Operator Theory: Advances and Applications Vol. 160 Editor: I. Gohberg Editorial Office: H. G. Kaper (Argonne) School of Mathematical S. T. Kuroda (Tokyo) Sciences P. Lancaster (Calgary) Tel Aviv University L. E. Lerer (Haifa) Ramat Aviv, Israel B. Mityagin (Columbus) V. V. Peller (Manhattan, Kansas) Editorial Board: L. Rodman (Williamsburg) D. Alpay (Beer-Sheva) J. Rovnyak (Charlottesville) J. Arazy (Haifa) D. E. Sarason (Berkeley) A. Atzmon (Tel Aviv) I. M. Spitkovsky (Williamsburg) J. A. Ball (Blacksburg) S. Treil (Providence) A. Ben-Artzi (Tel Aviv) H. Upmeier (Marburg) H. Bercovici (Bloomington) S. M. Verduyn Lunel (Leiden) A. Böttcher (Chemnitz) D. Voiculescu (Berkeley) K. Clancey (Athens, USA) H. Widom (Santa Cruz) L. A. Coburn (Buffalo) D. Xia (Nashville) K. R. Davidson (Waterloo, Ontario) D. Yafaev (Rennes) R. G. Douglas (College Station) A. Dijksma (Groningen) Honorary and Advisory H. Dym (Rehovot) Editorial Board: P. A. Fuhrmann (Beer Sheva) C. Foias (Bloomington) B. Gramsch (Mainz) P. R. Halmos (Santa Clara) G. Heinig (Chemnitz) T. Kailath (Stanford) J. A. Helton (La Jolla) P. D. Lax (New York) M. A. Kaashoek (Amsterdam) M. S. Livsic (Beer Sheva) Recent Advances in Operator Theory and its Applications The Israel Gohberg Anniversary Volume International Workshop on Operator Theory and its Applications IWOTA 2003, Cagliari, Italy Marinus A. Kaashoek Sebastiano Seatzu Cornelis van der Mee Editors Birkhäuser Verlag . . Basel Boston Berlin Editors: Marinus A. Kaashoek Sebastiano Seatzu Department of Mathematics, FEW Cornelis van der Mee Vrije Universiteit Dipartimento di Matematica De Boelelaan 1081A Università di Cagliari 1081 HV Amsterdam Viale Merello 92 The Netherlands 09123 Cagliari e-mail: [email protected] Italy e-mail: [email protected] [email protected] 2000 Mathematics Subject Classification 34, 35, 45, 47, 65, 93 A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. ISBN 3-7643-7290-7 Birkhäuser Verlag, Basel – Boston – Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2005 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Member of the BertelsmannSpringer Publishing Group Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Cover design: Heinz Hiltbrunner, Basel Printed in Germany ISBN 10: 3-7643-7290-7 e-ISBN: 3-7643-7398-9 ISBN 13: 978-3-7643-7290-3 9 8 7 6 5 4 3 2 1 www.birkhauser.ch Contents Editorial Preface .......................................................... vii T. Aktosun, M.H. Borkowski, A.J. Cramer and L.C. Pittman Inverse Scattering with Rational Scattering Coefficients and Wave Propagationin Nonhomogeneous Media ....................... 1 T. Ando Aluthge Transforms and the Convex Hull of the Spectrum of a Hilbert Space Operator ......................................... 21 W. Bhosri, A.E. Frazho and B. Yagci Maximal Nevanlinna-Pick Interpolation for Points in the Open Unit Disc ............................................... 41 M.R. Capobianco, G. Criscuolo and P. Junghanns On the Numerical Solution of a Nonlinear Integral Equation of Prandtl’s Type .................................................... 53 M. Cappiello Fourier Integral Operators and Gelfand-Shilov Spaces ................ 81 D.Z. Arov and H. Dym Strongly Regular J-inner Matrix-valued Functions and Inverse Problems for Canonical Systems ......................... 101 C. Estatico Regularization Processes for Real Functions and Ill-posed Toeplitz Problems ..................................... 161 K. Galkowski Minimal State-space Realization for a Class of nD Systems .......... 179 G. Garello and A. Morando Continuity in Weighted Besov Spaces for Pseudodifferential Operators with Non-regular Symbols ................................ 