Constructive Methods of Wiener-Hopf Factorization

417 Pages·1986·9.4 MB·English

by  I. Gohberg

OT21:
Operator Theory: Advances and Applications
Vol. 21
Editor:
I. Gohberg
Tel Aviv University
Ramat -Aviv, Israel
Editorial Office
School of Mathematical Sciences
Tel Aviv University
Ramat -Aviv, Israel
Editorial Board
A. Atzmon (Tel-Aviv) T. Kailath (Stanford)
1. A. Ball (Blacksburg) H.G. Kaper (Argonne)
K Clancey (Athens, USA) S. T. Kuroda (Tokyo)
L. A. Coburn (Buffalo) P. Lancaster (Calgary)
R. G. Douglas (Stony Brook) L. E. Lerer (Haifa)
H. Dym (Rehovot) M. S. Livsic (Beer Sheva)
A. Dynin (Columbus) E. Meister (Darmstadt)
P. A. Fillmore (Halifax) B. Mityagin (Columbus)
C. Foias (Bloomington) 1. D. Pincus (Stony Brook)
P. A. Fuhrmann (Beer Sheva) M. Rosenblum (Charlottesville)
S. Goldberg (College Park) 1. Rovnyak (Charlottesville)
B. Gramsch (Mainz) D. E. Sarason (Berkeley)
1. A. Helton (La Jolla) H. Widom (Santa Cruz)
D. Herrero (Tempe) D. Xia (Nashville)
M. A. Kaashoek (Amsterdam)
Honorary and Advisory
Editorial Board
P. R. Halmos (Bloomington) R. Phillips (Stanford)
T. Kato (Berkeley) B. Sz.-Nagy (Szeged)
S. G. Mikhlin (Leningrad)
Birkhauser Verlag
Basel . Boston· Stuttgart
Constructive Methods of
Wiener-Hopf Factorization
Edited by
I. Gohberg
M. A. Kaashoek
1986 Birkhauser Verlag
Basel . Boston· Stuttgart
Volume Editorial Office
Department of Mathematics and Computer Science
Vrije Universiteit
P. O. Box 7161
1007 Me Amsterdam
The Netherlands
Library of Congress Cataloging in Publication Data
Constructive methods of Wiener-Hopf factorization.
(Operator theory, advances and applications;
vol. 21)
Includes bibliographies and index.
1. Wiener-Hopf operators. 2. Factorization of
operators. I. Gohberg, I. (Israel), 1928-
II. Kaashoek, M. A. III. Series: Operator theory,
advances and applications; v.21.
QA329.2.C665 1986 515.7'246 86--21587
CIP-Kurztitelaufnabme der Deutsdlen Bibliothek
Constructive methods of Wiener-Hopf factorization
/ ed. by I. Gohberg ; M. A. Kaashoek. - Basel ;
Boston ; Stuttgart : Birkhiiuser, 1986.
(Operator theory ; Vol. 21)
NE: Gohberg, Israel [Hrsg.]; GT
All rights reserved.
No part of this publication may be reproduced, stored in a retrieval
system, or transmitted in any form or by any means, electronic,
mechanical, photocopying, recording or otherwise, without the prior
permission of the copyright owner.
ISBN-13: 978-3-0348-7420-5 e-ISBN-13: 978-3-0348-7418-2
DOl: 10.1007/978-3-0348-7418-2
© 1986 Birkhiiuser Verlag Basel
softcover reprint of the hardcover 1st edition 1986
v
7hih volume conhihth ot a helection ot pape~h conce~­
ning a new app~oach to the p~otlem ot &iene~-Hopt tacto~ization
to~ ~ational and analytic mat~ix-valued (o~ ope~ato~-valued)
tunctionh. It ih a ~ehult ot developmenth which took place du~ing
the paht ten yea~h. 7he main advantage ot thih new app~oach ih
that it allowh one to get t1e &iene~-Hopt tacto~ization explicit
ly in te~mh ot the o~iginal tunction. 7he hta~ting point ih a
hpecial ~ep~ehentation ot the tunction which ih taken t~om
~athematical SYhtemh 7heo~y whe~e it ih known ah a ~ealization.
10~ the cahe ot ~ational mat~ix-valued tunctionh the tinal
theo~emh exp~ehh the tacto~h in the tacto~ization and the indiceh
in te~mh ot the th~ee mat~iceh which appea~ in the ~ealization.
7hih took conhi~th ot two pa~th. Pa~t I conce~nh canon
ical and, mo~e gene~ally, minimal tacto~ization. Pa~t II ih
dedicated to non-canonical &iene~-Hopt tacto~ization (i.e., the
tacto~ization indiceh a~e not all ze~o). Each pa~t hta~th with
an edito~ial int~oduction which containh hho~t dehc~iptionh ot
each ot the pape~h.
