Functions of a-Bounded Type in the Half-Plane (Advances in Complex Analysis and Its Applications)

Functions of a-6ounded Type in the Half-Plane

Advances in Complex Analysis and Its Applications Volume 4

Series Editor: C.C. Yang The Hong Kong University ofScience& Technology, Hong Kong

Advisory Board: Walter Bergweiler Kiel University, Germany George Csordas University of Hawaii, U.S.A. Paul Gauthier University of Montreal, Canada Phillip Griffiths Princeton, U.S.A. Irwin Kra State University of New York, U.S.A. Armen G. Sergeev Steklov Institute of Mathematics, Russia Wolfgang Tutschke University ofGraz, Austria

Functions of a-Bounded Type in the Half-Plane

by ARMEN M. JERBASfflAN Institute of Mathematics National Academy of Sciences, Armenia

Springer

Library of Congress Cataloging-in-Publication Data A CLP. Catalogue record for this book is available from the Library of Congress. ISBN 0-387-23625-2

e-ISBN 0-387-23626-0

Printed on acid-free paper.

© 2005 Springer Science+Business Media, Inc.

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1 springeronline. com

SPIN 11161851

CONTENTS

PREFACE

vii

INTRODUCTION

ix

CHAPTER 1 THE LIOUVILLE OPERATOR AND HERGLOTZ-RIESZ TYPE THEOREMS 1. The Liouville integro-differentiation 2. Herglotz - Riesz type theorems 3. Proofs of Theorems 2.3 and 2.4

1 14 17

CHAPTER 2 BLASCHKE TYPE PRODUCTS 1. 2. 3. 4.

Elementary properties of factors Integro-differential properties of factors Blaschke type products A representation of the product

21 25 33 39

CHAPTER 3 EQUILIBRIUM RELATIONS AND FACTORIZATIONS 1. F. and R. Nevanlinna, Carleman and Levin type formulas 2. Some other formulas of Carleman and Levin type 3. Factorizations

45 55 65

CHAPTER 4 MEROMORPHIC FUNCTIONS WITH SUMMABLE TSUJI CHARACTERISTICS 1. Levin type formula and Tsuji type characteristics 2. Canonical factorizations 3. Descriptive representations

77 82 90

CHAPTER 5 BOUNDARY VALUES 1. Main results 2. Proofs of Theorems 1.1 and 1.2 3. Proof of Theorem 1.3

101 103 112

CHAPTER 6 UNIFORM APPROXIMATIONS 1. Main results 2. Proofs of Theorems 1.1, 1.2, 1.3

121 123

CHAPTER 7 SUBHARMONIC FUNCTIONS WITH NONNEGATIVE HARMONIC MAJORANTS 1. Main results 2. Representation in the semi-disc 3. Proofs of Theorems 1.1 - 1.4

129 134 141

CHAPTER 8 WEIGHTED CLASSES OF SUBHARMONIC FUNCTIONS 1. Green type potentials 2. Subharmonic functions with nonpositive Liouville primitives 3. The growth of functions from Sa

147 152 155

CHAPTER 9 FUNCTIONS OF o^-BOUNDED TYPE IN SPECTRAL THEORY OF NON-WEAK CONTRACTIONS 1. 2. 3. 4.

Factorization of regularized determinants Boundary behavior of regularized determinants 6p-perturbations of self-adjoint operators Completeness of operators in Cp

159 171 177 183

REFERENCES

189

INDEX

195

VI

PREFACE

This book is related to the theory of functions of a-bounded type in the halfplane of the complex plane. I constructed this theory by application of the Liouville integro-differentiation. To some extent, it is similar to M.M.Djrbashian's factorization theory of the classes Na of functions of a-bounded type in the disc, as much as the well known results on different classes and spaces of regular functions in the half-plane are similar to those in the disc. Besides, the book contains improvements of several results such as the Phragmen-Lindelof Principle and Nevanlinna Factorization in the Half-Plane and offers a new, equivalent definition of the classical Hardy spaces in the half-plane. The last chapter of the book presents author's united work with G.M. Gubreev (Odessa). It gives an application of both a-theories in the disc and in the half-plane in the spectral theory of linear operators. This is a solution of a problem repeatedly stated by M.G.Krein and being of special interest for a long time. The book is proposed for a wide range of readers. Some of its parts are comprehensible for graduate students, while the book in the whole is intended for young researchers and qualified specialists in the field. I am grateful to my father Prof. Mkhitar M. Djrbashian who in 1980's led me into the field of representations of regular functions. I am very glad to express my gratitude to Prof. Cabiria Andreian Cazacu and Prof. Chung-Chun Yang for encouraging me to write this book. Also, I am thankful to my wife Gohar Jerbashian for her valuable assistance in refining the text of the book.

ARMEN M. JERBASHIAN

INTRODUCTION

1. The fundamental role of integral representations and factorizations in complex analysis is well known. In this field, the basic results of Poisson, Cauchy, Weierstrass, and Hadamard where continued by outstanding achievements of Hardy, Blaschke, Fatou, Herglotz, F. and M. Riesz, Sego, R.Nevanlinna, V.I. Privalov and other classics of the contemporary complex analysis. These achievements partially were summarized in the widely known monographs of R. Nevanlinna [84], I.I.Privalov [86, 87], L.De'Branges [14]. A series of further results is due to Paley-Wiener [85], Hille-Tamarkin [47], V.I.Krilov [76], B.Ja. Levin [77], M.Tsuji [106] and others. The investigated non-weighted classes of functions mainly were connected with the following three classical problems, which depend on the specific geometry and technical apparatus of the disc and half-plane of the complex plane. Problem I. Describe some classes of regular functions, the growth of which is restricted by a rotation-invariant condition that does J^Q^ distinguish the boundary points of the unit disc, The unit disc on the Riemann sphere ^^ considers the whole boundary as a single point. The growth condition P2TT

sup / 0 0: n-\-oo -I-00

/

u'^[x -h iy)dx < -hoc, -00

(4)

-foo

at"

sup ?/>0

/

\u{x -f iy)\ dx < +CXD. -00

One can note the differences between the analytic apparatus peculiar to the disc and to the half-plane. Particularly, representation as a Laplace transform in the half-plane corresponds to the Taylor expansion in the disc. 2. Further development of representation apparatus was necessary for applications in different fields of science. Particularly, the apparatus was extended to different weighted classes of functions regular in different senses. The canonical representations of weighted classes of holomorphic and meromorphic functions, established by M.M.Djrbashian [16] in 1945 were among first results in the field. His results actually revealed a new way of application of fractional integ-

INTRODUCTION

xi

ro-differentiation - in the representation problem of weighted classes of regular functions. One can find a detailed survey of these and other methods of fractional integro-differentiation in the monograph of S.G.Samko, A.A.Kilbas and O.I.Marichev [92]. The initial work [16] (see also [17] for detailed proofs and additional results) mainly aimed at improving Nevanlinna's result [84, Sec. 216] related to description of the density of zeros and poles of the weighted class of functions meromorphic in the disc \z\ < 1, the Nevanlinna characteristics of which satisfy the rotation-invariant condition

I

1

(l-r)^T(r,/)(ir )\YM,

T^ir,f) =

mc,ir,f)+N„{r,f),

where ka = a X ] ^ i [ ^ ( ^ + ^)]~^' The class N^ (—1 < a < H-CXD) of functions of a-bounded type in |z| < 1 is defined by the condition sup Ta{rJ)

< 4-00,

0 0 (1 + |t|)"-^t/(t) e L i ( - o o , d - e).

(1.2)

Note that for any ai and a2 (0 < a i < a2 < +00) La2{~oo,d) C Lai(—oo,d).

(1.3)

Lemma 1.1. 1°. IfU{v) E La(—00, d) for some a G (0, +00), then the integral W~^U{v) is absolutely convergent for almost allv G (—oo,(i) and it represents a function summable on any finite interval (a^b) C (—oc,(i). 2°. If ai, a2 e (0,+oo) and U{v) e La,^a2{-oo,d), La2{-oo,d).

then

W-'^'lUiv)] G

Proof. 1°. Let - 0 0 < a < b < d. Then nb

I

oh

\W-'^U{v)\dv

Ja

< I

W-°'\U{v)\dv

Ja

^ ^ ^ I j d v j

{v-t)"-^\U{v)\dt+

f dv f (v -

t)''-^\Uiv)\dt

= p(YT^ (^fjb - tr - (a - tr] mv)\dt+jjb - trmv)\dt

2

CHAPTERi

On the other hand, for enough great C = C{a^ 6, a) > 0 0 < (6 - t ) " - (a - t)^ < C(l + |t|)"-^ and

( 6 - t ) " < C ( l + |t|)^-i

if if

- oo < t < a, a+0.

Now introduce the function pa

F„(cr) =

pv

U{v- a)da = / Jo

U{T)d7

Jv—cr

which is continuous for a > 0. Since !» G (—oo, d) is a Lebesgue point of 17, \Fv{cj)

U{v) 0) to a = 0 is given by the identity W^U{v) = U{v).

(1.10)

6

CHAPTERi

For defining fractional derivatives, we assume that a > 0 and p > 1 is the natural number deduced from p — 1 < a < p. Then we set

W"U{v) = W-^P-''^i^£^U{v)Y

(1.11)

preassuming the existence of the latter function for almost all v G {—oo,d). This we call the a-th derivative of U{v). Observe that by (1.10) (IP

W'^Uiv) = —U{v), if p is a natural number, i.e. WPU{V) oiU{v).

(1.11')

coincides with the ordinary p-th derivative

1.3. Henceforth we apply Liouville's operator by the variable v to some functions U{w) = U{u-^iv) given in several domains which along with inner points w — u + iv contain the rays r(oo,w) = {s = w-i(j

: 0 < cr < +00}

(1.12)

directed from oo to w. We call such domains starshaped in respect to oc, or briefly oo-starshaped. Unless otherwise stipulated, we shall assume the function U{w) harmonic in a 00-starshaped domain D.^ with a possible exception of a countable set of points {wm}' Besides, we shall assume that any r(oo,if) C D contains not more than a finite set of points from {ifm}, and also that on T[oo^w) the function U{w){w — Wm) satisfies Holder's condition in neighborhoods of the points {wm} C r{oo,w). Preassuming absolute convergence in any interval disjoint from the points {wm} C T{oo,w) and existence of Cauchy's principal value in the rest of the integration contour, we call W'Uiw)

'miy

=:p^-3 /

{v- t)°'-'^U{u + it)dt

J —00

1 f^^ = —— / cr""^C/(^ - ia)da, 1 W Jo

w = u-{-iv eD,

(1.13)

the a-th (0 < a < +00) primitive of U{w). For o^ = 0 we set W^U{w) = U{w),

weD.

(1.14)

For definition of a-th {a > 0) derivative of U{w)j we assume that the derivative gp ——U{w),

w = u-\-iveD

{p — I < a < p)

LiouviLLE O P E R A T O R AND H E R G L O T Z - R I E S Z T H E O R E M S

7

satisfies the above requirements. Then we set WU{w)

= T^-(P-^) I ^ U { w ) I ,

w^^u + iveD.

