Operator Theory Advances and Applications 244 Daniel Alpay Bernd Kirstein Editors Recent Advances in Inverse Scattering, Schur Analysis and Stochastic Processes A Collection of Papers Dedicated to Lev Sakhnovich Operator Theory: Advances and Applications Volume 244 Founded in 1979 by Israel Gohberg Editors: Joseph A. Ball (Blacksburg, VA, USA) Harry Dym (Rehovot, Israel) Marinus A. Kaashoek (Amsterdam, The Netherlands) Heinz Langer (Wien, Austria) Christiane Tretter (Bern, Switzerland) Associate Editors: Honorary and Advisory Editorial Board: Vadim Adamyan (Odessa, Ukraine) Lewis A. Coburn (Buffalo, NY, USA) Wolfgang Arendt (Ulm, Germany) Ciprian Foias (College Station, TX, USA) Albrecht Böttcher (Chemnitz, Germany) J.William Helton (San Diego, CA, USA) B. Malcolm Brown (Cardiff, UK) Thomas Kailath (Stanford, CA, USA) Raul Curto (Iowa, IA, USA) Peter Lancaster (Calgary, Canada) Fritz Gesztesy (Columbia, MO, USA) Peter D. Lax (New York, NY, USA) Pavel Kurasov (Stockholm, Sweden) Donald Sarason (Berkeley, CA, USA) Vern Paulsen (Houston, TX, USA) Bernd Silbermann (Chemnitz, Germany) Mihai Putinar (Santa Barbara, CA, USA) Harold Widom (Santa Cruz, CA, USA) Ilya M. Spitkovsky (Williamsburg, VA, USA) Subseries Linear Operators and Linear Systems Subseries editors: Daniel Alpay (Beer Sheva, Israel) Birgit Jacob (Wuppertal, Germany) André C.M. Ran (Amsterdam, The Netherlands) Subseries Advances in Partial Differential Equations Subseries editors: Bert-Wolfgang Schulze (Potsdam, Germany) Michael Demuth (Clausthal, Germany) Jerome A. Goldstein (Memphis, TN, USA) Nobuyuki Tose (Yokohama, Japan) Ingo Witt (Göttingen, Germany) Daniel Alpay • Bernd Kirstein Editors Recent Advances in Inverse Scattering, Schur Analysis and Stochastic Processes A Collection of Papers Dedicated to Lev Sakhnovich Editors Daniel Alpay Bernd Kirstein Department of Mathematics Mathematisches Institut Ben-Gurion University of the Negev Universität Leipzig Beer Sheva, Israel Leipzig, Germany ISSN 0255-0156 ISSN 2296-4878 (electronic) Operator Theory: Advances and Applications ISBN 978-3-319-10334-1 ISBN 978-3-319-10335-8 (eBook) DOI 10.1007/978-3-319-10335-8 Library of Congress Control Number: 2015935221 Mathematics Subject Classification (2010): 47A57, 93C05, 60B20 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.birkhauser-science.com) Contents Editorial Introduction .................................................... vii Part 1: Biographical Material and List of Publications of L.A. Sakhnovich L.A. Sakhnovich – Biography ............................................ 3 List of Publications of L.A. Sakhnovich .................................. 9 L.A. Sakhnovich My Teachers and Studies ........................................... 23 Part 2: Reserach Papers D. Alpay, F. Colombo and I. Sabadini Inner Product Spaces and Krein Spaces in the Quaternionic Setting ................................................ 33 D. Alpay, P. Jorgensen, I. Lewkowicz and I. Martziano Infinite Product Representations for Kernels and Iterations of Functions .............................................. 67 Y. Arlinski˘ı and S. Hassi Q-functions and Boundary Triplets of Nonnegative Operators ...... 89 S. Boiko, V. Dubovoy and A. Kheifets On Some Special Cases of the Radon–Nikodym Theorem for Vector- and Operator-valued Measures .......................... 131 A.E. Frazho, S. ter Horst and M.A. Kaashoek State Space Formulas for a Suboptimal Rational Leech Problem II: Parametrizationof All Solutions .................................... 149 vi Contents B. Fritzsche, B. Kirstein and C. M¨adler On a Simultaneous Approach to the Even and Odd Truncated Matricial Hamburger Moment Problems ............................. 181 F. Gesztesy and R. Nichols A Jost–Pais-type Reduction of (Modified) Fredholm Determinants for Semi-separable Operators in Infinite Dimensions ................. 287 K.A. Makarov and E. Tsekanovski˘i On the Addition and Multiplication Theorems ...................... 315 J. Rovnyak and L.A. Sakhnovich On Indefinite Cases of Operator Identities Which Arise in Interpolation Theory. II .......................................... 