195 G.J. Groenewald and M.A. Kaashoek A New Proof of an Ellis-Gohberg Theorem on Orthogonal Matrix Functions Related to the Nehari Problem .................... 217 vi Contents G. Heinig and K. Rost Schur-type Algorithms for the Solution of Hermitian Toeplitz Systems via Factorization ................................... 233 M. Kaltenba¨ck, H. Winkler and H. Woracek Almost PontryaginSpaces ........................................... 253 D.S. Kalyuzhny˘ı-Verbovetzki˘ı Multivariable ρ-contractions ......................................... 273 V. Kostrykin and K.A. Makarov The Singularly Continuous Spectrum and Non-Closed Invariant Subspaces ................................................. 299 G. Mastroianni, M.G. Russo and W. Themistoclakis Numerical Methods for Cauchy Singular Integral Equations in Spaces of Weighted Continuous Functions ......................... 311 A. Oliaro On a Gevrey-Nonsolvable Partial Differential Operator ............... 337 V. Olshevsky and L. Sakhnovich Optimal Prediction of Generalized Stationary Processes .............. 357 P. Rocha, P. Vettori and J.C. Willems Symmetries of 2D Discrete-Time Linear Systems ..................... 367 G. Rodriguez, S. Seatzu and D. Theis An Algorithm for Solving Toeplitz Systems by Embedding in Infinite Systems ................................................... 383 B. Silbermann Fredholm Theory and Numerical Linear Algebra ..................... 403 C.V.M. van der Mee and A.C.M. Ran Additive and Multiplicative Perturbations of Exponentially Dichotomous Operators on General Banach Spaces ................... 413 C.V.M. van der Mee, L. Rodman and I.M. Spitkovsky Factorization of Block Triangular Matrix Functions with Off-diagonal Binomials ......................................... 425 G. Wanjala Closely Connected Unitary Realizations of the Solutions to the Basic Interpolation Problem for Generalized Schur Functions ......... 441 M.W. Wong Trace-Class Weyl Transforms ........................................ 469 Editorial Preface Thisvolumecontainsaselectionofpapersinmodernoperatortheoryanditsappli- cations. Most of them are directly related to lectures presented at the Fourteenth InternationalWorkshop on Operator Theory and its Applications (IWOTA 2003) held at the University of Cagliari,Italy, in the period of June 24–27, 2003. The workshop, which was attended by 108 mathematicians – including a number of PhD and postdoctoral students – from 22 countries, presented eight special sessions on 1) control theory, 2) interpolation theory, 3) inverse scattering, 4) numerical estimates for operators, 5) numerical treatment of integral equations, 6) pseudodifferential operators, 7) realizations and transformations of analytic functions and indefinite inner product spaces, and 8) structured matrices. The program consisted of 19 plenary lectures of 45 minutes and 78 lectures of 30 minutes in four parallel sessions. The present volume reflects the wide range and rich variety of topics pre- sentedanddiscussedattheworkshop,bothwithinandoutsidethespecialsessions. The papers deal with inverse scattering, numerical ranges, pseudodifferential op- erators,numericalanalysis,interpolation theory, multidimensional system theory, indefinite inner products, spectral factorization, and stationary processes. SinceintheperiodthattheproceedingsofIWOTA2003werebeingprepared, Israel Gohberg, the president of the IWOTA steering committee, reached the age of 75, we decided to dedicate these proceedings to Israel Gohberg on the occasion of his 75th birthday.Alloftheauthorsoftheseproceedingshavejoinedtheeditors and dedicated their papers to him as well. The Editors Israel Gohberg, the president of the IWOTA steering committee OperatorTheory: Advances andApplications,Vol.160,1–20 (cid:1)c 2005Birkh¨auserVerlagBasel/Switzerland Inverse Scattering with Rational Scattering Coefficients and Wave Propagation in Nonhomogeneous Media Tuncay Aktosun, Michael H. Borkowski, Alyssa J. Cramer and Lance C. Pittman Dedicated toIsrael Gohberg on the occasion of his 75th birthday Abstract. Theinversescatteringproblemfortheone-dimensionalSchr¨odinger equationisconsideredwhenthepotentialisrealvaluedandintegrableandhas afinitefirst-moment andno boundstates. Corresponding to suchpotentials, forrationalreflectioncoefficientswithonlysimplepolesintheupperhalfcom- plex plane, amethod is presented to recoverthepotential and thescattering solutions explicitly.A numericalimplementation of themethod is developed. For such rational reflection coefficients, the scattering wave solutions to the plasma-wave equation are constructed explicitly. The discontinuities in these wavesolutionsandintheirspatialderivativesareexpressedexplicitlyinterms of the potential. Mathematics SubjectClassification (2000). Primary 34A55; Secondary 34L40 35L05 47E05 81U40. Keywords.Inversescattering,Schro¨dingerequation,Rationalscatteringcoef- ficients, Wavepropagation, Plasma-wave equation. 1. Introduction Consider the Schro¨dinger equation ψ(cid:1)(cid:1)(k,x)+k2ψ(k,x)=V(x)ψ(k,x), x∈R, (1.1) The research leading to this paper was supported by the National Science Foundation under grants DMS-0243673 and DMS-0204436 and by the Department of Energy under grant DE- FG02-01ER45951. 2 T. Aktosun, M.H. Borkowski,A.J. Cramer and L.C. Pittman where the prime denotes the x-derivative, and the potential V is assumed to have no bound states and to belong to the Faddeev class. The bound states of (1.1) correspondtoitssquare-integrablesolutions.Bythe(cid:1)Faddeevclasswemeantheset ofreal-valuedandmeasurablepotentialsfor which ∞ dx(1+|x|)|V(x)| is finite. −∞ Via the Fourier transformation (cid:2) ∞ 1 u(x,t)= dkψ(k,x)e−ikt, 2π −∞ we can transform (1.1) into the plasma-waveequation ∂2u(x,t) ∂2u(x,t) − =V(x)u(x,t), x,t∈R. (1.2) ∂x2 ∂t2 In the absence of bound states, (1.1) does not have any bounded solutions for k2 < 0. The solutions for k2 > 0 are known as the scattering solutions. Each scattering solution can be expressed as a linear combination of the two (linearly- independent) Jost solutions from the left and the right, denoted by fl and fr, respectively, satisfying the respective asymptotic conditions fl(k,x)=eikx[1+o(1)], fl(cid:1)(k,x)=ikeikx[1+o(1)], x→+∞, fr(k,x)=e−ikx[1+o(1)], fr(cid:1)(k,x)=−ike−ikx[1+o(1)], x→−∞. We have 1 L(k) fl(k,x)= eikx+ e−ikx+o(1), x→−∞, T(k) T(k) 1 R(k) fr(k,x)= e−ikx+ eikx+o(1), x→+∞, T(k) T(k) where L and R are the left and right reflection coefficients, respectively, and T is the transmission coefficient. Thesolutionsto(1.1)fork =0requirespecialattention.Generically,fl(0,x) and fr(0,x) are linearly independent on R, and we have T(0)=0, R(0)=L(0)=−1. In the exceptional case, fl(0,x) and fr(0,x) are linearly dependent on R and we have (cid:3) T(0)= 1−R(0)2 >0, −1<R(0)=−L(0)<1. When V belongs to the Faddeev class and has no bound states, it is known [1–5]thateitheroneofthereflectioncoefficientsRandLcontainstheappropriate informationtoconstructtheotherreflectioncoefficient,thetransmissioncoefficient T, the potential V, and the Jost solutions fl and fr. Our aim in this paper is to present explicit formulas for such a construction when the reflection coefficients arerationalfunctions ofk with simple poles onthe upper half complex planeC+. We will use C− to denote the lower half complex plane and let C+ := C+ ∪R and C− :=C−∪R. The recovery of V from a reflection coefficient constitutes the inverse scat- teringproblemfor(1.1).Therehasbeenasubstantialamountofpreviouswork[2,