) | | > < 11^^(^)11^ (1 - r) - ( p - i )
At last, taking as $n(^) the n-th sum of the Schmidt series of the operator DT and letting n —> oo we come to (1.3). P r o o f of T h e o r e m 1.1. Using a simple inequality for regularized determinants [31, Ch. XI, Sec. 22], from (1.1) we get log+ M T ( ^ ) | < mp\\I-WT{z)\\P {\z\ < 1). Therefore log+ \dT{z)\ < 2^M,{\\I
- Wi;\0)r,
+ \\Wi;\OW\\DT{I
-
ZT)-'DT\\I],
Consequently, by (1.3) /
D-^log+ |^T(re^'^)|^^ = Z ) - ^ | /
log+ \dT{re'^)W^^
Ci+C2(l-r)-^+i
for any a > p - 1. Hence dr G A"^ {a > p - 1). To prove that T>T G ^4° for any a > p — 1, observe that log+
\VT{Z)\
< Mp\\I -
WTiz)W}{z)rp
< 2PMp{||/ - WrizWp + < Ci + C2{\\DT{I
- zTr'Drr^
\\I-WS{z)\\p} + \\DT{I
-
zT^r'DTFp}.
164
CHAPTER 9
Application of (1.3) completes the proof. Corollary. If p > 3 is an integer, T e Cp and I — T'^T G Got for some a ^\p — l,p); then both (IT and J>T belong to ^ p _ i . Proof. As Lemma 1.1 is true for any p>2, using the above notation we obtain
- zTr'Dr\\;=^s^,
\\DT{I
which even may be not an integer,
= E4'"«i 3
3
<\\DT{I-zT)-^DTr-''Y.''i 3
-
<2^-"\\DT{I
zT)-^DTn.
Hence i- /
- zT)-'DT\\l\dz\
\\DT{I
< ^
^TT J\z\=r
/ ^^
\\DT{I
-
zT)-'DTUdz\
J\z\=r
<2P-"||D2^||^(l-r)-("-^).
Recalling the proof of Theorem 1.1 and using the fact that the class A° enlarges as a increases, we complete the proof. 1.3. The following assertion is another corollary of Theorem 1.1. Proposition 1.1. Let {zk} be a sequence of numbers from \z\ < 1, such that for a given integerp>2 ^ ( l - | z , | f <+oo.
(1.7)
k
Then the products
k
I fe=l •'
fp-l J J2 ^[l - Bk{z)]
Vo{z) = l[Bk{z)exp{
(1.8)
k^i^
k
where Zk — Z \Zk\
bzA^) = --, =1-7-
1 — YkZ Zk
^^^
^k{z) = bzj,{z)bj^{z),
are holomorphic functions in \z\ < 1, which belong to A^_i^^ for any e > 0. Proof. Consider the normal operator T = ^
Zk{'
,ek)ek.
FUNCTIONS OF a-BOUNDED TYPE IN ...
165
in our Hilbert space S^. Obviously, T eCphy (1.7). Further, for z ^ cr(T~^) U a{T*~^) the operators WT{Z) and WT{Z)W>}^{'Z) are normal and the sequences {bzk{z)} and {Bk{z)} are their eigenvalues. Hence ^0(^2^) = driz), ^o{z) = J)T{Z), and our assertion follows from Theorem 1.1. Now we shall check the precision of Theorem 1.1. First, it is easy to observe that the discrete spectrum of an operator T ^ Cp satisfies (1.7). Thus, the functions (IT and T^T in general do not belong to A^ when a < p — 1. For checking this, observe that the set of zeros of dx is the discrete spectrum of T*. Indeed, it follows from (1.1) that driz) — 0 if and only if ;2: = 0 is an eigenvalue of the operator WT[Z)^ and this is equivalent to noninvertability of WT{Z^ since / — WT{Z) G 6 p . On the other hand, applying the equalities z^ ai.T''),
T^r(^)W?(^"^) - W^(^-i)Wr(z) - / ,
\z\ < 1,
(see [104, 12]) we conclude that the operator WT{Z) {\Z\ < 1) is invertible if and only if z ^ a{T*). Similarly, {z^} U {zk} is the set of zeros of 2>T, where {zk} is the set of eigenvalues of T. Thus, if dx (or 7)T) belongs to A"^ for some a < p — 1, then
^(l-|z,|)i+«<+oo k
according to the general assertion on the density of zeros in ^ ° , and these series obviously does not converge for any T G Cp. It is not known weather the functions dr and X>T belong to Ap_i or even to M.M.Djrbashian's more wider class Np-i for any T E Cp. Also, it is not known weather the products do and I^o? constructed by a sequence satisfying (1.7), belong to Np-i. But we can state that there exists a sequence {2^fc}i° satisfying (1.7) and ^(l-|z,|)^-^<+oo k
for any £: > 0, such that do, ^ 0 G Ap_i for the corresponding products. However, we mention that Theorem 1.1 is as much precise as it is necessary for the further assertions. 1.4' Finding factorizations for C?T and J>T, we shall pay the main attention to the function T^T since its factorization will play more important role in our further considerations. First we note that 0<
VT{X)
which immediately follows from
< 1,
- 1 < a: < 1,
WT{X)W'}{X)
(1.9)
< I.
T h e o r e m 1.2. Ifp > 2 is an integer, T E Cp and {zk} is the discrete spectrum of T, then for any £ > 0
V^{z)=i:)o{z)e^^[-^j^
5p_i+,(e-^^z)d^,(i?)|,
\z\ < 1,
(1.10)
166
CHAPTER 9
where ips is a real, continuous function of bounded variation in defined by (1.8) and
[—TT^TT], 2>O
is
5p-i+.(C) = r(p + e ) { ^ 3 - A _ _ i | , Each factor of this representation satisfies (1.9) and if it is assumed that i/jsi—Tr) = 0, then the factorization (1.10) is unique for any s > 0. For proving this theorem we need the following Lemma 1.2. Let A and B be nonnegative contractions and let A = I — XP, where A G (0,1) and P is a one-dimensional orthogonal projector. Ifl—B G &p for some integer p>2, then
sp I E ^ [(^ - ^ ) ' + (^ - ^ ) ' - (^ - ^ ^ ) 1 1 ^ 0-
(1-11)
Proof. First, consider the case when A and B are acting in a finite-dimensional space and B is continuously invertible. Let {A^li', {l^k}^-, and {Z^A;}? be the sets of eigenvalues of the operators A, 5 , and AB correspondingly, numerated in decreasing order (note that the spectrum of AB is positive). It is well known [35, Addition] that TTl
771
n
n ^J ^ n ^^N 3= 1
(1 < m < n - 1) and
3= 1
71
H ^^' "" 11 ^^N3= 1
3= 1
In terms of these sequences, the inequality (1.11) takes the form n
p—1 ^
3=1
k=l
Observing that
3 = 1 k=l n
"^ \
= ^ ( l o g A,- + log iij - log I/,) - - log H - ^ 3=1
3=1
- 0,
^3
we conclude that (1.11) is equivalent to
EEfc{(i-^^)' + (i-^i)'-(i-'^i)'}^o3 = 1 k=p
(1-12)
FUNCTIONS O F Q-BOUNDED T Y P E IN ...