(1.15)

Definition 1.1. A function U{w) given in a oo-starshaped domain D is said to be of the class Mp{D) (0 < /? < +00)^ if there exists an angular domain A((5o, Ro) = {w : |7r/2 + 8iVgw\ < Jo, \w\ > RQ} (0 < JQ < 7r/2; 0 < RQ < +00) such that -I-00

a^-^\U{w - ia)\da < +00 (1.16) sup / weK for any compact K C D D A((5o,i?o)- The function U{w) is said to be of the class K^{So,D) (0 < 7 < +00), if \U{w)\ < C\w\-^,

wGDn

A(Jo, i?o),

(1.17)

for some 60 {0 < 60 < 7r/2) and RQ {0 < RQ < +00). It is evident that for any /?i and P2 (0 < /3i < /32 < +00) M0,{D) C Mp^iD). Besides, it is obvious that for any 71, 72 (0 < 71 < 72 < +00) and 60 {0 < SQ < 7r/2) Thus, the classes Mp and K^ narrow down as P and 7 increase. At the same time, the below lemma is true. L e m m a 1.5. Let /? G [0,+CXD) and 7 G (/?,+00) be any numbers. any 60 {0 < 60 < 7r/2) K^{6o,D)cMp{D).

Then for (1.18)

Proof. One can verify that (a + b)~^ < max{l, 2^~'^}(a'^ + 6'*)"^ for any positive numbers a, b and K. Consequently, \w-ia\-^

< {\w-ia\'^)-i

= [\w\'^ + a'^{l-^2a-'^\lm

< {\w\'^ + a^)-i

< max{l, 2^-^/^}{\w\^

w\)]~^ < + a^)-^

if Im w 0. Thus, for any w (Im it^ < 0) and cr, 7 > 0 '-ia\~^

It; •

0) and K is any compact in i^ fi A((5o, Ro). Then by (1.17) and (1.19) Ji

Ji


oo

N^-oo

L e m m a 1.8. Let U{w) be harmonic in a oo-starshaped domain D, and let U{w) G Mp{D) for some /? > 0. Then also W~^U{w) is harmonic in D for any a G [0,/3]. Proof. Our assertion is trivial for a = 0 since Ty^ is identical. Consider the case ^ > 0, a G (0,/3]. By Lemma 1.7, W~^U{w) is finite-valued and measurable in D. Therefore (see, for instance, [11], Addendum, §1, Subsection. 1), W~^U{w) necessarily will be harmonic in D, if we prove the equality W-^'Uiwo) = ^ f 27r Jo

W-''U{wo+re'^)d^

for any wo ^ D and r > 0 such that Or{wo) = {C '• |C ~ '^o| ^ ^o} C D. It follows from Lemma 1.7 and (1.20) that W~^\U{w)\ is finite-valued and measurable in D. Also, it follows that 27r

X

W-"" \U{wo +re'^)\d^

< +00.

Therefore /.27r

^- /

27r ^"TT Jo

W-'^U{wo-i~re''^)d^

1

P+OO

= : F ^ / r(Q^) Jo

cr'^-^Uiwo - i(j)d(T - W-"/7(^o),

10

CHAPTER i

where all integrals are absolutely convergent. Thus, the proof is complete. Remark 1.3. The requirements of Lemma 1.8 can be weakened. Namely, its assertions remain valid if instead the uniform boundedness of (1.16) in any compact K lying in Dr]A{5o,Ro) one require only its uniform boundedness on a family of continuous curves {Ik} C D such that for any K, C D there exists /u > 1 for which Re K C Re Z^;- We omit the proof of this statement. 1.5. The following lemmas also will be useful. Lemma 1.9. Let U{w) be continuous in an angular domain A{6o^Ro) (0 < So < 7r/2; 0 < Ro < +oo); and let \U{w)\ < Ci\w\-^''-^^\

w e A(^o,i?o),

for some positive a and K. Then for a constant Ci > 0 W-'^\U{w)\

< Ci\w\-^,

w G A((5o,i?o).

Proof. If ^ G A{So,Ro), then by (1.19) + 00

0. One can verify that if U{w) and u{z) are connected by (1.26), then the inclusions U{w) G Mp{D) and U{w) G K^{5Q^D) are equivalent to u{z) G M/3(D) and u{z) G K^{5Q^Y>) ( D = D~^) respectively, whatever be /3 G [0,+oo), 7 G (0,+cx)) and Jo G (0,7r/2], i.e. M^(D) = {u{z) = U{z-')

: U{w) G Mp{B-')}

K^{6o, D) = {u{z) = U{z-^) : U{w) G K^{6Q,

,

D"^)}

.

(1.28)

Therefore, all lemmas proved for Mp{D) and K^{So,D) can be equivalently stated for Mp(D) and i^^(Jo5D) (E) = Z)~-^). Omitting these statements, we only note that by (1.27) and Lemma 1.5 Mp{-D) C K^{5o,T>) for any P G [0, +oo), 7 G {p, +00) and Jo G (0,7r/2].

(1.29)

14

CHAPTER i

2. HERGLOTZ-RIESZ TYPE THEOREMS Below we give some well-known results on descriptive representations of several subclasses of the Herglotz-Riesz class by Cauchy integral. After that, we state the main results of this section. 2.1. Everywhere below R means the Herglotz-Riesz class in the lower halfplane G~ = {w : luiw < 0}, i.e. the set of functions holomorphic in G~, having there nonnegative real parts. Definition 2.1. R° is the set of functions holomorphic in G~, representable in the form n ^ ^ ^ l T M l „,G-, m J_^ w - t where fi{t) (—oo < t < +CXD) is a nondecreasing, hounded function. 7 < 2) is the set of those F{w) G R for which

I

(2.1) R7 (0
o 3)F{w)eIl, sup t Re F {-it) 0

lim F{-it)

= 0.

t-^+oo

The classes R'^ (0 < 7 < 2) were considered by LS.Kats [72] (see also [4], Addendum I). The next theorem follows from his results. Theorem 2.2. R'^ (0 < 7 < 1) coincides with the set of functions representable in the form F{w) - iC + - T " ^ ^ , weG-, 7nJ_^ w-t where C is a real number and fi{t) is a nondecreasing function ^+^ d/2{t) < +00. l + lth

I

(2.2) satisfying (2.3)

J —(

R e m a r k 2 . 1 . It is well known (see [4], Addendum I), that R° C R'^ C R for 0 < 7 < 2. One can see that by Theorem 2.1 formula (2.1), where /^{t) is a nondecreasing, bounded function, is a descriptive representation for the class of functions defined by any of statements 2), 3) of the same theorem.

LiouviLLE O P E R A T O R AND H E R G L O T Z - R I E S Z T H E O R E M S

15

2.2. For stating our theorems, we need the following definitions. Definition 2.2. R^ (—1 < OL < +CXD) is the set of those holomorphic functions in G~, which are representable in the form

F(nj) = £ ( i ± ^ e - ^ f (^+-) r ^ r ^ % - ,

w G G-,

(2.4)

where fi{t) (—oo 0 is the integer from p — 1 < a 0 , T^"^

(c)

/

weG-; df

W^-^Re f{-it)

— < +00,

if

a=p.

(2.5)

2°. / / - l < a < 0 , then (a) Im F'{w) e M i ; (b) T^-^Re F ( ^ ) > 0,

WGQ-.

Theorem 2.3. Let F{w) he holomorphic in G~. 1°. / / 0 < a < +00 and p > 0 is the integer from p — 1 < a < p, then the following statements are equivalent: (i)

F{w)GR^;

(ii) Re F{w) G Mp, T^-"Re F{w) > 0 for w e G', supt^^^\F{-it)\ t>o

< +00,

/ Ji

W - " R e F{-it)—

< +oc t

{if a y^ p, then the convergence of the last integral is not required); (iii) Re F{w) € Mp, W'^'Re F{w) >0 s u p t W ^ R e F{-it)

< +cx),

t>0

forweG', lim F{-it)

= 0.

t-).+oo

2°. If — 1 < a < 0; then the following statements are equivalent: (i) F{w) e R^; (ii) Im F'{w) G Ml,

W-'^Re F{w) > 0 for w e G',

supt^+^|F(-i^)| < +00,

supt^+^llm F'{-it)\

t>0

t>0

< +oo;

16

CHAPTER i

(iii) Im F\w)

e Ml, W - « R e F{w) > 0 for w e

sup^l^-^Re F{-it)

< +00,

t>0

G',

lim F{-it)

= 0.

t->+oo

T h e o r e m 2.4. H^ {—l0

^„.,C +

from p — 1 < a 0. Besides, gp _ V F - b - « ) ( ^ o ( ^ ^ ) = (p^{w),

W-'^^aiw)

= ^o(^^),

2°. If ae (-1,0) and ^i-oo +/

-oc

\du{t)\ 1

+ \t\1+a

< +00,

w=u+

iveG-.

LiouviLLE O P E R A T O R AND H E R G L O T Z - R I E S Z T H E O R E M S

19

then '^'a.{^) ^ -^1- Besides,

P r o o f is obvious. 3.2. P r o o f of T h e o r e m 2.4. 1°. Let F{w) G R^ for some a > 0. Then the function W~^Re F{w) is harmonic in G~ by Lemma 1.10(a) with D — G~. Besides, this function is nonnegative, and by Lemma 3.2 +°° /

rif W-^'Re F(-it)-rr

< +00.

Thus, the function ^{w) = W~^Re F{w) + iV{w) (where V{w) is the conjugate harmonic of ^ " ' ^ R e F{w)) belongs to R^+*^~^. By Theorem 2.2, ^{w) is representable in the form (2.2)-(2.3) for 7 = 1 -}- a — p. Therefore, the representation (2.6)-(2.7) follows by the assertions 1° of Lemmas 3.1 and 3.3. Conversely, let F{w) admit the representation (2.6)-(2.7) for some a > 0. Then, by Lemma 3.3(1°) Re F{w) G Mp. By the same lemma, also (2.10) is valid, and hence W-'^Fiw) > 0 for w e G'. On the other hand, by (2.10) the function ^{w) is representable in the form (2.2)-(2.3), where 7 == 1 + a — p. Hence (2.5) holds for integers a = p > 0 by Theorem 2.2. 2°. For — 1 < Q; < 0 our assertion and formulas (2.10) and (2.11) are proved similarly, using Theorem 2.2 and Lemmas 1.10 and 3.1, 3.2, 3.3. P r o o f of T h e o r e m 2.3. In both cases 1° (0 < a < +00) and 2° {-1< a < 0) we prove the implications (i) => (ii) ^ (iii) => (i). 1°. Let a > 0 and (i) is true, i.e. F{w) admits (2.4). Then, by Lemma 3.3 Re F{w) G Mp and (2.10) is valid. Hence W^-"Re F{w) > 0 for ^ G G". Besides, (2.5) follows from (2.12) and Theorem 2.4. Now observe that by (2.4)

t^+^\F{-it)\ < £ i l ± ^ / " dfi{t) < +00, ^

t> 0.

