Table Of ContentIntroduction to the Theory of Singular Integral Operators with Shift
Mathematics and Its Applications
Managing Edilor:
M. HAZEWINKEL
Centre for MatMmatics wuI Computer Scknce, A.msterdam, The Netherlands
Volume 289
Introduction to the Theory
of Singular Integral
Operators with Shift
by
Victor G. Kravchenko
Ukrainian State Hydrometeorologicallnstitute,
Odessa, Ukraine
and
Georgii S. Litvinchuk
Ukrainiall State U/Jiversity,
Odessa, Ukraine
SPRINGER-SCIENCE+BUSINESS MEDIA, B.Y.
Library of Congress Cataloging-in-Publication Data
Kravchenko. V,ctor G.
Introductlon to the theory of singular lntegral operators .,th
shlft I by V,ctor G. Kravchenko and G.orgll S. Lltvlnchuk.
p. cm. -- (Mathematlcs and ltS appl icatlons ; v. 289)
Includes blbllographlcal references and index.
ISBN 978-94-010-4515-5 ISBN 978-94-011-1180-5 (eBook)
DOI 10.1007/978-94-011-1180-5
1. Integral operators. 1. Lltvlnchuk. G. S. IGeorgli
Semenovlchl. 1931- Ir. Tltle. II!. Series, Mathematlcs and lts
appllcations (Kluwer Academlc Publlshers) ; v. 289.
OA329.6.K73 1994
515' .723--dc20
ISBN 978-94-010-4515-5
Printed on acid-free paper
The manuscript was translated from Russian by
O.S. Litvinchuk
All Rights Reserved
© 1994 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1994
Softcover reprint ofthe hardcover 1st edition 1994
No part of the material protected by this copyright notice may be reproduced or
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CONTENTS
Introduction ...........................................................•.........••. . .. xi
Chapter 1. Background information . . . . . . . . . . . . . . . . . . . I
§1. On NoetherilLfl operllton ................. 1
§2. On the operator of 8i"~ular integration .................... 15
§3. On the shift function and shift operator ............ 26
§4. On C··algebras .................... . . ................ 33
Chapter 2. No~therity criterion and a formula for the index of a singular integral
functional operator of first order in the continuous case ........... . .37
§1. Criterion of Noetherily for singular integral functional operators of first order
with orientation-preserving shift ...... . . .................. .40
1.1. Criterion of Noetherity for the singular integral operator with a Cauchy
kernel .......................................... . .......... 40
1.2. Ab5trl\ct ~cht:lIle for tin: study of tile Nodherily of paired operators ... .41
1.3. A criterion of NOf:lherily for singular integral {unctional operators of the
fint order in the cue of a Carleman shift ......... . ............... 44
1.4. A criterion of Noetherity for singular integral functional operators of first
order in the case of a onn-Carleman shift with a finite number of fixed
points .................... 46
1.5. Noetherity criterion for a singular integral functional operator of first Of
der in the case of a non·Carlem1\n shift having a finite number of periodic
points ...... . . ............. 55
1.6. No(;thcrity criterion for a singular integral functional operator of first order
with a shift having any non-empty set of periodic points ... 56
§2. The calculation of the index of a singular integral functional operator of the fiNlt
ord~ with a .bin prt:llervillg the orientation ................... . .. ..... 60
2.1. The index of a singular integl'al operator with a Cauchy kernel ......... 60
vi TA8LE: Of' CONTENTS
2.2. The index of a singular integra.l fun<:tiona! operator of tbe Kveselava,-Vekua
type............ . ....................................... 62
2.3. The calculation of the index of a singular integral functional operator of the
first order in the case of Carleman shift ...... . ................ 66
2.4. The calculation of the index of a. singular integro1l functional operator of the
fint order with a shift having a finite number of fixed points ........... 71
2.5. The calculation of the index of a singular integral functional operator of
the first order with a shift having a finite number of periodic points with
multiplicity k ........ . .......................... 72
2.6. The calculation of the index of a singular integral fundional operator of
the first order witb a shift having an arbitrary non-empty set of periodic