341 A. Sakhnovich and L. Sakhnovich Nonlinear Fokker–PlanckEquation: Stability, Distance and the Corresponding Extremal Problem in the Spatially Inhomogeneous Case ................................................ 379 Editorial Introduction Daniel Alpay and Bernd Kirstein This volume is dedicated to Lev Aronovich Sakhnovich, on the occasion of his 80th birthday. Lev Aronovich is an outstanding expert in operator theory and its applications, and his mathematical career is intimately related with the town OdessaintheUkraine.wherehestudiedmathematicsatthePedagogicalInstitute. V.P. Potapov, who was a Professorat the PedagogicalInstitute during that time, very early observed the extraordinary abilities of the young student Sakhnovich. The strong support of V.P. Potapov allowed Lev Aronovich to become a doctoral student at the Pedagogical Institute in 1953. His advisor was M.S. Livsic, one of the pioneers of operator theory. About three years later the main results of the candidatethesiswerepresentedinthefamousseminarofMarkGrigorievichKrein inOdessa.ThiscandidatethesiswasheldinextremelyhighesteembyM.G.Krein who had the opinion that the thesis even deserved the second doctorate degree (habilitation). The candidate thesis was the starting point of an extraordinary scientific career, the main steps of which are reflected in the biographic material which is contained in the first part of the volume. After the political changes in the SovietUnionLev Aronovichwasallowedto takepartinconferencesinforeign countriesandtowritemonographsinWesternpublishinghouses.Heisfamousfor his far-reaching method of operator identities which turned out to be a universal tool in several branches of analysis and stochastic processes (see in particular his monograph [14] on Levy processes). ItshouldbementionedthatLevAronovichwashonouredinaremarkableway byLeipzigUniversity.IntheWintersemester2007/2008hewasawardedtheLeib- nizguestprofessorshipofLeipzigUniversity.Uptonowthiswastheuniquetimein the history of Leipzig University that a mathematician was honoured in this way. The volume contains bibliographical material as well as a collection of ten selected and refereed papers. The ten papers can be divided into four main (over- lapping) families: Interpolation and Moment problems: The work Infinite product representations for kernels and iterations of functions, by Daniel Alpay, Palle Jorgensen, Izchak LewkowiczandItzikMartziano,containsinparticularanexampleofalinearcom- bination interpolation problem, where a linear combination of values at different viii D. Alpay and B. Kirstein nodes, is fixed, rather than the values at the nodes themselves. The work State space formulas for a suboptimal rational Leech problem II: Parametrization of all solutions,byA.E.Frazho,S.terHorstandM.A.Kaashoekisadirectcontinuation ofthe paper [6] where the authorsdiscussedthe maximumentropysolutionofthe interpolationproblem under consideration. It explicits connections between inter- polation, Leech’s factorization theorem (see [9, 10]) and the state space method. Next, the paperOn a simultaneous approach to the even and odd truncated matri- cial Hamburger moment by Bernd Fritzsche, Bernd Kirstein and Conrad Ma¨dler, continues the former investigations of the authors on matricial versions of power moment problems (see [4, 5, 7] and the papers in the volume [1]). The approach is based on Schur analysis, The main tool consists of an appropriate adaptation of the classical algorithmdue to I. Schur and R. Nevanlinna to the moment prob- lems under consideration. It should be mentioned that the truncated matricial Hamburger matrix moment problems with an odd or even number of prescribed moments will be handled in the most general case. Aspects of indefinite inner product spaces: Here one can find the paper Inner product spaces and Krein spaces in the quaternionic setting,byDanielAlpay,Fab- rizio Colombo, and Irene Sabadini, which lays the foundations of quaternionic Krein spaces (the Pontryagin space case had been considered in [2]). The paper On indefinite cases of operator identities which arise in interpolation theory. IIby J. Rovnyak and L.A. Sakhnovich, relates interpolation problems, operator identi- ties, andKrein–LangerrepresentationofgeneralizedCarath´eodoryfunctions. The