J-00

Hence (i) ^ (ii) is true. Let (ii) be valid. Then \F{-it)\ < Cot"^"" (0 < t < +CXD) for some Co > 0. Hence F{—it) = o(l) as t -^ +CXD and for any t > 0 tW-'^Re

F{-it)

^

< - ^

/

^ - r(a) Jo {1+x)

l+a

< +00.

This proves (iii). Now let (iii) be true. Then, by Lemma 1.10(a) the function Ty-^Re F{w) > 0 is harmonic in G' and W ^ R e F{-it) < Cit''^ {t > 0) for some Ci > 0. Consequently, for any 7 > 0 the above function ^{w) is of R^. Using Lemmas 3.1 and 3.3 we conclude that F{w) is representable in the form (2.6). As it follows from (2.10), +00

IL

dfi{a) = sup tW'^Re

F{-it)

< +00.

20

CHAPTER i

Thus, the function fi{t) is bounded. On the other hand, it is obvious that \C\ < \Fi-zt)\

+ ^

^

J_^

^^ _ 2 | / + . = 0(1)

as

t->+oo.

Hence C = 0 and the representation (2.4) is true, i.e. (i) is valid. 2°. The proofs of the case — 1 < a < 0 (including formulas (2.10) and (2.11)) are quite similar. NOTES. Some of the auxiliary statements on Liouville's fractional integro-differentiation, which are given in Section 1, are well known. One can find them, for instance, in [92].

CHAPTER 2 BLASCHKE TYPE PRODUCTS

We shall consider a family of Blaschke type products in the lower half-plane G~ = {w \ Iva w < {)} and investigate some of the properties of these products. The factors of our products depend on a parameter —\ 0}, which for a == 0 coincide with the widely known Blaschke product

^(^'^^^»=nH7l' ^^'^>^^'' ^

< +00.

(2)

Zk

1. ELEMENTARY PROPERTIES OF FACTORS 1.1. We start by another representation of hoc{w, (). Assuming C, — (^-[-ir] ^ G~ a fixed point, for any a G (—1,-foo) we introduce the function ^.KC)=/ Jo

—— -ZZ7TT^+ [r + i{w - C)j Jo

7=^ TiT^[i{w - C) - ^]

(1-1)

The changes of variables t = —{r + rj) and t = r-i-rj in the above integrals give »aKC)=r'

J-\n\

../'"^'"'"^"l..

[t{w

-l i\w - CI - IvW'^" for any t € [-|7?|, |??|]. Hence by (1.1') we come to (1.6). For proving (1.6'), one have to repeat the same argument after a differentiation of (1.1') by f = Im w.

(1.4)

24

CHAPTER 2

Lemma 1.2. If a e ( - l , + o o ) , then for any compact K C G~ 2 g - i f (l+a) |^|l+a

Oc('a;,C + i7/)

(it;-0^+'' 1 + a

+ 0(|77|2+«)

as

rj^-0

(1.7)

uniformly in respect to it; G K and Ro > 4|C|, then \w - ^\ - \ri\ > \w\ - 2|C| > \w\/2 since 2|CI > 1^1 + 1^1- Hence by (1.2), (1.6) and (1.6') 22+a

|log|6„(«;,C)|| < |i)a(ii^,C)l < f - — | r ? | ' + " k r ( ' + ° ^ l + a d_ < — log|6a(^,C)| dv Qa{w,C)

(1.8)

for any w G C\{(-\-ih : 0 < h < +00} {\w\ > RQ). More simple representations than (1.5) are true for ba{w,() for a = p, where p is an even or odd number (see [54], formulas (1.6)-(1.80). 1.4- Assuming that 5 = 5 + iA is an arbitrary fixed point from G^ and ( = ^ -\-ir] e G~ — s~^ (G G~), consider the ray r*[^, 00] = {w = ^ + ih: 0 0 and a \r]\. The proof of this estimate is similar to that of (1.6). Lemma 2.2. 1°. For any a € (—l,+oo) and any w € C\[C,C] M^-"log|6„(w,C)| = - R e

W-"naiw,(:)

_ 2Imw /"I"! \w-^\'^-t^ ~T{l + a)Jo |(w_^)2+i2|2
—1 these limits are continuous functions of Lip A on (—1^|,|^|) with some A G (0,1). 4°. If a > 0, then ^a{z) is continuous at the points —1?7| and \r]\, as a function of complex variable. 5°. If — 1 < Q; < 0, then the following representations are true in enough small neighborhoods of the points —\rj\ and \rj\:

where '^^(2:) and il^l,{z) are holomorphic in the same neighborhoods. In view of (2.9) and (2.9') we come to the following L e m m a 2.3. The function W~^ log \ba{wX)\ (—1 < a < +00), which is harmonic in C\[C,C]; ^^ continuous on [C,C] /^^ a > 0, and is continuous on iCX) for -1 < a 0

1- j ^ W-^\ogK{s-\-pe'\0\dd

Hence our assertion will follow by the maximum and the minimum principles of subharmonic and superharmonic functions. Before proving (2.15), observe that by (2.10) and (2.14) 1 C"^^ W-'^\og\b^{^ + pe'\C)\d^ ~ j

= W-'^\ogK{U)\=Q

(2.16)

for any a e (—1, +oo) and p > 0. Now assume that ^ < h < \r]\ and 0 < p < max{/i, |ry| - h}. Then by (2.2) 1

/"^^

= WT^^—T /'' (l^|-| p. Hence

^

rW-^\ogK{i-ih^pe'\0\dd 1 (['-' • /'^'^ r(i + a) l^y_|,, 7;,+,;

{\v\-\t\r (i^. t-/i

Consequently, by (2.2) we obtain that 1

/>27r /"^^

- y

27r

W^-" log |6„(^ - ih+pe''', 0\d^ - W-" log |64^ - i/i, C)| ir^M^^ r(i + a)Jh-p t-h

(2.17) ^

^

for a G (—l,+oo), 0 < /i < |7y| and 0 < p < max{/i, |r/| — /i}. One can be convinced that also 27r

T ^ - log |6,(C + pe^^ C)M^ - W

27ryo

log |6a(C, 01 1

p^

r(l+a) a

> 0.

(2.170

for any a > 0 and p G (0, Ir/I). Now observe that

''-"'{\v\-\t\r.._ Jh-p h-p

t-h i — II

r Jo Jo

i\v\-x-hr-{\v\+x-hr dx. X

Consequently, ^27r

^ y ' " w - ^ log |6,(^ -ih + pe'\ C)\d^ - W - ^ log |6«(^ - 2/1,01 27r./o M « - (\n\ - ^ _ M « 11 ZPP *^ n-nl 4- ^ - {\rj\-hx-h)^-{\rj\-x-h)'' -da: r ( l + a) Jo X > 0, n. 0n 1^1,

(signi;)|,^^^[|r7r-^--(M-|H)'-^1

if

(2.20) H < M-

As the function W'"^ log |6a('?i;, C)l ( 0) keeps its sign in G' and G+, (2.20) implies (2.19). For proving (2.19) with a G (—1,0), note that in view of (2.10) it suffices to prove (2.19) as t* —> —0. To this end, suppose v < 0 and

32

CHAPTER 2

denote by S^ the part of the Hne {w = u -}- iv : —oo Np -\-l. Besides, Wk G G^, then \w — Re Wk\ > |Im w\ > \p\ > |Im Wk\ for k > Np-\-l. Therefore, by (1.6)

for any Wk G G^ and k > Np-{-l. Together with (3.3), this implies the absolute and uniform convergence of the series oo

y^ logba{w,Wk) k=Np+l

oo

= -

^ fla{w,Wk) k=Na+l

(3.5)

34

CHAPTER 2

in G~^. Hence, also the product (3.4) is absolutely and uniformly convergent in G~^. Consequently, Ba{w^ {'^k}T) ^^ holomorphic in G~ and {wkj'i^ are its zeros. 2°. The function Ba{w,{wk}f) ^ 0 is holomorphic in G~ since (3.4) is assumed uniform convergent inside G~. Besides, Im Wk -> 0 3.8 k -> oo since the sequence {wk}i^ is bounded. Observe that the absolute convergence of the product (3.4) in any fixed point a G G~ is equivalent to the absolute convergence of the series (3.5) (where p = lin a and Np is chosen by p as above) in the same point. Besides, by our requirements sup^ |Re Wk\ < sup^ \wk\ = M < -hoc. Hence 2g-if(l+a)|j^^^|l+a

>(|a + M| + |a-M|)-^-"'l^"^"'=l'''"

(a-Reif;fc)^+^(l + a)

Since Im IC'A; —> 0 as fc -> oo, (1.7) and the last inequality lead to \fta{ci,Wk)\ > C\lm Wkl^'^'^,

k > fco,

for enough great ko > 1, where C = C{a, a, M) > 0 is a constant independent of k. Therefore, the absolute convergence of (3.5) in w = a implies (3.3), and the proof is complete. Remark 3.1. Considering the convergence oiBa{w, {tffc}i°) on compacts K C C, one can show that under (3.3) Ba{w, {^k}T) {^ > ~1) admits holomorphic continuation into the strip {w : a < Re w < b, 0 < Im w < +00} C G"^, through any interval (a, b) disjoint from the points Uk = Re Wk (AJ = 1, 2 , . . . ) . Let {wk} C G~ be a bounded sequence (finite or infinite) satisfying (3.3) for a given a G (—1, +00). Denote sup;. \wk\ = M (< 4-oo) and assume w G G~ a fixed point such that \w\ > 4M. Then, in view of absolute convergence of (3.5) and (1.8) \\0g \Ba{w,{Wk})\\


0) is subharmonic in G~ and satisfies (3.12), one can use the well-known Littlewood's theorem on boundary values of subharmonic functions (see, for instance, [105], Ch. IV, Sec. 10, Theorem IV.34) to prove that for almost all ix G (—oo, +oo) T^-" log \Ba{w, {wk})\ - 0,

lim

(3.15)

where l{uo) is the image of a radius for any conformal mapping of |2:| < 1 onto G-. Lemma 3.4. If {wk} C G~ is a bounded sequence {i.e. sup^ \wk\ = TQ < +CXD) satisfying (3.3) for a given a G ( - 1 , 4-oo)^ then for w e G~ {\w\ > 4ro)

\W-'^log\B^{w,{wk})\\ < |

^ ^ ^ E | I ^ ^ ^ I ' ^ 4 H " ' -

16) (3-

Proof. Assume w G G and \w\ > 4ro. Since \w — Re Wk\ — |Im Wk\ > \w\ - (|Re Wk\ + |Im Wk\) > \w\ - 2\wk\ > \w\/2 + 2(ro - \wk\) > \wk\/2, (2.4) implies 4

l+a.