167
Now set ^(t) = f {I- e-'^y-^dx, Jo
0
+CX),
Thus, denoting — log Aj = aj, — log/ij = bj and — loguj = Cj {I < j < n) one can write (1.12) in the form
3=1
i=i
As A = I — XP, where P is a one-dimensional orthogonal projector, we have ai = a2 = ''' = ctn-i = 0, an = — log(l — A) and also m
>.
/ m
^bj
= -log( YlfiA
y m
\
<-log(Y[uj]
m
=J2^J^
1 <m
and an + Zlj=:i ^j = S j = i ^J- Consequently
f]{$(c,) - ^bj)}=j2 r ^^ - ^"')'"'^^' 3=1
3=1 -^^^
where Cj > bj (1 < j < n) and Cn > an since for each fc(l < fc < n) we have i^k < Xk and Uk < /J^k [35, Addition]. Note that at least the first n — 1 intervals {^3^ ^j) (1 ^ J < ^ — 1) are disjoint. Indeed, let C be the Hnear hull of those eigenvectors of AB, which correspond to the eigenvalues ^'l, z/2,.. • , ^j-i, and let P = (• , e)e. Then, using the minimaximal properties of eigenvalues [37, Ch. II] we obtain that
(ABx.x)
Uj — max —
(Bx,Ax)
r— > max —
(Bx.x)
r— = max —
r^ > //j+i
for any j ( l < j < n — 1). Thus Cj < &j+i. Further,
3=1
73=1"^^^ = 1 ^^3
> r\l-e-^r-^dx Jo
= ^{an)
168
CHAPTER 9
since {bj, Cj) (1 < j < n) are disjoint and the sum of their lengths is a^. Thus, we proved (1.11) under the assumption that A and B are acting in a finite-dimensional space. It is clear that (1.11) is valid also when B is not invertible. For proving (1.11) in the general case, observe that in virtue of the formula detpAB
={detpA){detpB) x e x p | - S p (j2l[{I-A)'-^{I-B)'-{I-AB)']^^
(1.13)
and other simple properties of regularized determinants [37, Ch. IV] the lefthand side of (1.11) depends on the operators I — A and I — B continuously in the ©p-metric. Consider a monotonely increasing sequence of orthogonal projectors {Pn}T which strongly tend to / and PnA = APn (n > 1). As (1.11) is already proved for An = PnAPn and Bn = PnBPn, letting n -> oo we come to (1.11) in the general case. 1.5. Proof of Theorem 1.2. As we have proved, {zk} U {zk} is the zeroset of the function X>T which belongs to Ap_-^_^^ for any e > 0. In virtue of Proposition 1.1, also the product !Do constructed by the sequence {zk} belongs to Ap_i_^^. Hence the function 'DT{Z)/'DQ{Z)^ which has no zeros in \z\ < 1, belongs to A'^_ij^^ for any 5 > 0, and consequently it allows the representation
VT{Z)/V^{Z)
= e'^^ e ^ p { - ^ f
Sp-i+e{e-'^z)dM^)V
|z|
where Im 7^ = 0 and i/^e is a real-valued function of bounded variation in [-7r,7r] [19, Ch. IX]. In this formula 7^ = 0 since X>T(0),X>O(0) > 0. Thus (1.10) is true. For proving that -0^ is continuous, observe that log ( P T / ^ O ) is holomorphic in |2:| < 1 and hence it can be expanded in a power series: 00
log {VT{Z)/VO{Z))
= ^ 4 ^ ^
\z\ < 1.
A;=0
On the other hand, using the power expansion of the kernel 5p_i+£ we find
Hence
169
FUNCTIONS OF Q-BOUNDED TYPE IN ...
As the same equality is true for any ei G (0,6:), we have \dk\
Tjl + k)
^1
-iM
r ( p + 5 + fe) TT
#.iW
Tip + £i + fc) T{p + e + k)
0(1)
as fc -> cxD, i.e. the Fourier coefficients of V^e tend to zero and hence '^^ is continuous on [—7r,7r] [122, Ch. Ill] (note that more might be said on the differential properties of t/^e). The uniqueness of the function ijje follows from the results of [19, Ch. IX]. Moreover, the following inversion formulas are true:
V^(±) . (^2)
- ^ f ^(^i) = ^Hm J j ^ rD"" . - a .log .J^T(re^^)
(1.14)
M,
'0^(^) :='0(+)(^)-'0(-)(^) (-7r<7?i<^2<7r, - 7 r < ' i ? < 7 r ,
a=p-l-\-e),
where ijje {'&) are nonnegative, continuous, increasing functions defined as the positive and the negative variations of ijje on [—TT,-??]. Finally, we shall show that each term of the factorization (1.10) satisfies (1.9). It is well known [12] that to each invariant subspace S)i of the operator T, on which T is invertible, corresponds a factorization of the characteristic function: WT{Z) = WI{Z)W2{Z), (1.15) where Wi and W2 are holomorphic contractions in |z| < 1. Now, let zi be the first eigenvalue of T, Tei = ziei, ||ei|| = 1 and let ^ 1 be the one-dimensional invariant subspace born by ei. Then by (1.15) and (1.13) = detj,[W^ {z)Wi{z)W2{z)W^
VT{z)^detp[Wi{z)W2{z)W^{z)W^{z)] =
{z)]
det^[W^{-z)Wi{z)]det^[W2{z)W^{-z)]
xexp|-Sp(X^i[(/-A)^ + ( / - 5 ) ^ - ( / - A B ) ^ ] ) | ,
(1.16)
where A = Wi{j)W\{z) and B = W2(-2^)1^2*(^)- For finding the elementary factor corresponding to zi, note that detpW^{z)Wi{z)
=
detpWi{z)W^{z)
and use the well known formula [12] Wi{z)W^{z)
= 7 - (1 - Z^)DTPI{I
-
zTi)-\l-ZT^)-^PIDT,
where Pi is the orthogonal projection on i j i and Ti =T\S)i. (1 - \zif)h (7 - zTiy^h
= (7 - Ti*Ti)/i = Pi (7 - T*T)Pih = = (7 - zzi)-'^h
and
As
PiD^Pih,
(7 - zT;;)-'^h = (1 -
zziy^h,
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CHAPTER 9
for any h e S^i, we conclude that the spectrum of the operator Wi{z)Wi {"z) coincides with the spectrum of the operator / - (1 - z'^){I - zTi)-\l = I-{1-
-
-zT^y^PiDlPi
z^){I - zz,)-\l
- zzi)-\l
- |zi|2)Pi
having only one eigenvalue Tyi{z) that differs from 1: vx{z) = \ - {l-z^)il-\z^\^) {\ - zz{){\ - zz{)
_{z-z^){z-z^) (}. - zz{){\ - zzx)
= h.,{z)h-,,{z) = B^{z). Therefore, obviously r pp--ii - .
detpT^i(z)l^i*(^) = ^ i W e x p ^ E ^ [ l - ^ i ( ' ^ ) ] ' \3 = ^ 3 is the desired elementary factor. Further, denoting as ^/c(>3) the elementary factor corresponding to z^^ by (1.16) we obtain VT{X)I^X{X)
< detpW2{x)W^{x),
- 1 < X < 1.
