Operator Theory Advances and Applications 256 Daniel Alpay Fabrizio Colombo Irene Sabadini Slice Hyperholomorphic Schur Analysis Operator Theory: Advances and Applications Volume 256 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 (Orange, CA, USA) 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) More information about this series at http://www.springer.com/series/4850 Daniel Alpay • Fabrizio Colombo • Irene Sabadini Slice Hyperholomorphic Schur Analysis Daniel Alpay Fabrizio Colombo Department of Mathematics Dipartimento di Matematica Chapman University Politecnico di Milano Orange, California, USA Milano, Italy Irene Sabadini Dipartimento di Matematica Politecnico di Milano Milano, Italy ISSN 0255-0156 ISSN2296-4878 (electronic) Operator Theory: Advances and Applications ISBN 978-3-319-42513 - 9 ISBN 978-3-319-42514-6 (eBook) DOI 10.1007/978-3-319-42514 - 6 Library of Congress Control Number: 2016960537 Mathematics Subject Classification (2010): 30G35, 46C20, 47B32, 30E05 © Springer International Publishing Switzerland 2016 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 This book is published under the trade name Birkhäuser, www.birkhauser-science.com The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Contents Prologue ..................................... ix Part I Classical Schur Analysis 1 Preliminaries 1.1 Somehistory.............................. 3 1.2 Krein spaces, Pontryagin spaces, and negative squares . 6 1.3 TheWieneralgebra.......................... 10 1.4 TheNehariextensionproblem.................... 12 1.5 The Carath´eodory–Toeplitzextensionproblem........... 15 1.6 Various classes of functions and realization theorems . 16 2 Rational Functions 2.1 Rationalfunctionsandminimalrealizations............. 23 2.2 Minimalfactorization......................... 26 2.3 Rational functions J-unitaryontheimaginaryline......... 28 2.4 Rational functions J-unitaryontheunitcircle........... 31 3SchurAnalysis 3.1 TheSchuralgorithm.......................... 37 3.2 Interpolationproblems......................... 39 3.3 First-orderdiscretesystems...................... 41 3.4 The Schur algorithm and reproducing kernel spaces . 45 Part II Quaternionic Analysis 4 Finite-dimensional Preliminaries 4.1 Somepreliminariesonquaternions.................. 51 4.2 Polynomialswithquaternioniccoefficients.............. 54 4.3 Matriceswithquaternionicentries.................. 57 4.4 Matrixequations............................ 65 v vi Contents 5 Quaternionic Functional Analysis 5.1 Quaternioniclocallyconvexlinearspaces.............. 71 5.2 Quaternionic inner product spaces . 74 5.3 Quaternionic Hilbert spaces. Main properties . 79 5.4 Partialmajorants........................... 84 5.5 Majorant topologies and inner product spaces . 88 5.6 QuaternionicHilbertspaces.Weaktopology............. 96 5.7 QuaternionicPontryaginspaces.................... 100 5.8 QuaternionicKreinspaces....................... 105 5.9 Positive definite functions and reproducing kernel quaternionic Hilbertspaces............................. 109 5.10 Negative squares and reproducing kernel quaternionic Pontryagin spaces.................................. 111 6 Slice Hyperholomorphic Functions 6.1 Thescalarcase............................. 117 6.2 TheHardyspaceoftheunitball................... 137 6.3 Blaschkeproducts(unitballcase).................. 143 6.4 TheWieneralgebra.......................... 150 6.5 TheHardyspaceoftheopenhalf-space............... 157 6.6 Blaschkeproducts(half-spacecase)................. 161 7 Operator-valued Slice Hyperholomorphic Functions 7.1 Definitionandmainproperties.................... 165 7.2 S-spectrum and S-resolventoperator................ 171 7.3 Functionalcalculus........................... 175 7.4 Tworesultsonslicehyperholomorphicextension.......... 181 7.5 Slicehyperholomorphickernels.................... 183 2 7.6 The space HH(B) and slice backward-shift invariant subspaces . 189 Part III Quaternionic Schur Analysis 8 Reproducing Kernel Spaces and Realizations 8.1 Classesoffunctions.......................... 197 8.2 ThePotapov–Ginzburgtransform.................. 200 8.3 Schur and generalized Schur functions of the ball . 202 8.4 Contractive multipliers, inner multipliers and Beurling–Laxtheorem......................... 212 8.5 A theorem on convergence of Schur multipliers . 218 8.6 Thestructuretheorem......................... 220 8.7 Carath´eodory and generalized Carath´eodoryfunctions........................ 222 Contents vii 8.8 Schur and generalized Schur functions of the half-space . 227 8.9 Herglotz and generalized Herglotz functions . 235 9 Rational Slice Hyperholomorphic Functions 9.1 Definitionandfirstproperties..................... 241 9.2 Minimalrealizations.......................... 246 9.3 Realizations of unitary rational functions . 250 9.4 Rationalslicehyperholomorphicfunctions.............. 253 9.5 Linearfractionaltransformation................... 259 9.6 Backward-shiftoperators....................... 261 10 First Applications: Scalar Interpolation and First-order Discrete Systems 10.1TheSchuralgorithm.......................... 266 10.2Aparticularcase............................ 270 10.3Thereproducingkernelmethod.................... 274 10.4 Carath´eodory–Fej´erinterpolation................... 275 10.5 Boundary interpolation . 282 10.6First-orderdiscretelinearsystems.................. 295 10.7Discretesystems:therationalcase.................. 298 11 Interpolation: Operator-valued Case 11.1Formulationoftheinterpolationproblems.............. 309 2 11.2 The problem IP(HH(B)): the non-degenerate case . 312 11.3 Left-tangential interpolation in S(H1, H2, B) ............ 315 11.4 Interpolation in S(H1, H2, B). The non-degenerate case . 319 11.5 Interpolation: The case of a finite number of interpolatingconditions........................ 324 11.6Leech’stheorem............................ 328 11.7 Interpolation in S(H1, H2, B). Nondegenerate case: Sufficiency . 331 Epilogue ..................................... 333 Bibliography ................................... 335 Index ....................................... 355 Notation Index .................................. 361 Prologue Functions analytic and contractive in the open unit disk (also known as Schur functions) have applications to, and connections with, a host of domains, such as classical analysis (via for instance the theory of orthogonal polynomials), linear system theory, inverse scattering, signal processing and operator models, to name a few. Schur analysis can be defined as a collection of problems related to Schur functions (and to related classes of functions, such as functions analytic in the open unit disk and with a real positive part there) and their applications to these various fields. A key role in Schur analysis is played by reproducing kernel Hilbert spaces of functions, of the kind introduced by de Branges and Rovnyak (see, e.g., [106, 107, 157]), associated to these functions. It is of interest to consider Schur analysis in various other settings. Extensions have been made for instance to the case of several complex variables (the Schur– Agler classes; see [6, 7, 83, 84]), the case