1-1

Therefore, (3.16) follows from (3.11). 3.4- Now we pass to the properties of the products Ba{z, {zk}) = Y[ba{z, Zk),

-l< a < +00,

(3.17)

k

with zeros in the upper half-plane G^. Some of^these properties are described by means of the integro-differential operator W~^ (formulas (1.23)-(1.25) in Ch. 1). One can verify that the inversion w — z~^ {wk = z^^^ fc = 1,2,...) transforms Theorems 3.1, 3.2 and 3.3 into the following equivalent assertions. Theorem 3.1*. 1°. Let {zk^'i^ he a sequence of complex numbers in the upper half-plane G^ and let oo j

E

^ I l+a

Im —

< +00

(3.18)

Zk

for a given a G (—1, +oo). Then the infinite product Ba,{z,{zk}T)^\{K{z,Zk)

(3.19)

k=l

is absolutely and uniformly convergent inside G~^, and Bct{z^{zk}) morphic function in G^, with zeros {zk}-

is a holo-

BLASCHKE TYPE PRODUCTS

39

2°. / / an infinite sequence {zkyf^ C G"^ lies out of some neighborhood of the origin, and for an a G (—1,+CXD) the product (3.4) is absolutely and uniformly convergent inside G^, then {zk}^ satisfies (3.18). In the next theorem I/[5,Re s~^) {s G G"^) means an arc with the endpoints Zk and Re Zk~^, which is the part of a tangential to the imaginary axis circle centered on the real axis. Theorem 3.2*. 1°. / / a sequence {zk} C G^ satisfies (3.18) for a given is continuous and subharmonic in a € (0,-hcxD)^ then W~^\og\Boc{z^{zk})\ G^ and this function is harmonic in G^\ U^ L[zA;,Re Zk). Besides zeG^.

W-^\og\Bo^{z,{zk})\ |Im l/zk\ (/c = 1 , 2 , . . . ) . Theorem 3.3*. If {zk} C G+ satisfies (3.18) for an a E ( - l , + o o ) , then sup

/ \w-''log\Ba{Rsm^e'^,{zk})\\

d^ l+a < 4-00,

-r(2 + a)4lim

r

\w-^\og\Bo,{Rsm^e'\{zk})\\ -

(3.20)

Zk

^

= 0.

(3.21)

R^-\-oo

4. A REPRESENTATION OF THE PRODUCT 4.1. Below we shall prove a representation which will be used later, in Ch. 5, for investigation of boundary properties of our Blaschke type products. Theorem 4.1. Let a G (—1,1) be any number, and let {wk} C G~ be a sequence of complex numbers such that simultaneously ^^\lm

Wk\^^'^ Irj]"^ for a < 0. Consequently,

-i^riog

2\rj\+iu

> Ii{u)

if

- 1 < a < 0,



if if

0 0 (we cannot get rid of e in view of the below estimate for Re $«? which follows from (4.12)). Hence we come to the representation (4.14) and to the relation (4.3) with obligatory e > 0 (see [4], Addendum I, Sec. 3). 4.5. Lemma 4.5. Under the conditions of Lemma 4.4; W'^^aiw.iwk})

= log J^)^' e l l ^

Imw < -max|Imt 1) or pole {then da < —1) in A, and ' ^ ^ ( K ; , ^ ) is a harmonic function. Proof. Assuming e > 0 and p > 0 arbitrary numbers, introduce the domains D^ = {w e D : Oe{w) = {C :\C-w\ 0

SI ^ ^

^1

\>W-^\og\F{w)\,

«;G{U„r*[6„,oo)};

(b) if Q; G (—1,0), then in the same sense the function W~^\og\F{w)\ is superharmonic in the points w G {UA;r*[aA;, oo)\ {ak}} and subharmonic in the points W G {Unr*[&n,00)\{6n}}.

1.3. The following two lemmas will be necessary. Lemma 1.2. Let a function u{z) he harmonic in the semidisc G^{R) = {z : Im z > 0, |z| < R} {R > 0) and be continuous in its closure, with possible exception of a finite set of points {cm}i C dG^{R) {cm 7^ 0, 1 < m < g) m neighborhoods of which U{z) = dm log \Z - Cm\ + ^m{z)

u{z) =Em\z-

Or

Cm\~^\cmzr COS {7arg [i{z~^ - c~^] } -h ^^miz)

(0 < 7 < 1); where dm and Em CLTC real constants, and ^m{z), harmonic functions. Then for z G G'^{R)

^TT IdG+iR) JoG+(R)

2Tr Ja dG+{R)

«(C)

^" 1

[C-z

1

C-z

/

r

\R^-Cz

T'

R'^-Cz

(1-4)

^m{z)

are

M

dC,

(1.5)

where G{(,z) is the Green function ofG^{R), d/dn is the differentiation along inner normal, and ds is the element of the curve length.

48

CHAPTER 3

Proof. u{z) can be written as a Poisson integral in dG'^{R) since it can have only integrable singularities (1.4) on dG^{R). For a similar situation, one can find a detailed proof in [38] (Theorem 1.1, Sec. 1, Ch. 1). Lemma 1.3. Let U{w) he harmonic in the half-plane G~ {p < 0) and continuous in its closure C^ = {w : Im. w < p}, with possible exception of a finite set of points {Am}i C dG~ {Am ^ oo^ 1 <m < q) in neighborhoods of which U{w)

= dm log \W-Am\+

U{W) =Em\w-

Am\-^

(fm{w) COS {^[i{w

Or - Am)]}

+ ^m{w)

(0 < 7 < 1)

(1-40

where dm CLTid Em are real constants, and cpmi'w), i^mi'^) (^f^ harmonic functions. If + 00

/

\U{u + iv)\du < 4-00, -oo

then + CXD

/

\U{u + ip)\du< 4-00

(1.6)

-oo

and 7-T/ X

U{w)^^-

k-pl

/»+oo

f

U(u-\-ip)du

^ / ) ^^ TT j _ ^ {u - ty + {v-

.

T2 < + ^ ' w = u^iveGp.

^_

PY

P

,^ ^.

1.7

Proof. The function V{w) = U{w-\-ip),

weG-,

(1.8)

is harmonic in G~. Besides, by the properties of U{w) V[u^)JAr^

^^f

^,

w = u^-iveG-,

(1.9)

where p{t) is of bounded variation in (—00, +00) and +00

/

/'+00

f{u)V{u + iv)du^ -00

/

f{u)dii{t) J-00

for any f{u) continuous in (—CXD,+CXD) and such that \miu-^±oo f{u) — 0 (see, for instance, [36], Ch. 1, Theorems 5.3 and 3.1(c)). It suffices to prove that in our case the measure dp{t) is absolutely continuous and equals V{t)dt. To this end, for an arbitrary segment [a, 6] G (—00,+00) introduce a sequence {fn{u)}T of finite, continuous functions in the following way: fn{u) = 1 for u € [a, b], fn{u) = 0 for t^ G (—00, a — 1/n] and u £ \b -\- 1/n, +00), and fn{u)

EQUILIBRIUM RELATIONS AND FACTORIZATIONS

49

to be continued to the remaining intervals as a linear function. Then, by the properties of V{w) +00

/*+oo

/

fn{u)V{u)du= lim / for n > 1. On the other hand, pb

p-\-oo

fn{u)V{u^-iv)du= /'+00

I V{u)du — lim /

fn{u)d^{u)

ph

fn{u)V{u)du

= /

«/ —oo

*J a

I

dii{u).

*J d

Therefore, in (1.9) dii{t) = V{t)dt and V{t) G L^(—oo,+CXD). Hence our assertion follows by the definition of V{t). 1.4' Now we turn to F. and R. Nevanlinna type formulas. Theorem 1.1. Let f{z) he meromorphic in the closed half-plane G+\{0} = {z : Im z > 0, z 7^ 0}; and let for some RQ > 0 this function has no zeros and poles in the semidisc G+(i?o)\{0} = {z \ Im z > 0, \z\ < RQ, Z ^ 0}. Further, let a > —1 and K, > 0 be any numbers for which one of the following conditions is satisfied: r . log|/(z)| G ^ a + / . W 2 , G + } , if ae [0,+oo), 2^ 9/a(Im l/z)\og\f{z)\

G ^ i + a + 4 7 r / 2 , G + } , if a e ( - 1 , 0 ) .

+CXD^ any R > 0 and any z G G~^{R) which does not Then for any —l 0 and ( = ^ -\-ir] £ G (r) Ia{C.r) = l

lim \v\Uiv,C,r)

2 t^^-oo

= - [ ]v-^logba{re'\C)^inM^,

(1.16) (1.17)

L [2 -\- a)

Proof. The function W~^logba{s^C) is continuous by s in G~, with possible exception of the point ( G G~{r) (see Theorem 3.2 in Ch. 2). Therefore, using the equality lim \v\

1 s — IV

IV s -\-iv

IV

+ r^ +

isv

2i{

^

1

(1.18)

52

CHAPTER 3

we get laiCr)

= ^ J W-'^log\b4s,C)\

( ^ - l ) ds

= - / W-"\og\ba{re'^,0\smM'd. •^ Jo On the other hand, by formula (2.2) of Ch. 2 I

hm \v\W-^\og\bc,iiv,0\

I"! (1^1 - l^)|. Our next lemma shows that one can get rid of those additional restrictions if f{z) is meromorphic in a neighborhood of the origin. Lemma 2.3. Let f{z) be meromorphic in \z\ < RQ. Then, multiplying f{z) by definite rational functions R{z) one can attain any of inclusions r. 2°.

\og\R{z)f{z)\ G K^{n/2,G^} for a given (3 G [0,+oo), d/d{lm 1/z) log \R{z)f{z)\ G Kp{7r/2, G+} for a given (3 G (0,1).

Proof. It suffices to consider the case when f[z) (/(O) 7^ 0) is holomorphic in \z\ /3 be a natural number, and let f{z) = ^t^^^z^ (ao 7^ 0, 1^1 < RQ). Observe that, multiplying f{z) by the n-th partial sum Pn{z) of the Taylor expansion of l/f{z), one can find f{z)Pn{z) = 1-\- bnz'^ + 6n+i2:^+^ + • • • {\z\ < Ro). Then \og\Pn{z)f{z)\^ l^nJkr^ as ^ -^ 0, where ni > n. Therefore, 1° holds by the definition of Kp. 2\ If f{z) = Et^o^kz'

OlomzA d{lml/z)

_ ^^^

(ao 7^ 0, \z\ < i?o), then

L . a . + 2 a . . - f 3 a 3 . - + . . . | ^ ^^^^^^^^ ^^ ^ ^ ^^ \

ao-\-aiz-h

a2z'^-\-'

60

CHAPTER 3

where ni > 2. Thus, 2° automatically holds for any p G (0, 2). The following lemma contains a useful estimate. Lemma 2.4. Let F{z) he holomorphic in G~\{0} = {w : Im w < 0, w ^ 0}, and let this function have no zeros in {w : Im w r} for some r > 0. Further, let log \F{w) G Ma{G-\{0}} for an a e (0, 4-oo), and let \F{w) < e"l^l~^'^'''\

1^1 RQ}. Therefore, integration by parts gives Y^

1

f smijjn

1^

^^'^^\.M\ 0, z 7^ 0}; and let for some RQ > 0 this function have no zeros in the semidisc {z : Im z > 0, \z\ < RQ} U {—RQ^RO}- Further, let a G (—1,+oc) be any number and let 1°. log 1/(^)1 G M^{G+} if aj [0,+oo), 2°. a/5(Im 1/^) log 1/(^)1 G M^{G+} if a G ( - 1 , 0 ) .