(1.17)
Here we used also (1.11) with A = W^{x)Wi{x), B = W2{x)W^{x), which, as one can easily verify, satisfy the conditions of Lemma 1.2. Since W2 is also a characteristic function of a contraction from Cp [12], the above argument is applicable for X>2(^) = detpW2{z)W^(z). By (1.17) this gives 0 < 'DT{x)/[^i{x)^2ix)]
< detpW3{x)W^{x)
< 1,
- 1 < a: < 1,
since W3 is a holomorphic contraction. Taking away the zeros oiT>T in this way, we come to the desired statement, and the proof is complete. The following question naturally arises: in which case the exponential factor in (1.10) is absent? The answer is given by Proposition 1.2. Let T he a completely non-unitary contraction from Cp (p > 2). Then its regularized determinant T>T is exactly the product T^o of (1.10) if and only ifT is complete and normal. We omit the proof of this statement since it will not be used in the future. Note that, as it was stated in the proof of Theorem 1.2, each discrete factor of the factorization (1.10) is a regularized determinant constructed by a divisor of the characteristic operator-function WT^ corresponding to an eigenvalue of T. So, the considered Blaschke-type factors have a definite spectral interpretation and their application is reasonable in operator theory.
FUNCTIONS O F Q;-BOUNDED T Y P E IN ...
171
In [19] M.M.Djrbashian used Blaschke-type factors of different form, which permit to obtain inversion formulas (1.14) without division by I>o- Note that products of nearly same type as in (1.10) were introduced in the early works of M.M.Djrbashian [16, 17].
2. BOUNDARY BEHAVIOR OF REGULARIZED DETERMINANTS In this section we find a connection between the behavior of the function J>T near the boundary point z = 1 and some spectral properties of operators T G Cp. Namely, our aim is to prove Theorem 2.1. Let p > 2 be an integer, let 1 be not an eigenvalue for an operator T E Cp and let 7>T be the regularized determinant (1-2). Then the following statements are equivalent: (i)
DTS^ C (/ - T^)9)
(ii)
sup
and
{I - T*)-^DT
G 62^;
{l-r)-^\\I-WT{r)W^{r)\\p<+oo;
0
(iii)
sup (1 - r)-P[l - Drir)] < +00; 0
(iv) if {zk} and i/js are the parameters of the factorization (1.10), then E f h l ' ^ ' I ' V < + ° ° '^'^d r
5 p _ i + , ( e - ^ ' ' r ) # e W = 0 ( ( 1 - rf)
(2.1)
as r -^ 1 — 0; (v) the following limits exist lim (1 - r)-n\I
- WT{r)W^{r)\\P,
r—>-l—0
^
hm ( 1 - r ) - ^ r r-^l-O
lim (1 - r)-^[l - D r ( r ) ] , 7—>-l—0
Sp.i^e{e-'^r)dM^)'
J_^
Moreover, if any of statements (i) - (v) is true, then 11(7 - TT'DTWZ
= 2 - ^ lim (1 - r)-^I
- T^T(r-)W?(r)||^
= p2-P lim (1 - r)-P[l - VT(r)]
= T.{]^)\W
- '^"2^ £ Vi+.(e-*^)#. W- (2.2)
2.1. The following lemmas are to be used for proving Theorem 2.1. L e m m a 2.1. //(ii) is true, then I \ogVT{r)\
< Mp\\I - WT{r)W^{r)\\l,
0 < ro < r < 1.
(2.3)
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Proof. From (1.1) it follows that oo
^
oo ^
I logI?T(r)| = E E 7(1 - ^'^W]' = E [ l - ^^'WF E 7[1 - ^A,(r)F-^ /e j = p "
J=p"^
where {Afe(r)} is the set of eigenvalues of the operator WT{r)Wj^{r). if r is sufficiently close to 1, then E [ l - Mr)r
Further,
= 11/ - Wr(r)Wf (r)ll^ < M ; ( 1 - r f < 5 < 1.
Thus 1 — Afc(r) < VS for each fe, and we get I logI>T(r)| < b ( l - ^ ) ] " ' E [ l - ^fc(^)]'' = ^pll-^ - ^T(r)W?(r-) \'pL e m m a 2.2. T/ie following assertions are equivalent: (a)
(b)
sup(l-r)-^[l-l^o(r)]<+oo, 0
^ ( i l l M y ^V|l-^A:r/
<+oc.
Proof. If (a) is true and r is sufficiently close to 1, then clearly 1
I
"^ 1
' k
j=p
|logX>o(r)|
n
Y^{l-r)-P[l-Bk{r)r
for any n > 1. As dr
Bk{r]
letting r ^ 1 — 0 we get
t{}0tr.. Zk
which implies (b) by arbitrariness of n > 1. Conversely, if (b) is true, then observe that l-B,{r)=^-^—p^^—p^ |1 - Zkr]''
and
\1 - z^rl >2-'\l
- Zk\
FUNCTIONS O F a-BouNDED T Y P E IN ...
173
for I^A;! < 1 and 0 < r < 1. Hence
(l-r)-.|l-B.(r)l=(l + r ) l ^ < 8 l ^ and by (b) J2[l~Bk{r)r
Consequently \\ogVo{r)\<M^Y}^-Bk{r)r, k
and finally we come to 1 - X>o(r) < | logX>o(r')| < (74Mp(l - r)^. Lemma 2.3. If the statement (i) of Theorem 2.1 is true, then s4im (/ - T ) ( / - rT)-^ = I
s4im (/ - T*)(/ - rT*)'^
and
1—)>1—0
= I,
r-^l—0
Proof. One can easily verify that the left relation holds if s4im (1 - r)(J - rT)-^ = 0.
(2.5)
1—)>1—0
To prove (2.5), note that for every h G 9) \\{I-rt)-^DTh-{I-T)-^DTh\\
< \\{1 - r)T{I - rT)-^{I <
Consequently, lim (l-r){I-rT)-^DTh
-
T)-^DTh\\
\\{I-T)-^DTh\\. = 0 and lim {l-r){I~rT)-'^f
1—>-l—0
=0
r—>-l—0
for any vector / of the form n
f=Y.