7hih took ih a ~ehult ot ~ehea~ch which to~ a la~ge
pa~t wah done at the V~~e linive~hiteit at Amhte~dam and wah
hta~ted atout ten yea~h ago. It ih a pleahu~e to thank the
depa~tment ot ~athematich and Compute~ Science ot the V~~e
linive~hiteit to~ ith huppo~t and unde~htanding du~ing all thohe
yea~h. &e alho like to thank the Economet~ich Inhtitute ot the
E~ahmuh linive~hiteit at Rotte~dam to~ ith technical ahhihtance
with the p~epa~ationh ot thih volume.
Amhte~dam, June 7986 I. 90hte~g, ~.A. Kaahhoek
VII
TABLE OF CONTENTS
PART I CANONICAL AND MINIMAL FACTORIZATION •..•.•.•... 1
Edi torial introduction ..•..........••..•.•...•.......•. 1
J.A. Ball and A.C.M. Ran: LEFT VERSUS RIGHT CANONICAL
FACTORIZATION •.•••••.•••••.•••••..••..•......•..•• 9
1. Introduction ••.•..•••....•••..•••...••..•••.•.. 9
2. Left and right canonical Wiener-Hopf
factorization ....••....•••..•.•••.••••.••••..•• 11
3. Application to singular integral operators ..••• 19
4. Spectral and antispectral factorization on the
unit circle ......•••....•••....••.••••...••.... 22
5. Symmetrized left and right canonical spectral
factorization on the imaginary axis ...•.•.•.... 33
References .........•.......•...••••...•.•......... 37
H. Bart, I. Gohberg and M.A. Kaashoek: WIENER-HOPF
EQUATIONS WITH SYMBOLS ANALYTIC IN A STRIP •...•••. 39
O. Introduction .••..••...••..•.••.•..•••••.•..•••. 39
I. Realization •.•.•...•.•....•.•.....•...•...•.... 41
1. Preliminaries •.•..••••...••.••..•...•••.••.. 41
2. Realization triples •....••.......••.•.••••.• 43
3. The realization theorem ..••...•••.••.••••••. 47
4. Construction of realization triples ..••••••. 49
5. Basic properties of realization triples •..•• 51
II. Applications .••...•.....••.......•.•.••••..... 55
1. Inverse Fourier transforms ..•.•••••..••••••• 55
VIII
2. Coupling ................................... . 57
3. Inversion and Fredholm properties .......... . 62
4. Canonical Wiener-Hopf factorization ........ . 66
5. The Riemann-Hilbert boundary value problem .. 71
References ....................................... . 72
I. Gohberg, M.A. Kaashoek, L. Lerer and L. Rodman: ON
TOEPLITZ AND WIENER-HOPF OPERATORS WITH CONTOUR-
WISE RATIONAL MATRIX AND OPERATOR SyMBOLS......... 75
O. Introduction................................... 76
1. Indicator...................................... 78
2. Toeplitz operators on compounded contours...... 81
3. Proof of the main theorems..................... 84
4. The barrier problem............................ 100
5. Canonical factorization........................ 102
6. Unbounded domains.............................. 107
7. The pair equation.............................. 112
8. Wiener-Hopf equation with two kernels.......... 119
9. The discrete case.............................. 123
References........................................ 125
L. Roozemond: CANONICAL PSEUDO-SPECTRAL FACTORIZATION
AND WIENER-HOPF INTEGRAL EqUATIONS................ 127
O. Introduction................................... 127
1. Canonical pseudo-spectral factorizations....... 130
2. Pseudo-f-spectral subspaces.................... 133
3. Description of all canonical pseudo-f-spectral
factorizations................................. 135
4. Non-negative rational matrix functions......... 144
5. Wiener-Hopf integral equations of non-normal
type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6. Pairs of function spaces of unique solvability. 149
References........................................ 156
IX
I. Gohberg and M.A. Kaashoek:MINIMAL FACTORIZATION OF
INTEGRAL OPERATORS AND CASCADE DECOMPOSITIONS OF
SySTEMS........................................... 157
O. Introduction................................... 157
I. Main results................................... 159
1. Minimal representation and degree........... 160
2. Minimal factorization (1)................... 161
3. Minimal factorization of Volterra integral
operators (1)............................... 164
4. Stationary causal operators and transfer
0,',.............