If liminf - i - / R^+OOTTR

Jo

lw-°'log\f(Re''^)\y L

J

sinM^ < +00

(2.18)

and (2.19) then f{z) G A^{G+}, 2°.

/'+00 r-\-oo\

-RQ p-Ro

(

di_

+ fRo

/

+/

^-00

JRo

(2.20)

< +00,

W-"log|/(i)| J '

lim R-^-\-oo -foo

/

W-'^\og\f{t)f{-t)\

dt

(2.21)

63

EQUILIBRIUM RELATIONS AND FACTORIZATIONS

3°. The following relations hold

1+a

lim

(2.22)

VF~^log|/(i?e^^)|sin^rf^ = / i ^ o o , _4^ / R Jo

— lim

.

Im

,?.ir(2 + a)

\zk\!"

)(^i5){"'"°'°*l«"l}"^'

-Ro


+oo

d

:^'')|};

27rJo {W^-"log|/(iZoe**)|i?o" + ^ W ^ - " l o g | / ( i ? o e ^ ^ ) | } sin7?d,9,

EQUILIBRIUM RELATIONS AND FACTORIZATIONS

65

where the right-hand side integral is absolutely convergent. Hence (2.22) follows by (2.9) and the finiteness of the limits (2.21), (2.23). The proof of the next theorem is quite similar. Theorem 2.4. Let F{w) be holomorphic in G~ = {w : Im w < 0}, and let for any p < 0 this function have not more than a finite set of zeros in G~ = {w : Im w < p}. Further, let a E (—1, -{-oo) and let r . log\F{w)\ e Ma{G-} if ae [0,+oo), 2°. a/a(Im w)log\F{w)\ e Mi+a{G~} if a G (-1,0). +00

/

IW'"^ log \F{u + ip) I du < -f oo,

- o o < p < 0,

(2.25)

-oo

then: 1°. Zeros {wk} C G~ of F{w) satisfy ^\lm

Wk\^'''' < + o o .

(2.26)

k

2°. The following relations hold:

Im Wk 0, and a ^ I m l/z)\og\fo.{z)\ G i^i+c.+7W2,G+} (7 = min{/^,l}) if - 1 < a < 0 by (1.2), (1.2') and our requirements 1°, 2°. Therefore, by Lemma 1.1 and Theorem 3.2 of Ch. 2, the function u{z) - W-^ log \f^{z)\ = W-^ log \f{z)\ - W-^ log \B^{z, {zk})\

(3.12)

is harmonic in G+\{Q}, except the points {cm}i^ = {(Re l/^m)""^}!^ ( ^ < oo, {zm}i^ C G+\{0}, Re Zm 7^ 0) which are the zeros of f{z) in (—CXD,+OO). Besides, u{z) has the representations (1.4) in the neighborhoods of these points, and by Lemma 1.9 of Ch. 1 u{z) is continuous at the origin, besides u{0) = 0. By (1.16), (1.19)-(1.190 and Remark 3.2 in Ch. 2 lim

4 /

W-''\og\Bc,{Re'^,{zk})\smM^

= 0.

(3.13)

Hence by (2.11') of Ch. 2 u{z) satisfies the conditions of Theorem 3.1 and therefore is representable in the form (3.1)-(3.3). Besides, by (3.15) the number h in (3.1)-(3.3) can be deduced by (3.9). Further, by (3.10) the integral (3.6) is convergent since the last term in (3.12) vanishes on the real axis. Thus, W--\og\Uz)\

= hy+^

r^T'^l^^f^^?^^ TT J_^ [X - ty + y^

z = x + iyeG^.

(3.14)

For proving the remaining formula (3.8), introduce the function U{w)=\og\fa{w-^)\

(3.15)

which is harmonic in G~. As W~^U{w) ^ W~^ log\fa{w~^)\,

[ ZW

7TI J_^

W-t

J

(3.14) becomes

68

CHAPTER 3

On the other hand, (3.6) impUes r+oo 1^—iog|/(t-iMl

1 + 1*2-/« 1

/ J —(

dt < +0C,

where 2 — / ^ < l + a — p i f Q ; > 0 and 2 — / ^ < l + Q^if—1 0 {j = 1,2) and U{w) - Ui{w) - U2{w). Besides, U{w) = U'^{w) — U~{w), where Ui{w) > U'^{w), U2{w) > U~{w). Therefore, setting ^{w) — U\{w) — U"^{w) we conclude U-{w) + ^{w) = U2(w),

w G Q-.

(3.20)

But 0 < ^{w) < Ui{w), and for Ui the integral of (3.19) is convergent in view of Theorem 1 in [72]. Therefore, the same integral is convergent also for ^{w). Further, by (3.20) the integral (3.19) is convergent also for f72. Hence, by same theorem a = 0 and fJ^2{t) satisfies the second requirement in (3.18''). The converse statement is obvious since f^"^

0
1 is a natural number, then by (3.22) and nonpositivity of W-^'loglBaiw)] in G- (see Theorem 3.2 in Ch. 2) + 00

n + OO

7.

1.

/

[u{~it)r j<j^

[w-'^iog\F^{-it)\y I < +00.

If Of > 0 is not a natural number, then by Lemma 3.2 of Ch. 1

J^^^ m-itr ^

< ^^°° w"^ iiog \F4-it)\\ ^

< +^.

Thus, U{w) e it+_„ since all requirements of Definition 3.2 are satisfied. Hence, U{w) is representable in the form (3.18), (3.18"), where a = 0, i.e.

W-'^log\F.{w)\=Re

{-

r ° ° ^ } , W ^ / '

weG'

74

CHAPTER 3

where /x(t) is as required in (3.25)-(3.25"). In view of Lemmas 1.10 and 3.1 of Ch. 1, applying (dP/dvP)W~^^~^^ to both sides of the last formula we come to (3.25). Now let - 1 < Of < 0. Then the proof of formulas (3.26)-(3.260 and (3.26") is quite similar to the previous case. The only difference is that instead of nonpositivity of W~^ log \Bot{w)\ one has to use its continuity in G~\{wk} and the fact that the points {wk} (i.e. zeros of F{w)) are integrable singularities. Also the inclusion W~^ log \Ba(w)\ G KI{7T/2, G~} has to be used, which is a consequence of (1.2') and Lemma 1.9 of Ch. 1. For proving the converse statements, first assume that a > 0 and (3.25)(3.250 is valid. Then log|F('?i;)| G Mp{G-} by Lemma 3.3 of Ch. 1. The function U{w) defined by (3.27) is harmonic in G~ and representable in the form (3.18)-(3.18'0, and U{w) G il^_p by Lemma 3.1. Therefore, by the estimate (3.14) of Ch. 2 sup /

[W-''\og\F{u

+

iv)\]^du

v — 1 (where A = 0, U / CK = 1 and CA = Co — /(O)). Replacing a by a — 1 in that formula, we get

iog/(.).^// (i-icpr-y^^^^ — 1 the function aa{w, () is holomorphic in G~ and vanishes only at w = ( which is a first order zero. Besides, the product Ba{w,{wk})

= Y[^c^{'^^'^k),

{wk}cG~,

-l

{w - ip) - (C - ip)

Besides, -1

[w - ip){C^ - ip) {w - ip) - (C - ip)

"R w -ip

(C - ip)

< i?i \f ,„w_L^r=i j i z i ^ 0, 0 < /3 < 1 + a). R e m a r k 2.2. The argument used to prove the case 0 < a < 1 in Theorem 2.2 permits to extend the assertion of Theorem 2.1 (i.e. formula (2.13)) to the case 0 < a < 1. 2.5. Below we prove a useful estimate for the exponential factor of (2.18). L e m m a 2.7. If F e 0^^^ (a > 0, /3 G [0,1 + a]), then for any w e G'

^

J J a-

[i{w-c.)y+'^

I

Proof. Obviously |w - CI > \^^ w\ + |Im C| for any t«, C S G~. Therefore, using the inequahty a + b> o(l + b)/ (1 + a) (a, 6 > 0) we get 1

^

1

(l + |Inm;|)^

l ^ - C I ' ' ~ |Im«;|^ (l + | I m C | ) ^ ' Further, (1 + xf/

(1 + x/^) e [1,2'^-!] {x > 0, P > 0). Thus, 1_ ^ ,|g_i| 1 + |Im ^/^ \w-C\0\lmw\0 l + |ImC|^

and ^ ^lg_il 1 + |Inm;|^ |Im w|i+" l + |ImC|''' y,_^|i+a-

90

CHAPTER 4

Using the last inequality we come to (2.20). Remark 2.1. If {wk} C G that

satisfies (2.8) for an a; > —1, then one can show-

|logB„ {w,Wk})\ < ^

1 ^ |Im « ; , r + 4 |Im w\-^'+'^^

(2.21)

for Im w < —5maxfc |Im Wk\. By (2.20) and (2.21), (limt->+oo F{—it)) the factorization (2.18), i.e.

= 1 in

CF = lim [F{-it)]-^

(2.22)

= ±1

/;—>-+oo

for any F G OT^^ (o^ > 0, /3 G [0,1 + a)). 2.6. Definition 2.2. A function F holomorphic in G~ is of the class OT^^ (0 < a < +CXD) if for

any p

0 is a constant depending solely on a. Proof. It suffices to prove (3.2) for a single factor of Ba- To this end, we consider two cases in assumption that ( = ^ + irj G G~ is fixed, (a) Let 2|r/|/|i(; — CI > 1/5, then we use the recurrent formula aa{w, C) = aa-p{w, C) exp \ ^

-——— ( — ^ ) \ ,

(3.3)

where p > 0 is the integer from p — 1 < a < p (this formula is easily derived from (2.6) by integration by parts). By (3.3)

log lap (^,01 < X I n=l

(T ^'

if a = p > 0 is an integer. But in our case

vi«.-ci/

-

*i ^ " > f J 2 L | . i 2 and 1+.5

Re

1-

1+5

1-

1+ t

cos (1 + ^) arg

l+t

1 -

1+t

>

Vs

Consequently, —

2|r?|

log|a,KC)| - ,

a > —1.

(b) Now let 2|77| /\w — C\ < 1/5. Then one can verify that 4r?Im w w~ ( = 1——=:z w—C \w-C\'

, and

2|Im w\ 2\rj\ T^ =^ < -,—^-^ \w-C\ \w-C\ + 2.