T'^Drhk,
n > 0, {kkjlr. C Sj.
k=—n
As 11(1 - r){I - rT)-iII < 1 (0 < r < 1), (2.5) holds if T is completely nonunitary [104, Ch. 1]. The relation (2.5) is true also in the case when T has a unitary component U. Indeed, 1 is not an eigenvalue for f7, as it is not an eigenvalue for T. Therefore, by the spectral decomposition of U s4im (1 - r)(I - rU)-^ = 0 . 2.2. P r o o f of T h e o r e m 2 . 1 . For proving (i) =^ (ii), we use the following well-known equalities for characteristic operator-functions [104, 12]: Wr(r/)W?(0 = / - ( ! - vODril - vT)-\l W^iOWriv) _ _ = / - {l-'n£,)W:^^{Q)DTT*{I - iT*)-^{I
-
1,T*)-^DT,
(2.6) - »7T)-iT£>TWf 1(0).
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CHAPTER 9
Hence / - WT{r)W^{r) and consequently
- (1 - r'^)DT{I ~ rT)-\l
(1 - r2)-i||7 - WT{r)W^{r)\\r, =
- rT'^y^Dr
- rT)-\l
\\DT{I
=
-
(0 < r < 1) rT^'DrW^
\\iI-rTn-'DT\\l.
On the other hand, II(/ - rT^r^Dr
- (J -
T^)-^DT\\^^
<(i-r)||(/-rr*)-i||.||(/-rT'i^T||2,<||(/-n-^i^T||2,. Thus, ||(/ - rT*)-iJ9T||2p < C < +00 (0 < r < 1), and (ii) holds. (ii) => (hi). By Lemma 2.1, for ro < r < 1 1 - -DTir) < I logX>T(r)| < M^\\I - WT(r)W?(r)||^ < M'^{1 -
rf.
(hi) ^ (ii). If {\k{T)} is the sequence of eigenvalues of the nonnegative operator WT{T)Wj^{r)^ then 1 ^
I k
\ k
"^ 1
- I log X>T(r) I
j=p'^
< M[l - Vrir)]
< M i ( l - rf,
and it suffices to see that ||/ - WT{r)W^{r)\\P = J ^ J l (ii) => (i). From (ii) it follows that for ro < r < 1 (1 - r)-^\\I - WT{r)W^{r)rp
(1 - r)-^I
0 < r < 1, M^W-
- W^{T)WT{r)\\l < K.
Therefore, using the formulas (2.6) we obtain ||(/ - vT^r^Drhp
< Ku
\\T{I - rT)-'DTW^\0)hp
< Ki
and moreover sup | | ( / - r r * ) - i D T / i | | 2 p < H - o o
and
0
sup \\{I - rTy'^Drh\\
<+oo
0
for any vector h E'DT- Since 1 is not an eigenvalue for T, we have DT9)C{I-T*)S)
and
DT^C{I-T)S)
(2.8)
by the well-known theorem of B.Sz.-Nagy and C.Foias [104, Ch. IV]. Besides, by the same theorem the following strong limits exist: s4im (/ - rT*)-^DT r—>1—0
= {I - ry^Dr,
s4im (I r—>1—0
VTY^DT
= {I -
T)-^DT,
FUNCTIONS OF a-BOUNDED TYPE IN ...
175
and using (2.7) we come to the inclusions [37, Ch. Ill] {I-T*)-^DTee2p
and
{I-T)-^DT
eG2p
(2.9)
which mean that (i) is true. For proving the equivalence (iii) <^ (iv), we denote the exponential factor in (1.10) as G{z) and observe that (iii) is equivalent to the following pair of conditions: sup (1 - r)-^[l - X>o(r)] < +oo,
sup (1 - r)-^[l - G(r)] < +oo. (2.10)
0
0
The last condition obviously is equivalent to the second condition of (2.1). Therefore, application of Lemma 2.2 proves the equivalence (iii) 4^ (iv). (i) ^ (v). By Lemma 2.3, s4im (/ - T*)(/ - rT'')-^! s4im (J - T){I - rT)-\l
- T*)-^DT
= {I -
T'^y^Dr,
- T)-^DT
= (I -
T)-^DT-
r—>1—0
Consequently [37, Ch. Ill] ^hmJ{I-rT)-^DT-{I-T)-'DT\\,^
=
^i4Y'J^I-rTT'DT-iI-T*r'Dr\\,^=^0,
and by the equalities (2.6) lim \\{I-rT*)-^DT\\l
= Hm (1 =
r^y^ll-WT(r)W}{r)\\
\\{I-T*)-'Dj
\2p'
Further, if {Xk} is the set of eigenvalues of the operator WT{t)Wj^{r), using the already proven equivalence (i) <^ (ii) we obtain
then
oo
lp+1 k
j=p-hl
•"
^
' ^
k
= C'\\I - WT{r)W}{r)\\
. \\I -
WT{r)W}{r)\\l.
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CHAPTER 9
Thus, by (ii)
^(l-r)-4logA,(r) + X:7[l-A.(r)FJ
{l-r)-P\logVT{r)\ oo
.
= E E 7(1 - '•)"''[i - ^-^ wi' = E ^(1 - ^)"'ti - ^*= w]' k j=p-'
k p
00
^
+ E E 7(l-0-"[l-A.(r)F = i ( l - r ) - P | | / - W r ( r ) W ^ ( r ) | | ^ + o(l)
as r - > 1 - 0.
Now (2.11) implies the existence of the hmits lini
^-^TJr)
r->l-0
{l-r)P
^
j . ^ r->l-0
llog'DTir)] ^ ^.^ 11^ - ^ T ( r ) W ^ ( r ) ||g (l-r)P
r ^ l - 0
(1 - r ) ^
Similar to the above argument (but Bk{r) taken instead of A^), we prove that p lim ( l - r ) - P [ l - X > o ( r - ) ] r—^-l—0
lim V ( l - r)-^[l - 5 , ( r ) f , r->l—0-^—' A;
and using the factorization (1.10) and Theorem 2.1 we obtain ,. 1-I?r(r) ,. I-Voir) p lim -^r— = p lim -7- r r-^1-0
(1—r)P
r-»l-0
[l—r)P
, ,. 1 - G{r) h p lim -7- r—. r-^l-0[l—r)P
As the equality (2.4) is true,
.]!?LoE(i-)-^[i - ^^Wf = 2^ E ( y i ^ ) ' Consequently
and the implication (i) =^ (v) is proved. The converse implication is an obvious consequence of previous results. Further, formula (2.2) follows from (2.11)(2.13), and the proof of Theorem 2.1 is complete.
