Operator Theory, Operator Algebras, and Matrix Theory PDF
Preview Operator Theory, Operator Algebras, and Matrix Theory
Operator Theory Advances and Applications 267 Carlos André M. Amélia Bastos Alexei Yu. Karlovich Bernd Silbermann Ion Zaballa Editors Operator Theory, Operator Algebras, and Matrix Theory Operator Theory: Advances and Applications Volume 267 Founded in 1979 by Israel Gohberg Editors: Joseph A. Ball (Blacksburg, VA, USA) Albrecht Böttcher (Chemnitz, Germany) Harry Dym (Rehovot, Israel) 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) B. Malcolm Brown (Cardiff, UK) J.William Helton (San Diego, CA, USA) Raul Curto (Iowa, IA, USA) Marinus A. Kaashoek (Amsterdam, NL) Kenneth R. Davidson (Waterloo, ON, Canada) Thomas Kailath (Stanford, CA, USA) Fritz Gesztesy (Waco, TX, USA) Peter Lancaster (Calgary, Canada) Pavel Kurasov (Stockholm, Sweden) Peter D. Lax (New York, NY, USA) Vern Paulsen (Houston, TX, USA) Bernd Silbermann (Chemnitz, Germany) Mihai Putinar (Santa Barbara, CA, USA) Harold Widom (Santa Cruz, CA, USA) Ilya Spitkovsky (Abu Dhabi, UAE) 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 Carlos André • M. Amélia Bastos Alexei Yu. Karlovich • Bernd Silbermann Ion Zaballa Editors Operator Theory, Operator Algebras, and Matrix Theory Editors Carlos André M. Amélia Bastos Departamento de Matemática Departamento de Matemática Faculdade de Ciências Instituto Superior Técnico Universidade de Lisboa Universidade de Lisboa L isboa Portugal Lisboa Portugal Alexei Yu. Karlovich Bernd Silbermann Departamento de Matemática Fakultät für Mathematik Faculdade de Ciências e Tecnologia Technische Universität Chemnitz Universidade Nova de Lisboa Chemnitz, Germany L isboa, Portugal Ion Zaballa Departamento de Matemática Aplicada y EIO Universidad del País Vasco ( UPV/EHU) L eioa, Spain ISSN 0255-0156 ISSN 2296-4878 (electronic) Operator Theory: Advances and Applications ISBN 978-3-319-72448-5 ISBN 978-3-319-72449-2 (eBook) https://doi.org/10.1007/978-3-319-72449-2 Library of Congress Control Number: 2018949649 Mathematics Subject Classification (2010): 47-xx, 15-xx, 46-xx © Springer International Publishing AG, part of Springer Nature 2018 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. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG part of Springer Nature. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Contents Editorial Introduction .................................................... vii C.A.M. Andr´e, F. Gomes and J. Lochon Indecomposable Supercharacters of the Infinite Unitriangular Group .............................................................. 1 M.A. Bastos, C.A. Fernandes and Yu.I. Karlovich A C∗-algebra of Singular Integral Operators with Shifts and Piecewise Quasicontinuous Coefficients .............................. 25 N. Bebiano, J. da Providˆencia and J.P. da Providˆencia Non-Hermitian Quantum Mechanics of Bosonic Operators ........... 65 C. Carvalho, V. Nistor and Y. Qiao Fredholm Conditions on Non-compact Manifolds: Theory and Examples ...................................................... 79 K. Demirci and S. Orhan Statistical e-Convergence of Bo¨gel-Type Continuous Functions ...... 123 K. Demirci, S. Orhan and B. Kolay Weighted Statistical Relative Approximation by Positive Linear Operators ................................................... 131 M. Dodig Descriptor Systems Under Feedback and Output Injection ........... 141 R.G. Douglas and R. Yang Hermitian Geometry on Resolvent Set .............................. 167 F. G´omez-Cubillo and S. Villullas Spectral Algorithms for MRA OrthonormalWavelets ................ 185 C.R. Johnson, C. Mariju´an, P. Paparella and M. Pisonero The NIEP .......................................................... 199 vi Contents A.Yu. Karlovich, Yu.I. Karlovich and A.B. Lebre Semi-Fredholmness of Weighted Singular Integral Operators with Shifts and Slowly Oscillating Data ............................. 221 A.B. Lebre and J.S. Rodr´ıguez Factorization of Singular Integral Operators with a Carleman BackwardShift: The Vector Case ................................... 247 S. Roch Extension-Restriction Theorems for Algebras of Approximation Sequences ........................................................... 261 S. Roch and B. Silbermann Toeplitz and Hankel Algebras – Axiomatic and Asymptotic Aspects ............................................................. 285 P.A. Santos More Than 40 Years of Algebraic Techniques in Numerical Analysis ............................................................ 317 K.V. Sklyar, G.M. Sklyar and S.Yu. Ignatovich Linearizability of Multi-Control Systems of the Class C1 by Additive Change of Controls ........................................ 359 I.M. Spitkovsky A Distance Formula Related to a Family of Projections Orthogonal to Their Symmetries .................................... 