authors studied the case of Nevanlinna functions in [11]. Operator-valued functions: Here we find the paper: Q-functions and boundary triplets of non-negative operators, by Yu.M. Arlinskii and S. Hassi, where the no- tion of Q-function is used in the setting of non-negative operators. In particular a result of Krein and Ovˇcarenko (see [8]) is made more precise. The paper On some special cases of the Radon–Nikodym theorem for vector- and operator-valued measures, by S. Boiko,V. DubovoyandA. Kheifets, studies operator-valuedmea- sures. Such measures play an important role in representation of operator-valued functions appearing in operator theory. See for instance [3]. Also in this category are the papers: A Jost–Pais-type reduction of (modified) Fredholm determinants for semiseparable operators in infinite dimensions by Fritz Gesztesy and Roger Nichols, and On the addition and multiplication theorems by K.A. Makarov and E. Tsekanovskii. Non linear differential equations: Non linear equations have always been an im- portantresearchtopic for Lev Sakhnovich(see for instance [13, 12]) and it is very fitting that the paper Nonlinear Fokker–Planck equation: stability, distance and the corresponding extremal problem in the spatially inhomogeneous case, written by Alexander Sakhnovich and Lev Sakhnovich, appears in the volume. These various papers cover a wide range of the interests of Lev Sakhnovich, and contain material which appears for the first time in print (as opposed to survey papers). Editorial Introduction ix References [1] D.Alpay and B. Kirstein (eds.): Interpolation, SchurFunctionsand Moment Prob- lems II,Operator Theory: Advancesand Applications, Volume226, Springer,Basel 2012. [2] D. Alpay and M. Shapiro. Reproducing kernel quaternionic Pontryagin spaces. In- tegral Equations and Operator Theory, 50:431–476, 2004. [3] M.S.Brodski˘ı. Triangular and Jordan representations of linear operators. American Mathematical Society, Providence, R.I.,1971. Translated from theRussian byJ.M. Danskin,Translations of Mathematical Monographs, vol. 32. [4] Yu.M.Dyukarev,B.Fritzsche,B.Kirstein,andC.M¨adler,andH.C.Thiele.Ondis- tinguished solutions of truncated matricial Hamburger moment problems. Complex Analysis and Operator Theory 3(4):759–834, 2009. [5] Yu.M. Dyukarev, B. Fritzsche, B. Kirstein, and C. M¨adler. On truncated matricial Stieltjes typemoment problems. Complex Analysis and Operator Theory, 4(4):904– 951, 2010. [6] A.E. Frazho, S. ter Horst, and M.A. Kaashoek. State space formulas for a subop- timal rational Leech problem I: Maximum entropy solution. Integral Equations and Operator Theory, 79:533–553, 2014. [7] B.Fritzsche,B.Kirstein,andC.M¨adler.OnHankelnonnegativedefinitesequences. thecanonicalHankelparametrization,andorthogonalpolynomials.Complexanalysis and Operator Theory, 5(2):447–511, 2011. [8] M.G. Kre˘ın and I.E. Ovˇcarenko. Inverse problems for Q-functions and resolvent matrices of positive Hermitian operators. Dokl. Akad. Nauk SSSR, 242(3):521–524, 1978. [9] R.B. Leech. Factorization of analytic functions and operator inequalities. Unpub- lished manuscript.Available at: http://www.people.virginia.edu/∼jlr5m/papers/leech.ps. [10] R.B. Leech. Factorization of analytic functions and operator inequalities. Integral Equations Operator Theory, 78(1):71–73, 2014. [11] J. Rovnyak and L.A. Sakhnovich. On indefinite cases of operator identities which arise in interpolation theory, volume 171 of Oper. Theory Adv. Appl., pages 281– 322. Birkha¨user, Basel, 2007. [12] L.A. Sakhnovich. A hyperbolic sine-Gordon equation. Izv. Vyssh. Uchebn. Zaved. Mat.,(1):54–63, 1991. [13] L.A.Sakhnovich.Integrablenonlinearequationsonthesemi-axis.Ukrain. Mat. Zh., 43(11):1578–1584, 1991. [14] L.A. Sakhnovich. Levy Processes, Integral Equations, Statistical Physics: Connec- tions and Interactions, volume225 of Operator Theory: Advances and Applications. SpringerBasel, 2012. Daniel Alpay Bernd Kirstein Department of Mathematics Mathematisches Institut Ben-Gurion Universityof theNegev Universit¨at Leipzig P.O.B. 653 Augustusplatz 10/11 Beer-Sheva,Israel D-04109 Leipzig, Germany e-mail: [email protected] e-mail: [email protected]