FUNCTIONS O F ^ - B O U N D E D T Y P E IN ...
177
We close this section by two remarks related to Theorem 2.1. R e m a r k 2 . 1 . Theorem 2.1 remains valid if we add to (i) - (v), for instance, (vi)
sup(l - \z\)-^I zer
-
(vii)
sup(l - |2:|)~^|1 - VT\ < +00, zer
WT{Z)W}{Z)\\P
< +00,
where F is an angle of opening < TT in |z| < 1, symmetric with respect to the real axis and with vertex at z = 1. Moreover, all the limits in Theorem 2.1 exist as z —> 1 non-tangentially. The proof of this extension of Theorem 2.1 is simple and it needs no new idea. R e m a r k 2.2. Formula (2.2) will be discussed also in the next section. One has to note that it is not known weather (2.2) can be observed as a regularized trace formula for operators of Cp (p > 2). The previous considerations may be used to obtain essentially more general relations of (2.2)-type, which are called trace formulas in the case p = 1 [75]. The existence of e in (2.2) brings some dissatisfaction. The way in which e appears was explained in Sec. 1. One may get rid of it by letting e —> 0 but the properties of the limit function ipo (which may be even a distribution) would be hard to analyze. At last. Theorem 2.1 has been proved earlier for the case p = 1. 3. 6p-PERTURBATI0NS OF SELF-ADJOINT OPERATORS The meaning of the condition (i) of Theorem 2.1 becomes more clear for dissipative unbounded operators A whose Cayley transforms belong to Cp. Namely, it appears that (i) is equivalent to the representability of A as a sum of a selfadjoint operator and an operator from &p. 3.1. Denote by Qp the set of those operators A whose Cayley transforms T={AiI){A -\-iI)-^ belong to Cp. Then A e Qp ii and only if:
1)
±i^a{A),
2)
Im ( A / , / ) > 0 for a l l / G i ^ ( A ) ,
3)
The operator iR^i - iR*_. - 2R-iR'Li belongs to ©p.
(where Rx = {A-
A/)"^)
Besides, an operator T G Cp appears to be the Cayley transform of another operator from Qp if and only if 1 is not an eigenvalue for T. In addition, the following statement is true. P r o p o s i t i o n 3.1. Let A G Qp be an arbitrary operator. transform satisfies (i) of Theorem 2.1 if and only if A = AR+iAi
Then its Cayley
(3.1)
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CHAPTER 9
where AR = A'^ and Aj is nonnegative and belongs to ©p. Proof. Let T satisfies (i). Then by (2.8) (/-T*r)i3C(/-T*)^
and
{I-T^T)^
(l{I-T)^,
(3.2)
Since / - T'^T = 2{iR.i - iR:_. - 2i21^i?_i), D{A) = {I - T)S) and Z}(^*) (/ - T'')S^, the inclusions (3.2) mean that {iR-i - iR'Li - 2R1^R-i)^ C D{A) and {iR-i - iRL^ - 2R'^_-R-i)S) C D{A*). By the first of these inclusions, R'LiTh e D{A) for any h e S), and D{A') C D{A) since T is invertible. Similarly, the second relation gives D{A) C D{A*), and so D{A) = D{A*). Consequently, we can write i{A* - iI)R-i
- i l - 2R-i = 2-\A*
and if we take h = {A + il)f 2 - \ A * - iI)Dl{A
+ il)f
- il){l - T^'T) = 2-\A*
-
iI)D^,
(/ G D{A), then = i(A* - il)f - i{A -I- il)f -2f
= i{A* -
A)f.
Thus Aj = ^ ~ ^ * ={I-
T'^y^Dlil
- T)-\
(3.3)
By the left relation in (2.9), Aj G &p. In other words, the closure of Aj (which initially was defined on D{A)) is a nonnegative operator from (5p. On the other hand, AR = {A -{- A*)/2 is a self-adjoint operator. Indeed, since A is a closed symmetric operator, it suffices to show that its defect index is (0,0). If we contrarily suppose that this defect index is {n,m) {n -\- m > 0), then introducing the operator A - ^ © {-A)
=AR
+ ZAI,
AR = ARe
{-AR),
AI
=
AT
e (-^/),
acting in the space M. = S) ^ f) we conclude that the defect index of AR is {n + m,n-]- m). Thus, AR has an extension in H. Since A/ is bounded, also A has such an extension. The latter is impossible since A and therefore also A have ±z as regular points. Suppose now that A is represent able in the form (3.1). Then similarly - T)-^f for any vector / G D{A) we obtain that Ajf = {I - T*)-^D^I ( - D{A'')). Besides, for any / G D{A) \\DT{I
- T ) - V | r = ((/ - Tn-'DUl
- T)-'f, f) = {Ajf, f) < M l l / p
since Aj is bounded. Consequently, the operator S = DT{{I — T)~^ can be extended by taking its closure to a continuous operator in i^. As A / G &p and Ai = 5*5, one can easily deduce that DTS) C (/ - T * ) ^ and 5* = (/-r*)-iL>TG62p.
FUNCTIONS O F Q;-BOUNDED T Y P E IN ...
179
3.2. Now we shall convert the statement of Theorem 2.1 to an assertion for operators A £ Qp. It is convenient to connect such operators with the functions DA{W) = DT (l^^^ , \i — w J
T={A-
iI){A + iI)-\
lmw<0,
(3.4)
which are holomorphic in the lower half-plane G~ = {w : Im. w < 0} and for which the factorization (1.10) takes the form VA{w)=Vo{w)g{w),
(3.5)
where ^ is a non-vanishing function in G~, corresponding to the exponential factor in (1.10), and X>o is the product jj—i
V"
^f^i^
J
1-Xkwl
+ XkW
(Im \k > 0) constructed by the discrete spectrum {A^;} G G~ of A. Note that a formal representation of the function g can be obtained from the formulas g(w) = G f ^ i ^ ) ,
Giz) = exp { - ^
r
Vi+,(e-^''z)dV.w|
I —W
which are true for w E G~. Nevertheless, g{w) has another, more natural representation given in the below similarity of Theorem 1.2 for unbounded operators. T h e o r e m 3 . 1 . Let p > 2 be an integer, and let A G Qp be an arbitrary continuously invertible operator. Then for any e G (0,1/2) 2..M=I>oHexp{4/;j^^^^^^},
lm.<0,
(3.6)
where //^(t) is a real-valued function of locally bounded variation, satisfying
I
(l + |t|)2e
< +00.
3.3. For proving Theorem 3.1, we need the following purely technical lemma which we give without a proof. L e m m a 3.1. Let G{z) ^Q be a holomorphic function in \z\ < 1, and let G{z) be holomorphic also in a neighborhood of z = —1, where it may be written as oo
G(z) = l + ^ a f t ( 2 + l)'= k=p
{p>2,
|z + l | < r o < l ) .
