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Constructive Methods of Wiener-Hopf Factorization PDF

417 Pages·1986·9.4 MB·English
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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 functions ••.•....•.••.•....• ~'. 0,',............. 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- goin~ data.................................. 285 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

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