functions ••.•....•.••.•....• ~'. 168
5. SB-minimal factorization (1)................ 172
6. SB-minimal factorization in the class (USB). 174
7. Analytic semi-separable kernels............. 175
8. LU- and UL-factorizations (1)............... 175
II. Cascade decomposition of systems.............. 178
1. Preliminaries about systems with boundary
conditions.................................. 178
2. Cascade decompositions...................... 182
3. Decomposing projections..................... 182
4. Main decomposition theorems................. 184
5. Proof of Theorem II.4.1..................... 186
6. Proof of Theorem II.4.2..................... 191
7. Proof of Theorem II.4.3..................... 195
8. Decomposing projections for inverse systems. 198
III. Proofs of the main theorems.................. 202
1. A factorization lemma....................... 202
2. Minimal factorization (2)................... 203
3. SB-minimal factorization (2)................ 208
4. Proof of Theorem I.6.1...................... 211
5. Minimal factorization of Volterra integral
operators (2)......... .•.••••••••••••. .•••.• 215
6. Proof of Theorem I. 4 .1. . . • . • • . • • • • • • • • • . • . • • 220
7. A remark about minimal factorization and
inversion. • • • . • • . •• • . . • • • . • . . . . • • • • • • • .. • •. • 222
8. LU- and UL-factorizations (2)............... 222
x
9. Causal/anticausal decompositions............ 225
References. . . . . . . . . . . . . . . . • . . . . . . . . • . . . . . . . . . . . . • . 229
PART II NON-CANONICAL WIENER-HOPF FACTORIZATION ..... . 231
Edi torial introduction................................. 231
H. Bart, I. Gohberg and M.A. Kaashoek: EXPLICIT WIENER-
HOPF FACTORIZATION AND REALIZATION................ 235
O. Introduction................................... 235
I. Preliminaries.................................. 237
1. Peliminaries about transfer functions....... 237
2. Preliminaries about Wiener-Hopf
factorization............................... 240
3. Reduction of factorization to nodes with
centralized singularities................... 243
II. Incoming characteristics...................... 254
1. Incoming bases.............................. 254
2. Feedback operators related to incoming bases 262
3. Factorization with non-negative indices..... 268
III. Outgoing characteristics..................... 272
1. Outgoing bases.............................. 272
2. Output injection operators related to out-
going bases................................. 277
3. Factorization with non-positive indices..... 280
IV. Main results.................................. 285
1. Intertwining relations for incoming and out-
data.................................. 285
goin~
2. Dilation to a node with centralized singula-
ri ties. . •• . • . . . . . ••. • .. • 291
3. Main theorem and corollaries................ 303
References. . . . . . . . . . . . .. ... . . . . . . . . . .. . . .... . . . .. . . . 314
H. Bart, I. Gohberg and M.A. Kaashoek: INVARIANTS FOR
WIENER-HOPF EQUIVALENCE OF ANALYTIC OPERATOR
FUNCTIONS...... ........ ...•.••... .•..••.••••....•• 317
XI
1. Introduction and main result................... 317
2. Simple nodes with centralized singularities.... 322
3. Multiplication by plus and minus terms......... 326
4. Dilation....................................... 334
5. Spectral characteristics of transfer functions:
outgoing spaces................................ 338
6. Spectral characteristics of transfer functions:
incoming spaces................................ 343
7. Spectral characteristics and Wiener-Hopf equi-
valence. . . . • • . . . • . . • . . . . . . . . . . . . . • . . . . . . • • • • . • • 352
References. • . . . . . . . . . . . . . . . . • . . . . . . . . • • • . . . . . . . • . . 354
H. Bart, I. Gohberg and M.A. Kaashoek: MULTIPLICATION
BY DIAGONALS AND REDUCTION TO CANONICAL FACTOR-
IZATION................ ..•.•....... .•. ••• .•••..... 357
1. Introduction........................ .•• .•. .•... 357
2. Spectral pairs associated with products of
nodes. . • • . • . . . . . • . . . . . . . . . • • . . • . . . . . . • . . . . . . . • . 359
3. Multiplication by diagonals.................... 361
References........................................ 371
M.A. Kaashoek and A.C.M. Ran: SYMMETRIC WIENER-HOPF
FACTORIZATION OF SELF-ADJOINT RATIONAL MATRIX
FUNCTIONS AND REALIZATION......................... 373
O. Introduction and summary....................... 373
1. Introduction................................ 373
2. Summary..................................... 374
I. Wiener-Hopf factorization...................... 379
1. Realizations with centralized singularities. 379
2. Incoming data and related feedback operators 381
3. Outgoing data and related output injection
operators................................... 383
4. Dilation to realizations with centralized
singulari ties............................... 385
5. The final formulas.......................... 395