(3.7)
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CHAPTER 9
If for some a e {p — l,p) I=ff
il-\z\^r-'\\og\Giz)\\daiz)<+oo J
(3.8)
J\z\
{where d(j{z) is the area element), then for each 7 G (a—_p-f-l,l) the function g{w) = G ( f z ^ l ; which is holomorphic in G~, satisfies
P r o o f of T h e o r e m 3.1. Evidently, - 1 0 a{T) (where T {A-iI){A-\-iI)-^) since A is continuously invertible. Using the similarity of Theorem 2.1 for the point z — — 1 (i.e. for the operator —T for which all remains true with the change z -^ —z), we obtain lim
(1 + x)~^[l - DT{X)]
=bp
lim
(1 + x)~P[l - G{x)] =ap
{0 < bp < +00),
and moreover (0 < a^ < +00),
where G is the exponential factor in (1.10). In our case G{z) is holomorphic in a neighborhood of 2: == —1. Hence, G{z) is representable in the form (3.7). Besides, for any a > p— 1 and any r G (0,1) 1
f\r
- i)"-i [ r
r(a) Jo
log+ \G{te'^)\d^] tdt
U~n < f
i D-" log+ |G(re^'')|di? <M < +00,
J —TV
where D~^ is the Riemann-Liouville integro-differentiation and M is a constant independent of r (the latter was seen in Proof of Theorem 1.1). Thus, sup 0
/
/
{r~t)'^-'^\og-^\G{te'^)\tdtd^<M.
J-TV
Further, the equilibrium relation between Nevanlinna characteristics of G{z) leads to (3.11) since G{z) 7^ 0 in \z\ < 1. Thus, G{z) satisfies the requirements of Lemma 3.1. From (3.7) it follows that there exists a neighborhood of it; = 00, where I log |^(if;)|| < C\w\~P. Assuming a=p—l-{-€{0<€< 1/2) and using Lemma 1.9 of Ch. 1, we conclude that for w G G" with enough great modulus \W-"\og\g{w)\\
(3.10)
FUNCTIONS O F a-BouNDED T Y P E IN ...
181
where W~^ is Liouville's integro-differentiation and Ci is a constant. Hence by (3.9) (where 7 = 2e) we find that for any v <0 +°°
du
L
-j
< sup—-
nv
n-\-oo
+°° I log \g{u +
{v-tr-'dt
it)\\-
du
+ KP^
J
r(a)
< +00.
On the other hand, by (3.10) Hm ^ /
\W-''log\g(Re'^)\\sinMi}
= 0.
R-^+00 i t Jo
But the function W~^ log \g{w)\ is harmonic in the lower half-plane (see Lemmas 1.5 and 1.8 in Ch. 1). Consequently, by the results of Ch. 7
„,-.,„,i,,„+,„,i=_i_£|;j_i^, „<„,
,3.„)
where ji^ is a measure with the required properties. Using these properties and the corresponding assertion in Ch. 1, one can verify that dP 1 V dii,{t) _1 7; /*+^ r W~^^~^^ dvP T{p) 4- 5) TT J_, J_^ {u - t)2 + i;2 = - R e i r , / ^ - ( ; )
•
(3.12)
On the other hand, the properties of G{z) provide that the holomorphic function Log g{w) is expandable in a neighborhood of ty = oo by a power series: L o g 5 H = ^/lfe(^u;)-^
(3.13)
k=p
As W-^P-''^^W-''{iw)-^
= {iw)-^
{k > p),
successively applying the operators W~^, d^/dv^ and W~^^~^^ to both sides of (3.13) (the termwise application of these operators does not change the uniform convergence of the sum) and then taking real parts, we get W-^f-''^—W-''log\giw)\
= log|5(w;)|,
Im w = z; < 0.
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CHAPTER 9
Thus, g{w)=exp\--
I
r . / ^ ' S z ^ + e +^^f^
lmit;<0,
where (7 is a real number. It remains to observe that C = 0, since g{w) as If; -> oo and ^™
/
= 0.
TTT
Hence, our assertion follows from (3.5), and the proof is complete. 3.4- The next theorem is a consequence of above results. Theorem 3.2. If p > 2 is an integer, then for any operator A G Qp the following statements are equivalent: (i) A is representable in the form A = AR + iAi, where AR = A}^ and Aj >0 (ii)
sup
belongs to
6p;
\v\~P[l - DA{iv)] < +oo;
-l
(iii) The limit
lim |t'|~^[l — DA{iv)]
exists.
V—>—0
Besides, if A is continuously invertible, then (i) - (iii) are equivalent to (iv)
V^(Im Xk)^ < -foo and the following limit exists:
d,s)
lim - ^ r v^-0 4P7r J-oo
\v\P{\v\-it)P+^'
where {Xk} cind /i^ are the parameters of the factorization (3.5). In addition, if A is invertible and at least one of the statements (i) - (iv) is true, then
= V ( I m A.)- + lim / -
r ^
, , , y ^ , .
(3.14)
Proof directly follows from Theorem 2.1, Proposition 3.1, and Theorem 3.1, if one uses the formulas (1 _ ^)-P[i _ Zk =
DT{X)] T——:,
Xk+i
= (1 - ^)^(2|7;|)-^[1 - DA{iv)l 71
To = I m Xk
\^-Zkr
^ < 0,
FUNCTIONS O F a-BouNDED T Y P E IN ...
and the equality (3.3) by which | | ^ j | | ^ = ||(/ -
183
T*)-^DT||2P-
Note that formula (3.14) is an improvement of the inequality
j2{irciXkr<\\Ai\\i, k
which is well known at least for bounded dissipative operators. Further, note that for p = 1 the quantity Sp {Aj) was calculated by the parameters of the factorization of bounded holomorphic functions in a half-plane in [42]. 4. COMPLETENESS OF OPERATORS IN Cp The aim of this section is to show how the properties of the regularized determinant X>T may be applied to the completeness problem of non-weak contractions. Recall that an operator is called complete if the closed linear hull of its root subspaces corresponding to the eigenvalues of that operator coincides with the whole space. 4.1. In the case of dissipative operators, the following theorem improves a well-known result due to M.V.Keldisch (see [74] or [37, Ch. V]). T h e o r e m 4 . 1 . Let p > 1 be an integer, and let T G Cp be any operator for which 1 is not an eigenvalue. Further, let the spectrum of T be a sequence {zk} having 1 as its unique limit point, and let Vrir) liminf l o g - ^ ^ = 0
(4.1)
for the functions J>T{r) and X>o(r) defined by (1.2) and (1.8). Then the operator T is complete. Before proving this theorem, we shall show that it really is a generalization of the mentioned result of M.V.Keldisch. Indeed, (4.1) is satisfied "with considerable reserve" if sup (1 - r)-P[l - G{r)] < +00,
where
G{z) =
VT{Z)/VQ{Z).