Therefore w—( \w — (

>

^

12\lmw\

1 /

2\rj\_

b\w-C\

^\\w-C\

^\ J

>

14 25

and consequently \w — ( w—(

w-C 2i\r]\x > w—( w—(

* '

H-CI

>i

2'

^e[0,l|.

(3.7)

FUNCTIONS WITH SUMMABLE TSUJI CHARACTERISTICS

93

Thus, for a > - 1

log|aa(w,C)ll
Np-\-l. Then we write 'No

\\og\Ba{w,{Wk})\\
Amo

Jo

(|u|-2mo)i+-

a ( l + a) '

^

^

and /2 ^ ^ock{i^) is a continuous function of t'(—oo < !» < 0). Besides, lim^^oo ^oi,k{v) = 0 in view of (2.21). Consequently I2 < max

max "^akM < +00.

l 0 be arbitrary. Then by (3.2) and (3.13)-(3.13') p-\-oo j.a—1

/

/'+00 j.a—1

Y^^^-t^Ba)dt=

r+oo

YTJp^^ ^

log^\Bao{u-it,{wk})\du

,

p+00

j.a—1

jj.

One can verify that A fl

1+t/3(t+|Im«;,!)" ^ i i fO-l

^+

/.|Imto| 1

70 l + i ' 5 ( t + |Imwfc|)" ^ / o

t^~(3' x°'''^dx

(l + :r)^ 1

Jo /o

Im^fcl 2/(1 + 2/)'' Jllm

a are representable in the form

where Bao{w,{a^}) and Bao{w,{bjy}) are convergent Blaschke type products with zeros {a^}; {bj,} C G~ satisfying

J2 |I«i ^MI^"^"" < +0C and Yl \^^ ^^l^"^"" < + ^ '

(3-1^)

and /i(C) = Mao(C) ^5 ^ function of bounded variation in any compact from G , such that llm Cl"^'^ ^' CI'' |^MC)l a, and dii{() = log |F(C)|(icr(C). For proving the converse statement, note that Bc.,{w, {a^}), Ba^,{w, {K}) e ^ZP

(3-17)

by the assertion 1° of Lemma 3.3. Thus, it suffices to prove that also

To this end, observe that i|a —1

/ / .G- 1 + |Im w\l^

-^?~II

|ImCr-'/(a,ao,/3,OMMOI,

(3.18)

where ff |Imt/;|"-i da(w) J JQ- l-\-\lmwf \w-C\^-^'^''

,^ _ „

For a G (0,1] a suitable estimate of the last integral follows from the inequality

(which is true for any a > 0). If a > 1, then one can be convinced that

Jo (1 + t0)it + a)"o - "

Jo a-f^ + xP-""

'A-.

1 +t0 a—ao—fd

Besides,

A

(i+t^)(t+a)-o ^ y.

(l + ax)^o

~ ao - a + Z?'

Hence, for a > 1 J{a,cxo,l3,a)


^)]-^"^°'°(l + 2:^)} is holomorphic in the z-plane, except the cut {z : Re z = Re 5, Im z < Im s) C G~, and ^s(z) has simple poles at z = ±i. Calculating the integral of this function along V R — {Re^^ : 0 < ^ < TT} U {[-i?,i?]} and letting R -> +00, we obtain la^^s) = 7r(l + is)~^-^+"°^. Inserting this expression into (3.21) and using the well-known formula for Euler's ^S-function we find G{w) = (1 -[- iw)~^~^. Consequently, F{w) = exp{(l +iK;)~2 ^j iov w = u-\-iv e G~ and \og\F{w)\ = \l-\-iw\

2

^cos1

(3-a\ ( u 2—y arctan VI+ 1^1

Besides, log \F{w)\ > 0 since 1/2 < 1 - (/? - a ) / 2 < 1. Therefore, 1

C{t,F)^—

/"-hoo

j cos < —

\og^ \F{u + w)\du

27r

/; _ , , [(i + | t | ) 2 + ^ 2 ] | -

= +00

J —c

for any t < 0, and hence (3.20) is true. The above counterexample shows that for a < /? < 1 -|- a (3.14)-(3.16) is not a descriptive representation for OT^^^ (0 < a < +00), in contrast to the case 0 < yS < Q; considered in Theorem 3.1. The case j3 — a remains open. NOTES. A similar method of mapping and successive passage has been used earUer by M.M. and A.E.Djrbashian [24] for proving integral representations of holomorphic functions in G from L^(G , y^dxdy). One can observe that the functions (1.3) are the inversions of a special, infinite case of Tsuji characteristics considered in [38] (Ch. I, Sec. 5). As F.A.Shamoian noticed, the passage of this chapter can be used for proving the representation (2.18) for holomorphic functions from somewhat larger classes (the suitable density condition for zeros must be presupposed). But there were some difficulties in finding descriptive representations. Later a version of the descriptive representation (3.14) was proved [5] for subharmonic extensions of the particular classes OT^Q (a > 0). Namely the approach of F.A.Shamoian [94] was used to prove some descriptive representations over the real axis, where d^ is replaced by (p(t)dt and (^(t) is from a definite O.V.Besov space.

CHAPTER 5 BOUNDARY VALUES 1. MAIN RESULTS 1.1. In view of results of Chapters 1 and 3 the below definition is natural. Definition 1.1. A function F{w) meromorphic in the lower half-plane G~ is of a-bounded type in G~ or, which is the same, of the class Ncc{G~} (—1 < a < +oo), if it can be represented in G~ in the form ^, , Ba{w,{an}) i /"+^ - ^ H = J. ) \ {{ exp 0) and C are any real numbers, a{t) is representable as the difference a = ai — a2 oi two nondecreasing functions satisfying

/ where p > 0 is the integer deduced hyp—l M sup / w-ty weG-J-00

, ^^ , = +00. \w-ty

The next assertion easily follows by Fatou's lemma and a passage to a limit. Lemma 2.2. Let 7 G (0,1); let E C (-00, +00) be a B-set such that C^{E) > 0, and let fi ^ E be a measure from Definition 1.2. Then f^^

dfi{t)

^ ^

/^+^ dfi{t)

,

^

.o.x

104

CHAPTER 5

Lemma 2.3. Let (3 G (—1,0) be any number, and let a{t) be a nondecreasing function such that

Further, let f{w) be a holomorphic function given by the formula

/H=r°°,/"^'L,^ ^eC-. J-co

(2.3)

[l{w - t)\ ^

Then the set of those u G (—oc, 4-cx))^ where the nontangential boundary value f{u) exists as a finite limit, is of zero 1 -{- (3-capacity. Proof. From (2.2) and (2.3) it follows that the function exp{—/(^)} is holomorphic and bounded in G~. Therefore, by Lindelof's theorem the existence of a finite angular boundary value f{u) in a point u G (—oo, -\-OQ) is equivalent to the existence of the finite limit lim / ( ^ + i^) {=f{u))

(2.4)

V—>—0

in the same point. For proving the existence of this limit, write f{u + iv) = f{u + ivo) -f- i / f{u + iy)dy

(2.5)

Jvo

for any u G (—00, +00) and v, VQ (—00 < VQ < v < 0). One can verify that for any w = u-\-iv e G~, t G (—00, +cx)) and K > 0

\w-t\^ where (2.3)

CK,{W)

- {\u -1\ + \v\)'^ - ( l 4 - | t i ) « '

is a constant depending only on K and w. Therefore, by (2.2) and +00

\nu

+ iy) \dy 0 leads to a contradiction. Indeed, if Ci-\-p{Eo) > 0, assumption CI^^{EQ) then there must exist a nonnegative 5-measure //o -< ^0 satisfying (1.4)-(1.5). By (1.4), one can fix an enough great A> 0 such that / dfio{t) = I diio{t) > 0. /[-A,A]nEo J-A J[-A,A]nEQ Then

A /

pO

-A

diio{u) / \f{u + iv)\ dv = +00. If \u\ > 2A and \t\ < A,-A then J—00 < 77—rr^TlTIT' - ( i + H ) ( i + |t|)'

\u-t\

C,{a) = {l + A-'){1

(2.8)

+ 2A).

Hence for Iwl > 2A ^1+/?/

J-A \U - t|l+^ - (1 + \v\Y^f J^A (1 + 1*1)^+'' {l + A-^)^+>^{l + 2Ay+^ On the other hand, if \u\ < 2A, then by (2.1) ^

diXQJt)

5 i ( l + 2A)i+^

/ Hence, particularly for any u € (—00, +00) ^ /

rfMo(^)

|u-t|i+/'

, [(1 + A-'y+0

+ 5 i ] (1 + 2^)1+/^

(i + iuDi+z^

C2(^)

{i + \u\y+0'

,...

106

CHAPTER 5

But for any u G (—00, -f-00)

r \fiu+iv)\ dv < 2^+^/2 r°° -^

da{t)

J—00

J—00

1^

1+/3'

where the integrals can be finite or infinite. Therefore, by (2.9)

< +00

which contradicts (2.8). Lemma 2.4. Let /3 G (—1,0) he any number, and let the sequence {wk} = {uk + ivk} C G~ he such that ^kl'+^ 0 enough great to provide that the integral (1.4) (with 7 = 1 +/3) taken over [—A, A] equals a number M > 0, we consider the measure M~^fi\r_^ ^. = Jl ^ E. It is evident, that this measure satisfies (1.4) (with 7 = 1+/?). On the other hand, it is obvious that Ji satisfies also (2.11). But L \w

(2^+^ + A - ^ - ^ ) ( l + A)i+^

BOUNDARY VALUES

107

for IK;I > 2 ^ and \t\ < A. Consequently H-oo = M s u p \{l + \w\)^-^^ f oo

djl{t) r \w-t\

.A

< ( 2 i + ^ + A - i - ^ ) ( l + ^ ) i + ^ 4 - ( l + 2A)i+^ sup

/

\W\ 0. Then integration by parts gives

Al^l+!^i)/2

J

(l+/3)(l + ^ - a )

From these estimates it follows that \W~^\og\Goi{u-[-iv^{wk})\\du

< H-oo.

-oo

Hence Got ^ ^ J ^ C Ar^{G~}, and it remains to see that 3^ G Np{G~} by the inclusions Go. G iV/3{G~} and H^ G iV/3{G~}. L e m m a 2.6. Let a G (0, +oo) and let the sequence {wk} = {uk^-ivk} that

he such

Then Ba{w, {wk}) = Ba-i{w, {wk}) [Ha-i{w, {wk})]-'

,

(2.15)

where Ha-i is a function holomorphic in G , having the form (2.13'). 0 < a < 1; then Hot-i is a hounded function in G~.

If

110

CHAPTER 5

If a G (l,+oo) and the sequence {wk} = {uk -hivk}

satisfies

k

then Ba{w, {Wk})=Ba-l{w,

where Ra-i

{Wk}) [Hc,-2{w, {Wk})f

[Ra-l{w,

{Wk})f

,

(2.16)

is holomorphic in G~ and hounded for 1 < o; < 2.