371 OperatorTheory: Advances andApplications,Vol.267,vii–viii (cid:2)c SpringerInternational PublishingAG,partofSpringerNature2018 Editorial Introduction This book is dedicated to the International Workshop on Operator Theory and Operator Algebras – WOAT 2016, which took place in Lisbon at Instituto Superior T´ecnico,fromJuly 5 to July 8 of2016.This workshopcontinueda series ofconferencesorganizedin Lisboa IST since 2006andaimedto promoteresearch exchanges among Operator Theory, Operator Algebras and Matrix Theory areas. Thebookconsistsof17chaptersthatcoverresearchfieldsinOperatorTheory and Operator Algebras as well as in Matrix Theory and Representation Theory. The researchfield in OperatorTheory and Operator Algebras is mainly rep- resented in chapters that cover the following different topics: • Fredholmtheoryfornon-localC∗-algebrasofsingularintegraloperatorswith piecewise quasicontinuous coefficients and the local trajectory method. Ex- plicit Fredholm conditions for classes of pseudodifferential operators on sin- gular and non compact spaces and the new concept of Fredholm groupoid. • A Korovkin type approximation theorem. Weighted statistical relative ap- proximation by positive linear operators. • Studyofgeneral,possiblynon-commuting,tuplesusinggeometricideasbased on the newly emerged concept of projective joint spectrum. • Factorization of singular integral operators with a Carleman backward shift of linear fractional type, on Ln. Sufficient conditions for semiFredholmness p on Lp(R ) of weighted singular integral operators with shifts and slowly + oscillating data with discontinuities at 0 and ∞. • The structure ofeveryseparableC∗-algebraofapproximationsequencesand theC∗-algebraofthefinitesectionsdiscretizationforToeplitzoperatorswith continuous generating functions. Banach algebras of Toeplitz like operators definedinanaxiomaticwayandthe classicalToeplitz andHankeloperators. Overview of the historical development of more than 40 years of algebraic techniques in Numerical Analysis. • A formula for the distance from an orthogonal projection, on some Hilbert space, to a set of orthogonal projections. The researchfieldinMatrixTheoryis coveredinseveralchaptersdevotedto the following topics: • Spectral analysis of non-Hermitian operatorsappearing in quantum physics. The diagonalizationof such operators and their adjoints is proved. viii Editorial Introduction • Simultaneous feedback and output injection on descriptor linear system de- scribed by a quadruple of matrices. Description of the possible Kronecker invariantsoftheresultingpencil.Aconstructiveandexplicitsolutionisgiven over algebraically closed fields. • Spectral representations of the dilation and translation operators used to constructwavelets acting on the Hilbert space of square integrable functions on the real line. The spectral analysis concerns some particular orthonormal bases: the Haar basis, the Walsh–Paley basis and the trigonometric basis. • A survey on the nonnegative inverse eigenvalue problem (NIEP), and its several variants, with an emphasis on recent results. • Linearizationofnonlinearcontrolsystems.Necessaryandsufficientconditions arefoundinaparticularcase(localA-linearizability)ofthemoregeneralcase (local feedback linearization). Finally, in the research field of Representation Theory, the notion of an in- decomposable (extreme) supercharacter is defined for infinite algebra groups and a description of these is given for the infinite unitriangular group in terms of su- percharacters of the finite unitriangular groups. The editors of the volume are grateful for the support of the Portuguese Foundation for Science and Technology and the Center for Function Analysis, LinearStructuresandApplications.TheyalsodeeplythanktheBirha¨user’sedito- rial team, Dorothy Mazlum and Sabrina H¨ocklin, for their availability during the preparation of the volume. The editors December 2017 OperatorTheory: Advances andApplications,Vol.267,1–24 (cid:2)c SpringerInternational PublishingAG,partofSpringerNature2018 Indecomposable Supercharacters of the Infinite Unitriangular Group Carlos A.M. Andr´e, Filipe Gomes and Jocelyn Lochon Abstract. Let U∞(k) be the locally finite unitriangular group defined over a finite field k with q elements. We define the notion of an indecomposable supercharacter and describe these indecomposable supercharacters in terms of the supercharacters of thefiniteunitriangular groups Un(k). MathematicsSubjectClassification(2010).Primary20C15,20G40;Secondary 05E10, 43A35. Keywords. Unitriangular group; indecomposable supercharacter; branching graph; multiplicative graph; multiplicative linear function. 1. Introduction Let G be a group. A complex-valued function φ: G → C is said to be positive definite if 1. φ(g−1)=φ(g) for all g ∈G, and 2. F(cid:2)or any fi(cid:3)nite collection g1,...,gm of elements of G, the Hermitian matrix φ(g g−1) is nonnegative. i j 1≤i,j≤m A function φ: G → C is said to be central (or a class function) if it is constant on conjugacy classes, that is, if φ(ghg−1) = φ(h) for all g,h ∈ G; and it is said to be normalized if φ(1) = 1. We will denote by Ch(G) the set consisting of all central, positive definite, normalized functions on G (if G is a topological group, we additionally require the functions to be continuous). If φ,ψ ∈ Ch(G), then This research was made within the activities of the Group for Linear, Algebraic and Combi- natorial Structures of the Center for Functional Analysis, Linear Structures and Applications (University of Lisbon, Portugal), and was partiallysupported by the Portuguese Science Foun- dation(FCT)throughtheStrategicProjectUID/MAT/04721/2013. ThesecondandthirdauthorswerepartiallysupportedbytheLisbonMathematicsPhDprogram (funded by the Portuguese Science Foundation). The first half of the work is part of the third authorPh.D.thesis,whereasthesecondhalfispartofthesecondauthorPh.D.thesis.
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