0
If we additionally demand supo<^
< +00,
0
and the requirements of Theorem 2.1 will be fulfilled. Thus, by Theorem 3.2, the corresponding operator A (which is the inverse Cayley transform of T) will be representable as A = AR + iAj, where AR = A^j^ and Aj > 0, A/ € &p. Further, A and AR will have the same continuous spectrum since a compact
184
CHAPTER 9
perturbation does not change that. By the conditions of Theorem 4.1, the continuous spectrum of AR will be located in oc, i.e. AR will have only discrete spectrum. In this particular case Theorem 4.1 states: a 6p-perturbation of a self-adjoint operator with discrete spectrum is a complete operator. Namely this statement was established by M.V.Keldisch [74]. If we put p = lin the formulas for VT and 2>o, then Tyri^) will become the usual determinant of WT{Z)W'}{'Z) and 2^o(^) will become a Blaschke product. Thus, for p = 1 the condition (4.1) is also necessary for the completeness of T if this operator is a weak contraction. The latter fact immediately follows from the well-known completeness criterion for weak contractions [37, Ch. V]. 4.2. Proof of Theorem 4.1. Suppose our statement is not true. Then it is clear that S)i = Clos span{/^ : \zk\ < 1} ^ S), and denoting S)2 = ^ Q ^1 we conclude that the following triangulation given by the invariant subspace S) 1 is true:
Here T2 is an operator whose spectrum consists of the single point z = 1 which is not an eigenvalue. Thus, the characteristic function WT{Z) admits the factorization [12] WT{Z)
= WI{Z)W2{Z)
(4.2)
the factors of which are defined by the formulas Wi{z)W^{0)
=1-
DTPI{I
-
ZTI)-^PIDT,
W2{z)W^{0) =1-
Wr\0)DTP2{I
W2{0)
W{-\0)WT{0),
=
-
ZT)-^P2DTW^~\0),
where Wi{0) is any invertible solution of the equation Wi{0)Wi{0) = I — DTPIDT [12]. In the above formulas Pi and P2 are the orthogonal projections onto S)i and 9)2 correspondingly. As the operator Ti is complete, its characteristic function Wi can be represented by the product Wi(z) = W,,{z)W,,(z).-.
W,„(^) • • • = n W , , ( z ) ,
where Wzj^{z) are operatorial Blaschke factors. This can be checked by the standard methods introduced into operator theory by M.S.Livsic [78]. Note that this infinite product is convergent in the sense that the operator-function -1
An{z)=(flW.,{z))
Wi{z)
FUNCTIONS O F Q-BOUNDED T Y P E IN ...
185
uniformly converges to / inside \z\ < 1 in ©p-norm (i.e. ||/ — An(2r)||p —> ) 0 as n -^ oo). Thus, from (4.2) it follows that as n -> oo
W^''\z)^(flW,,iz))
WT{Z) = Y[
W.AZ)W2{Z)-^W2{Z)
in 6p-norm, uniformly in \z\ < 1. Consequently, by the properties of regularized determinants [37, Ch. IV] detpW^"'\z)W^''^'"{z)-^detpW2{z)W;{z)
as
n - > oo.
(4.3)
On the other hand, < detpW2{x)W^{x),
VT{X)/VO{X)
- 1 < x < 1.
(4.4)
Indeed, by Lemma 1.2 and relations in the end of the proof of Theorem 1.2 VT{X)
= detpWT{x)W^{x)
=
detpW^,W^^\x)W^^^*{x)W*^{x)
= detpW^^{x)W,,{x)W^^\x)W^^^*{x)
<
^idetpW^^\x)W^^^''{x),
where $ i , as before, is that elementary factor of the product I^o? which corresponds to the eigenvalue Zi. Evidently, one can apply the same argument to detpWj. \x)Wj^ {x) and so on. Thus for any n > 1 VT{X)
< n$fe(x)detpV4"^(a;)W^"^*(x).
Letting here n —> oo and taking into account (4.3) we come to (4.4). Now denote V2{z) = detpW2{z)W^{z). If {Xk} are the eigenvalues of the operator W2{x)W2(x), oo
then one can see that
oo
|logD2(a;)|=EE7[l-^'^(^)l' k=lj=p^ OO
^
oo
^
= E7ll(^-^2(x)l^2W)ii>E7ll^-^2(a;)T^;(^)ir'j=p ^
3=P "^
Further, by (4.4) and the second condition of our theorem oo
^hmJlogl>2(r)H^hm^X^^||/-T^2(r)W2*Mir-^0, J=P
(4.5)
186
CHAPTER 9
and consequently lim 11/ - W2(r)W:{r)\\
= 0.
(4.6)
Let r be the angle with vertex at z = 1, defined by the inequality 11-^1 < a
{\z\ < 1,
1-N
a> 1).
li z eT, then by formulas (2.6) ((7 - W2{x)W;{x))h, where Q2 =
P2DTWI~^{0).
(7 - zT^)-^Q2h
h) = {l-
\z\^)\\{I-zTi)-'Q2h\f,
h G 2)T,
Besides, it is obvious that
- (7 - rT*)-^Q2h
= {z - r)T;{I - zT^)-\l
-
rT^)-^Q2h
- r\\\{I -
rT^)-'Q2h\\
for \z\ — r and \l — z\ \z — r\ ^ 1 — r <+ 1—r 1— r 1 -
+a = K
for 2: G r . Consequently 11(7 - rr^)-^Q2h\\
< 11(7 - vT;)-^Q2h\\
+ (l - rr^\z
< K{{I - W2{r)Wi{r))h,h)
\im\\l-W2iz)W^{z)\\=0,
for any h € S T ,
zer.
Thus, in any angle F of the mentioned type 00
\\{W2{z)WUz)r'\\
< Ylp-W2iz)WUz)\\''
< C„ < +00,
where Ca is a constant depending solely on a. Henceforth, we shall suppose that a is enough great to provide that the opening of F is greater than 7r(l — 2/p). Now we may state that for any /ii,/i2 ^ 2 ) T the function f{z) = {W^^{z)hi,h2) is holomorphic in the whole complex plane, except the point z = 1, and is bounded in F. Besides, the entire function F{w) = f{z),
z = {w-i)/{w
+ i)
is bounded in the upper half-plane, except two angles adjoining the real axis, with openings less than n/p (since W2{z) is a unitary operator when |z| = 1
FUNCTIONS O F Q;-BOUNDED T Y P E IN ...