P r o o f follows from formulas (2.12), (2.13), (2.13') by successive integration by parts. Besides, it is clear that |i5/^a-i| < 1 in G~ for 0 < a < 1. In addition, one can show that

Roc-i{w,{wk}) ==exp^-——y^n—~—vi^^ r{

l-\-a^

[i{w - Uk)r

J

Using this, one can prove that |i?a-i| < 1 in G~ for 1 < a < 2. 2.3. P r o o f of T h e o r e m 1.2. 1°. Let (1.11) be valid with a fixed 7 G (0,1). First consider the case 7 — 1 < a < 7. If we denote /3 = 7 — 1, then — 1 < ^ < 0 and /3 < a < ^ + 1. Therefore B^c e Np{G-} = iV^_i{G-} by Lemma 2.5, and our assertion follows from Theorem 1.1. Next, let 7 < a < 7 + 1. Then the representation (2.15) of Lemma 2.6 is valid, and 7 — l < a — 1 < 7 . Therefore -B^-i, Ha-i G Nry_i{G~} by Lemma 2.5. Hence Ba is of the same class, and our assertion follows from Theorem 1.1. At last, let a = 7 + 1. Then, by (2.15) and (2.16) Ba = ^ 7 + 1 = ^7[-f^7]~ 5

^ 7 = [Hy-iRy-.i]~

,

(2.17)

where, by proven above, B^ has non-zero, finite boundary values at all u G (—cxD, +CXD), with possible exception of a set of zero 7-capacity. By Lemma 2.5, Hj-i G Nj-i{G~}. Therefore, by Theorem 1.1 also the set of those u, where H^-i has not non-zero, finite boundary values is of zero 7-capacity. Thus, if we also prove that lim HJu + iv,{wk})y^O (2.18) v—>—0

out of a set of zero 7-capacity, then, by Lindelof's theorem, non-zero, finite boundary values of i?^_i will exist out of such a set. By Lemma 2.1, H^ will have the same boundary property. And returning to the first equality in (2.17), we shall arrive at the desired assertion for -B7+1. Thus, for completion of the proof of assertion 1° it suffices to show that (2.18) exists as a finite limit outside a set of zero 7-capacity. One can prove this

BOUNDARY VALUES

111

quite similar to Lemma 2.3. The following estimates will be the only difference: r\'^k\

^ogH^{u + iv, {wk})\'\ < (7 + 1) 5 1 k

^ A

/

Jo

{\vk\-t)^dt [i{w - Uk) + t]2+7

{\u-uk\ + \v\y

pQ

dfJ.o{u) / -A


0. Further, ^{w) > 0 (it; G G~) by the Phragmen-Lindelof principle, and applying Theorem 1.4 of Ch. 1 we find that Boc^{w,{wk})=Boc2{'^'>{^k})^y^v\-

/

dG{t) [z(^-t)]l+"2

weG-

where a{t) is a nondecreasing function satisfying (2.2) with /3 = 0^2. Hence (1.15) follows.

112

CHAPTER 5

3. PROOF OF THEOREM 1.3 3.1. First note that in our case there is no similarity of formula (2.14) in the disc. Therefore, it is not evident how to prove the similarity of the inequality (1.15) for Blaschke-M.M.Djrbashian products. Nevertheless, for our aims it is enough to replace (2.14) by the following L e m m a 3.1. For any (3 G (—1, +oo) and a> (3 — 1

f

^-^'6,= /o (l-tz)2+«

^ '-&^ ., (l_^)l+a-/3'

..C,

Jo

where Ja,p{z) is such that Ma,p{Ro)

=

s u p |Ja,^(2:)| < + 0 0 , \z\ (1 + Ro)~^ > 0, one can choose this contour to be distant from —1 not less than (1 + -Ro)""^- Concretely, we choose the integration contour in the following way. If the point w = z/{l — z) is out of Ai^o = {w : \ a,Tgw\ > n — arcsin(l +i?o)~'^, Re li; < —1 + (1 -\-Ro)'^}, then we choose the contour to be the intercept [0,w]. li w G ARQ and \w\ < 2, then we choose the union of the intercept [O, |t(;|e*^^^)] (where (p{w) = [TT — arcsin(l + i?o)~'^]sign(argt 2, then we unite the intercept [0,2e^^^^)], the arc [2e^^(^), 2e^^"s^] of a circle with the radius 2 and the intercept [2e^^'s^, tf;]. Let \z\ < 1/3. Then w ^ AR^ and |^| = \z/{l - z)\ < S\z\/2, Therefore

- « | < H F ^ P " ^ = ^ H ^ ( I ) ' - ' (^•'i If 1/3 < \z\ < Ro and w 0 AR^, \W\ < 2. Then \JaA^)\

< 3^+^(1 + i?o)'+" / JQ

x^dx = f — ( 1 + Ro) 2 + a i

H-p

If 1/3 < 1^1 < RQ and w ^ AR^ but \w\ > 2, then obviously

Jo

J2

\X — i~) "^

BOUNDARY VALUES

113

Thus, if z/[l — z) remains out of Ai^^, then sup

(3.10

\Jot,(5{z)\ < + 0 0 .

1/^ 0

y

C{x,y)

~7

1

/

\2~^ 2 - 7 " ; 2 '

-00

^, ''

/i,2 = -

w-u + ivGG-,

(2.8)

J —C

/

.

TT y _ ^

NO .

{x -uf

odu,

V |/l-/2|. {x — uY + 2/2

Together with (2.7) and (2.10), this leads to the conclusion that the quantity y_ / - + d^{t) ^ y r ^ TT J_^ (X - ty +2/2 TT J_^

c//ii(t) y P^^ _ {X - ty 4- 2/2 TT J . ^ (x- t ) 2 + 2 / '

is equal to zero. Hence ii{t) = const by arbitrariness of a: G (—00, +00) and 2/ > 0, and consequently the representation (1.1) of Ch. 5 can be written in the form (1.3). 2.3. For proving Theorem 1.2 we shall use Lemma 2.2. Let a G (—1, +00)^ let N > 1 be a natural number and let

where fi{t) is a continuous, nondecreasing function on [—N,N]. Then there exists a triangular matrix ofpairwise different complex numbers {w^i^^{^, N)}^^^ CG- (A: = 1,2,...) such that: 1°. For any k >1 Im w^,'\fi,N)=rjk{fi,N)

( / = 1,2,... ,fc),

(2.11)

and

k(M,iV)|^+00

(2.13)

126

CHAPTER 6

Proof. For any fc > 1 we split [-N,N] tk^i = N in a, way to provide

as -N

< t[^^ < 4^^ < • • • < tf^
1 \f{w,Nj)-f.^{w,Nj)\<j,, weKj.

(2.17)

128

CHAPTER 6

By Lemma 2.2, for each j >1 there exists a triangular matrix of pairwise different numbers {wi (t^j^Nj)} (fc = 1, 2 , . . . ; Z = 1,2,... ,fc)satisfying (2.11) and (2.12) and such that (2.13) is uniform in respect to w inside G~. Consequently, one can extract a sequence of natural numbers {fej}i^ such that ,S^j)(

ife)

for any j > 1. Hence by (2.15) and (2.17) we come to the desired assertion.

CHAPTER 7 SUBHARMONIC FUNCTIONS WITH NONNEGATIVE HARMONIC MAJORANTS

1. MAIN RESULTS 1.1. It is well known, that Nevanlinna's factorization of the class N of meromorphic functions of bounded type in the unit disc [82] (see also [84], Ch. VII) leads to the following complete characterization of the growth of functions u{z) which are subharmonic in \z\ < 1 and have nonnegative harmonic majorants: P'ZTT

sup

/

P'2'K

u-^{re'^)dd ^limmi

tx+(re^^)d?? 0} [83] completely characterize the growth of such functions only in the particular case when they are subharmonic i n G + = {^ : l m z > 0 } (see also [10], Sec. 6.3-6.5, where it is assumed that u{z) can somehow be continuously extended to the real axis). 1.2. The main results of this chapter are the following three theorems particularly containing a complete characterization of the growth of functions subharmonic in G"^, having there nonnegative harmonic majorants. Besides, these theorems somehow improve Nevanlinna's uniqueness theorem ([84], Ch. Ill, Sec. 38) and also the Phragmen-Lindelof type theorem which follows from a result due to M.Heins and L.Ahlfors [1], by replacing the condition limsup z—>t, Im z > 0

u{z)+oo

and /or any 1? G (0, TT); except at most for a set of outer capacity zero, /i^sinT?-

lim R-^u'^iRe'^).

(1.17)

Remark 1.1. It is well known that the Nevanlinna class of functions (which are subharmonic in G+ and have there nonnegative harmonic majorants) coincides with the set of functions representable in the form (1.6)-(1.8) where ft{x) = (l + x^)~^, i.e. coincides with S{{1-{-x'^)~^). Thus, the pair of conditions (1.2) and (1.5) presents a complete characterization of the growth of such functions when Q>{x) = (1 + x'^)~^. Besides, if u{x) is subharmonic in G+, then it is obvious that the condition (1.5) with fl{x) = (1 + x^)~^ is equivalent to (1.3). The relations (1.5)-(1.7) are well known even in the most general case when fl{x) = (1 + x'^)~^ (these relations are true for h, h^ and /i~, since u{z) and u'^{z) have the same least nonnegative harmonic majorant). Remark 1.2. Using a result from [101] one can verify that the subset of functions of 5(1) {fl{x) = 1), for which /i < 0, coincides with the class of those functions subharmonic in G"^, for which +00

sup

y>o /

u'^{x + iy) dx < +00.

(1.18)

-00

T h e o r e m 1.2. Letu{z) he subharmonic in G^, and let there exists a sequence Rn t +00 such that for any n > 1 /»7r

I u^{Rne^'^)sm'dd'd Jo

pRn



=-oo,

(1.20)

where , ^

( 2-Ha;-2 - i?-2)

for

Ro < \x\ < R,

i?-2)

for

|a;| < Ro,

then u{z) = —oc. R e m a r k 1.3. Theorem 1.2 is an improvement of Nevanlinna's uniqueness theorem, since NevanUnna's conditions (1.4) and Hminfi^-^M(i?)--oo

{M{R) = sup

R^+oo

u{Re'^))

0 0, \z\ < R} has a nonnegative harmonic majorant in G^. Before stating our theorem we consider the function T-»

,

.