poinls ........... . ............... 73
§3. The Noetherily criterion and the index formula for a singular integral functional
operator of the fint order with a shift changing the orientation .. ..... . ... 81
3.1. Reduced and associated operators. The conneetion between the Noetherity
of a singular integral functional operator .... ith a shift changing the orienta.·
tion and the Noetherity of a reduced operator ... 81
3.2. Noetherity criterion and index forllluill. for a singular integral functiollaJ op
erator of the first order with a Carleman shift changing the
orientation . . . .. .. . . . ......... 84
3.3. Nocthcrity criterion and index formula for /I. singular integral functional op
erator of the flut order with a. non-Carleman shift changing the
orientation ... 86
§ 4. References and a survey of similar or closed rt.'lIulh .... 89
4.1. On weighted-shift operators ... 89
4.2. On the one-sided invertibility of functional operators and on the semi-Noetherity
of singular integral functional operators of the first order in the space
4(r).. . ............. ~
4.3. On functiona.l a.nd singular integra! functional operators of the firet order in
Holder and Orlicz spaces ..... . ..... 94
4.4. On functional and singular integrlll functional operator. of the first order with
a non-invertible shift ...................................................... 96
TABLE OF CONTENTS vii
4.5. On singular integral functional operators of the first order with a shift whicb ba.s
an empty set of periodic points ........................................ 97
4.6. For the question of invertibility and one-sided invertibility of singula.r integral
functional operators on a dosed contour ......... . .................... 98
Chapter 3. The Noether theory of a singular integral functional operator of fl.nite order
in the continuous case.......... .................. . ............... 101
§ J. The Noetherity criterion and the index formula for a system of singular in
tegral equa.tions with a. Cauchy kernd and continuous coefficientB on a dooed
contour ... .................................................... 104
§ 2. Theorems concerning decreasing the order of fundional and singular integral
operators . . . . . ... . . ........... 107
§ 3. Noetherity criterion and a formula for the index for systems of singular integral
equation with a Carlcman 5hift ........................................... Il3
3.1. The case of an orientation-preserving Carleman shift .. 113
3.2. The case of an orientation-changing Carleman shift ... 117
§4. An invertibility criterion for a matrix functional operator with a non-Carleman
shift... ............................... .. ........... 118
4.1. The reASOII$ for the ne«sury conditions of invertibility .... 1I8
4.2. Invertibility conditions in the cas~ K(a): 0 and I(a) ... n ... 120
4.3. On the invertibility of a canonical functional operator with a matrix function
a-reducible to the normal form ." .... ,.,.,', ................... , .... 121
4.4. a-reducibility of a boclk triangular matrix to the normal form ........ 123
4.S. A normaliuble matrix of solutions of a functional equation system. A con-
nection between a-reducibility and normalizability ................... 124
4.6. Lemma on the existence of a normalizable matrix of continuous solutions of
a functional equation .......................................... 127
4.7. Local a-reducibility .. . ........... 133
4.8. On local properties of fundamental solutions .. 134
4.9. A fundamental system of solutions and its connection with the invertibility
of the canonical operator ................ . . .................... 137
4.10. A connection between the nOl"mali1.ability and existence of a rundamental
system of solution ... . ........................... 140
viii TABLE OF CONTENTS
4.11. A criterion of invertibility for a canonical functional operator ......... 141
4.12. A criterion for the invertibility of a functiona.l opera.tor of the order m. The
C&'!e of a. finite set of fixed points of shift ............. 142
4.13. A malrix functional operator of the order m belongs to the set
,4>
M(S) ...............................................................