187
[12]). Aimed to apply the Phragmen-Lindelof principle in that angles, we shall show that F is of order p. To this end, we first note that if \z\ < 1, then ii^2~H^)ii = ii^2*(^-')ii < ^ 1 + C 2 i i ( / - z - ^ T ^ r ' i i = Ci+C2\z\-\\{T2-zI)-% Now introduce the operator A2 G Qp which is connected with T2 by the equality T2 = {A2 — iI){A2 -{-il)~^. li z = {w — i)/{w + i), then a simple calculation gives {T2 - zl)-^
= {2i)-\w
+ i)I 4- {2i)-\w
+ i){i - w){A2 +
wI)-\
On the other hand, if B2 = A ^ \ then B2 = -i{{I - T2){I + T2)~'^ and B2 - B* e 6 p since B2 - B^ = -2i{I + r 2 ) - i ( / - T2T^){I + T^)-\ As the spectrum of the operator A2 is an empty set, B2 is quasinilpotent and hence it is a compact operator [37, Ch. I]. Besides B2 G &p according to a theorem due to V.I.Macaev and, as it is well known, the order of the resolvent {A2 — wl)~^ is equal to p [37, Ch. IV]. Thus, the function F is of order p. Therefore, applying the Phragmen-Lindelof principle we conclude that F is bounded in the whole upper half-plane. By the elementary properties of characteristic operator-functions [12], F is bounded also in the lower half-plane. Consequently, the operator-function W2 is constant in the unit disc, i.e. T2 is a unitary operator. As cr(T2) = 1, we come to the equality T2 = I which contradicts the conditions of our theorem. Thus, Theorem 4.1 is proved. 4.3. Examining the proof of Theorem 4.1, one can see that a more general result could be proved. Namely, instead of the condition (4.1) one could require the function | \og{7yT{x)/7>o{x))\ to be bounded by a constant (which can be calculated) depending solely on p. Of course, stronger criterions of completeness given in terms of the regularized determinant T>T are possible. These can be obtained using more delicate Phragmen-Lindelof type theorems. For instance, the following result is true for the case p = 2 being of special interest. T h e o r e m 4.2. Let T be a contraction from C2, for which 1 is not an eigenvalue. Further, let the spectrum of T be a sequence {zk} with the unique limit point at 1, and let liminf (1 - r) log § ^
= 0.
(4.7)
Then T is complete. Moreover, for any given a < 0 there exists a non-complete operator in C2, for which the lower limit (4.1) is equal to a. Proof. It follows from (4.4) that liminf (1 - r)T>2(r) = 0 , r—>1—0
188
CHAPTER 9
where 1^2 is the function considered in the proof of Theorem 4.1. Suppose T>2{r) = exp{—(f{r)} {(p{r) > 0). Then {l — r)(p{r) -^ 0 as r -> 1 —0, according to Theorem 1.2. Further, (4.5) can be written in the form
^{r)>-log{l-\\I-W2{r)W^{r)\\)-Y^-:\\I-W2{r)WUrW. Using this we get (l — ||/ — W2(r)W2*(^)ll))~'^ ^ Cexp{(f{r)}
which impHes
oo
\\{W2{r)Wi{r)-^
3=0
Turning as ear her to the entire function F , one can see that hm sup y-'^ log \F{iv)\ < 0
(4.8)
and F is of second order and minimal type [37, Ch. V]. Application of the Pragmen-Lindelof principle to the function Fi{w) = F{w)exp{iw} separately in both quadrants of the upper half-plane gives that F is of first order and, consequently, it belongs to M.Cartwright's class [37, Ch. V]. Thus, by (4.8) the indicator of F is non-positive and hence F = const. The end of the proof of completeness is quite the same as that of Theorem 4.1 and we omit it. Now introduce an obviously non-complete operator T € C2 as the orthogonal sum T = Ti 0T2, where Ti is a complete operator from C2, for which (4.7) is true, and T2 is a contraction with one-dimensional defect, the characteristic function of which is of the form
T^2W = exp{|i±i}, a<0. Then
T>T{Z)
=
'DTI{Z)W2{Z)
exp{l — W2{z)} and consequently
Vrir)
limsup ^ , . = lim (1 — r) logl^2(^) = a < 0. r-,1.0 Voir) r->i-o^ J s zy J
NOTES. As it became clear from the later work [5], the measure dlp^ appearing in the similar to (1.10) factorization of the regularized determinant dx(z), is absolutely continuous and belongs to definite O.V.Besov class.
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INDEX
a-bounded type, xv, 101 Qf-characteristics in \z\ < 1, xiii a-kernels in \z\ < 1, xiii (1 + Q:)-capacity, xiv a-mean type, 68 ^ a { G + } , 62 ^ g (in \z\ < 1), xii Cp, 159 dT, 159, 160, 165 X>T, 158, 165, 168 dQ, 164 X>o, 164, 165, 168 G+(i?), 45 r*[^,oo), 46 7-capacity, 101, 102 HP (in \z\ < 1), ix iJ^ (in Imz > 0), X, xvi, 133 HP{n{x)dx), 133 iC^((5o,-D)-class, 7 i^^((5o,^)-class, 13 La(—oo,d)-class, 1 £c.(p,F^\81 M{R), 132 M/^(L)), Mp, 7, 147 Mp{D), 13 Blaschke-Djrbashian products, xi, xiii, XV, 103, 112 Blaschke type products, xiv, 21, 25, 33, 38, 84, 120, 121 M.M.Djrbashian factorization, xi, xiii
01, 9^"", X, 76, 131 AT, Nevanlinna class, 101, 120, 131 iVc-class, xiii, 113, 120, 160 91«, 0 1 ^ , 70, 71, 72, 75 Na{G-}, 101, 121 ?n^^, 82, 85, 86, 97 DA\W),
179
Qp, 111, 182 0t(p,F±),80 -R, Herglotz-Riesz class, 14 i?°, Nevanlinna class, 14, 23 i?T, I.S.Kats class, 14 Ra, 15 i?°, 15 Sa, 152, 154 Soc, 154, 155 6 p , Neuman-Scetten ideal, 159 it, il+, 69 il+,69 oo-starshaped domain, 6 oo-starshaped continuation, 45 W - " , 1, 6, 7, 20 W-^, 12 WT{Z), characteristic function, 159 na(^,C), 21 Ahlfors-Heins theorem, 132 Nevanhnna factorization in Im z > 0, ix, XV, 65, 129 Nevanhnna type characteristic, 61,82 Nevanhnna uniqueness theorem, 132
196
Eequilibrium relation, 61, 81, 82 Carleman type formula, 54,58 Cay ley transform, 177 Green type potential, 147, 149 Levin formula, 80 Levin type formula, 55, 59, 61, 62, 82
INDEX
Nevanlinna weighted class, xi, 82 R. and F.Nevanlinna type formula, 49,51 Phragmen-Lindelof Principle, x Riesz type representation, xvi Tsuji characteristic, xiv, 80, 100 Tsuji type estimate, 90 Tsuji type characteristic, 61, 62, 82