\

TV/a'

Rp + z-ip^ Rp- z + ip

^piC^z)

Rp-z-ipJ

I

where 0 < p < R, Rp = y/R? —/9^, and a = arccos (p/i?) and prove some lemmas. Observe that LUp{(, z) gives a conformal one-to-one mapping of the segment G j = {^ : Im ^ > p, |(^| < R} onto the unit disc and ijjp{z^ z) = 0. Therefore gR^p{(^z) = — log |cjp(C, z)| (z, ( G G'^ ) is the Green's function of G J . If u{z) is a function subharmonic in a semi-disc G^* (i?* > R), then by Riesz' theorem in G^

[

u{Re'^)ipRAd,z)dd+

[ ' u{t-\-ip)xljRAt,z)dt,

(2.1)

//3

where i^(^) is a nonnegative Borel measure in G j * , finite in any domain D compactly contained in G^*, and (fR^p, i/^R^p are the expressions for the Poisson kernel of G^^^, written on the arc {( = Re^'^ : (3 < d < -K — f3} {(3 = di.vcsin{p/R) = -K12 — a) and on the interval {C^ = t -\- ip \ —Rp < t < Rp}. Using the well known formula

^^p(C,^)

dC ^p(C,^)'

SUBHARMONIC FUNCTIONS WITH NONNEGATIVE . . .

135

where dn is the differentiation along the inner normal, one can calculate Tv/a^

-KIOL '^

{R, + Re'^ - ipyl-

- {R, - Re'^ + ipyl'^

i ^ ± ± - ^

^ f Rp -{- z-\-ip X J (i?p + Re'^ - ipyl'^ - {Rp - Re'^ + ip^^' Rp — I — ip

-K/a

-1

(2.2)

(2.2') 0 (/? < ?? < TT - /3) and il)R,p{t, z)>Q {-Rp Rp). The below lemmas relate to the behavior of these functions when p —>• +0. Lemma 2.1. If R> 0 and z G C^ are fixed numbers and p>0 small, then Ci
0. Next, observe that Rp^z-ip \Rp + Re'^ - %p\''l'' - \Rp - R^^ + i p r / ^ Rp- z-\-%p > I i?p + i? cos (5r/ 0 is arbitrary, choose 5 > 0 such that ' -(R-S)

R

V +VP^i'=^< 8C2*' where the constant Q is that of (2.6). Then, by (2.21) -{R-5)

V +V|A.<J^. -R

R-Sj

if n > 1 is sufficiently large. Therefore, J^ < s/2 by (2.6). On the other hand, it follows from (2.7) that the integrand of J^ tends to zero uniformly in [-{R - 6), {R - S)] as n -> oo. Therefore, by (2.21), J^ < e/2 for sufficiently large n > 1. Consequently, the relations (2.19)-(2.24) give lim /

u^{t + ipn)ijR,pAt^z)dt=

/

^R{t,z)dii±{t).

(2.25)

It follows from (2.3) that 0 < (pR,p{'d, z) < C2 sini? for sufficiently small p > 0. Hence, using the relation (2.16) (still proved only for RQ = R), we obtain lim /

u^{Re'^)^Rp{d,z)d^ ?1!B(JJ'-

W')'" 7o

\Re''

«"('''">™'' zmRe-i^-z

jdi}.

(2.26)

For the next passage, observe that the function ^{z)=gR{(,z) — gR,p{C^z) is harmonic in the closure of G'^ whatever be C ^ G^ . Besides, $(i?e*^) = 0 (/? < 79 < TT - /?) and $(t + ip) '= 9R{C, t + ip)>0 {-R^ < t < Rp). Therefore,

SUBHARMONIC FUNCTIONS WITH NONNEGATIVE . . .

141

^{z) > 0 in G+^^, and gniCz) > gR,p{C,z) for any z, C ^ G^^^. It is easy to prove that the condition (2.11) is sufficient for the convergence of the integral

//^gniC,z)du{C) ^ ff J JG+

iog\'~^ll~^j\du{C).

J JG+

\Z-CR^

-CZ\

Using this, one can prove that

The relations (2.25)-(2.27) permit to obtain the representation (2.10)-(2.12) by letting p = pn iO in (2.1). On the other hand, the representation (2.10)-(2.12) is necessary and sufficient for the existence of a nonnegative harmonic majorant of u{z) in G^, since using a conformal mapping this representation is derived from the similar representation of subharmonic functions which are of the same type in the unit disc. Thus, u{z) has a nonnegative harmonic majorant in G^, and, as its majorant is the same in any half-disc G J {0 < Ro < R), the relation (2.16) is true for any Ro {0 < Ro < R). Finally, the relations (2.13) and (2.15) without the indices ± can be easily verified using directly the representation (2.10). But also u'^{z) is a subharmonic function having the same nonnegative harmonic majorant. This proves the relations (2.13) and (2.15) with the index +, hence follows their validity with the index —. 3. PROOFS OF THEOREMS 1.1 - 1.4 Proof of Theorem 1.1. Let u{z) ^ —oo be a function of 5(f^), and let i?^ t oo be a sequence on which the lower limit in (1.2) is attained. Then the hypotheses of Theorem 2.1 are true for any R = Rk, and hence the representation (2.10) is valid for any R = R^. From (2.10) we subtract the same representation written for a smaller half-disc G^ {Q < Ro < R)^ put z = iy, divide the obtained equality by 2y and let y -> +0. This gives the following Carleman type formula:

+ (if - i^) / I **'> - i : [ "'^°'"'^'"'''"'- *=*•" The right-hand side of this formula remains bounded from above as R = Rk -^ -hoo. This follows from (1.2), (2.16) and from the relation +00

/

pR

ft{x)dfi-^{x) =

lim liminf /

u'^(x-i-iy) ft(x) dx +00, then there exists lim R~^M{R) = +00 as i? —)> +00, and a + == /3 = +00. In our case lim

^ /

R^-^00 R JQ

u-^(Re'^)sinM'i}

= +oo

144

CHAPTER 7

since otherwise u{z) would be representable in the form (1.6) - (1.8), where /x_i_(t) = 0, and obviously a < h < +oo which is a contradiction. Now consider the case when \iui sup R~^M{R) < +oo as i? -> H-cx). In this case u{z) is representable in the form (1.6)-(1.8) with /i+(t) = 0, besides, (1.4) is true and there exists lim R~^M{R) =(3 = a-^ as i? -^ +oo, according to the result of M.Heins and L.Ahlfors. Besides, from the representation (1.6)-(1.8) immediately follows that a < h. On the other hand, a > \ixnsupy~^u{iy) a.s y ^ +oo by the first of relations (1.16). Thus, a"*" = /i"^ = /3 and, additionally, (1.15) is true. 2^. U (3 = a^ < -foo, then u(z) is representable in the form (1.6)-(1.8). For such functions the relations (1.15)-(1.17) with h = fS = a'^ hold. Proof of Theorem 1.4. P . Let f{z) G HP{n{x)dx). Then \f{z)\P has a harmonic majorant in G"^ by Theorem 1.1. Thus, f{z) belongs to the conformal mapping of Hardy's class, and it has nontangential boundary values f{x) almost everywhere on (—OO,+CXD). Hence R

pR

X < +00, i? > 0,

/

\f{x)\Pn{x)dx

< liminf /

-R

2/-^+0

(3.3)

\f{x + iy)\Pn{x)d:

J-R

by Fatou's lemma, and f{x) € Lp{Q,{x)dx). Now let f{z) be from the conformal mapping of Hardy's class H^ {\z\ < 1), and let f{x) G Lp{fl{x)dx). Then, using the factorization of f{z) one can obtain

On the other hand, it can be easily verified that for almost all t G (—oo, +00) Q,{x)dx

-1

< C*nit),

0 0 depending only on R and Q,{x). Consequently, R

/

pR

\f{x + iy)\Pn{x)dx< / \f{t)\Pn{t)dt (3.5) -R J-R by Lebesgue's theorem. The relation (1.27) follows from (3.4). 2^. If / ( z ) G HP{Q{x)dx), then f(z) is from the conformal mapping of Hardy's class. Using the factorization of this function we obtain (1.26). Now let (1.24) be replaced by (1.26). Then evidently log"^ \f{z)\ has a harmonic

SUBHARMONIC FUNCTIONS WITH NONNEGATIVE . . .

145

majorant in G+, i.e. f{z) is of bounded type in G^. Therefore, |/(i?e*'^)|^ is continuous in 0 < i? < TT for almost all i? > 0. Besides, liminf / ?/->+0 J-R

dx \f{x + iy)\P- —^ 1

0. Therefore, by Theorem 2.1, \f{z)\^ has a harmonic majorant in any C^ (JR > 0), i.e. in any G^ the function f{z) is from the conformal mapping of Hardy's class. The transformation of the corresponding factorization by means of the conformal mapping of |z| < 1 onto G^ gives

log|/WI= E l°g z,eG+

z — Zk R — zzk z - ZkB? - zzk I d^

Here ZA; are the zeros of f{z) and dii{t) = \og\f{t)\dt — duj{t), where uj{t) is a nondecreasing function such that (jo'{t) = 0 almost everywhere in {—R^R). But we have already proved that log"^ \fi^)\ has a harmonic majorant in G^. Therefore it is obvious that the passage R -^ +oo in (3.6) leads to the factorization

(2: e G+), where £i/i(i) = log |/(t)|di - duj{t), Im C = 0 and /i=-

lim

4 /

log|/(i?e*'')|sin^di? 0. Hence J ^ 0. Then the Green type potential (1.2) has the following properties: 1°. Ioc{w) G M/3 for any /? G (0,1 + a). 2°.

W~^Iot(w) is a continuous superharmonic function in G",

W-^Ia.{w) = - j j

W-"log|6.(^,C)IMC)>0,

weG-,

(1.10)

where the integral is uniformly convergent in any half-plane G~ (p < 0) and

150

CHAPTER 8 7,

+00

W-^Ia{-it) J < +00.

(1.11)

/ is representable as an ordinary Green potential:

3°. W~^Ia{w) W-''Ia{w)

=- J j

log\bo{wX)\diya{C).

weG-,

(1.12)

where I'aiC) ^^ ^ nonnegative Borel measure satisfying (1.1) for a = 0. Proof. 1°. Let po < 0 and w = u-{-iv e C^Q. Assuming a < ^ < 1 + o; set I - / Jo

a^-' da

I log \bc,iw - ia, QWdi^iO J JG-

^ \IL G-^IIG-I

(/^°° -00.

2.1. Our aim is to find Riesz type representations for functions from SaL e m m a 2 . 1 . / / U{w) e Sa {c^>0), then its associated measure u{Q satisfies (1.1) and U{w) = l j

log\ba{w,0\du{0

+ U4w),

weG-,

(2.1)

W E I G H T E D CLASSES O F SUBHARMONIC F U N C T I O N S

153

where U^{w) is a harmonic function of SaProof. Let a > 0, and let J9 be a bounded domain such that D C G~. Then Uoiw) = U{w) - 11

logK{w,0\

dv{Q

is a function subharmonic in G~ and harmonic in D, and by Lemma 1.3 UD{W) e Mp. Therefore W-'^UDiw) = W-^'Uiw) - II

PF-^log|6a(^,C)l ^KC), ^ e G",

(2.2)

by Lemma 1.2, and consequently W^Uoiw)

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

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{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+«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,|)^-^ 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
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|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(-)(^) (-7ro(r)|

o(r)] 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)ro(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
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).

174

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\\^^

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)

178

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)-\

lmwo 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

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{irciXkrT 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).

02(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