4.14. Au illvertibility criterion for a. functiollal operat.or of the order m. The Cale
of a finite set of periodic points of the shift ... . ............. 146
§ 5. Noether theory for singular integral functional operators of superior
orden ..... 148
5.1. The case of a shift that preserves the orientation on rand hlUl i!L finite
number of fixed points ............................ 148
5.2. The ca.se of a ,hift preserving the orientation on r with periodic
points 15'
r
5.3. The cue of a shift changing the orientation on .152
§ 6. References and a ,urvey of similar results ........... . . ..... 154
6.1. The invertibility of the matrix functional operator and its connection with
the spectral theory of dynamics systems and the multiplicative ergodic the-
orem ........... . . .. .. 15'1
Chapter 4. The Noether theory of singular integral functional operators with contin-
uous coefficients on a non-closed contour ...................... 161
§ I. The Nodhcl"ity criterion ami the index formulll for Il singular integrlll operillor
with continuous coefficients on a non·closed conto.n ...................... 1M
1.1. An abstract scheme for the study of the Noethel"ity of paired operators on
a non·closed contour .164
1.2. The reduction of the (I - S')·Noetherity of a singular integral operator
with Cauchy kernel and continuous coefficients on ~he segment [0,1) to
the continuous invcrtibility of a. singular integral operator with constant
coefficients ......... 168
1.3. The spectral representation of the operator S. An invertibility criterion fOf
singular integral operators with conslant coefficients... . ..... 169
1.4. The Noetherity criterion and the index formula for a singular integral oper·
ator with Cauchy kernel ......... . .... 172
TABLE OF CONTENTS
1.5. Open,tors with fixed singularities .. 177
1.6. The Noetherity criterion and the index formula for o~rators of the
algebra L .................................. . .... 180
1.7. Systems of singul&T integral equations on a non·dosed contour ..... 184
§ 2. The Noetherily criterion and the index formula for singular integral functional
operators with continuous coefficients on a non·c1osed contour 186
2. L On the application of the abstract scheme for the study of
Noetherity 186
2.2. Some more results about operators with fixed singularities. .. ........ 188
2.3. The Noetherity criterion for the operator N .. 192
2.1. The index formula for the oper&tor N ..... . ..191
2.5. The Kveselav&·Vekua operator on a non·c1osed contour .. 198
§ 3. Systems of! Iinglllu integral operator.'! with shift on a non·c1osed (".onlour .. 202
3.1. The criterion (or Noetherity ................. . . ........... 202
3.2. The ca.\cul&tion of the index .20'
3.3. The cas<: of orientalion.changing shift ...... . .207
§ 4. R.eferente'l &nd & survey of dosely related .·t$ults ........... 208
4.1. On the question of invcrtibility and one-sided invertibility for singular inte-
gral functional operators on a non-closed conlour ... _........ . .... 209
Chapter 5. The Noether theory in algebras of singular integral functional
operators ...... 211
§ L C·-algebras of singular integral operators .... . .... 214
1.1. General scheme ....... . ..214
1.2. Algebras of an SIO on a closed contour .. 217
1.3. Alscbras of an SIO on a nOll-dosed contour. __ . . .. _ 218
§ 2. C·-algebras of singular integral operators with a Carleman shift .. _. _._. .. _2 21
2.1. The case of a dosed oontour and an orientation-preserving shi(~ .. _ .2 21
2_2. The case of a closed contour and &n orientation.chuging ,hift ........ ..222
2.3. The ca&e of a non-dosed con lour and an orien~ation-changing CarlemiLJI
shift .. _. ... _. .......... 223
x TABLE OF CONTENTS
§ 3. C· .algebras of singular integral operators with non-Carleman shift which bas periodic
points ..... . ............. 228
3.1. The caae of Ito clmed contour ......• _. ....... 228
3.2. The case of a non-dosed oontour .. 232
3.3. Discrete operators ............. ', ...... 235
§ 4. Further development of II. local method for studying the Noethcrity of bounded linear
operators of non· local type and its applications. Commentaries to the
literature ............. . ..238
4.1. The loca.l method (see Ju. J. I{arlovich· V. G. I<ravchenko 8)) .... 239
4.2. On the local methods of Ju. I. Kllrtovic:h and their applications .... 248
4.3. On the ractoribility of almost.periodic matrix (unctions alJd the Noctherity of
certain classes of convolution lype opcraton
References .............. . ..265
Subject Index. ..287
