An Overview of Some Recent Developments on the Invariant Subspace Problem
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Constructive Closed Range and Open Mapping Theorems in This Paper
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Indag. Mathem., N.S., 11 (4), 509-516 December l&2000 Constructive Closed Range and Open Mapping Theorems by Douglas Bridges and Hajime lshihara Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand email: d. [email protected] School of Information Science, Japan Advanced Institute of Science and Technology, Tatsunokuchi, Ishikawa 923-12, Japan email: [email protected] Communicated by Prof. AS. Troelstra at the meeting of November 27,200O ABSTRACT We prove a version of the closed range theorem within Bishop’s constructive mathematics. This is applied to show that ifan operator Ton a Hilbert space has an adjoint and a complete range, then both T and T’ are sequentially open. 1. INTRODUCTION In this paper we continue our constructive exploration of the theory of opera- tors, in particular operators on a Hilbert space ([5], [6], [7]). We work entirely within Bishop’s constructive mathematics, which we regard as mathematics with intuitionistic logic. For discussions of the merits of this approach to mathematics - in particular, the multiplicity of its models - see [3] and [ll]. The technical background needed in our paper is found in [I] and [IO]. Our main aim is to prove the following result, the constructive Closed Range Theorem for operators on a Hilbert space (cf. [18], pages 99-103): Theorem 1. Let H be a Hilbert space, and T a linear operator on H such that T* exists and ran(T) is closed. -
Operator Algebras Generated by Left Invertibles Derek Desantis [email protected]
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Dissertations, Theses, and Student Research Papers Mathematics, Department of in Mathematics 3-2019 Operator algebras generated by left invertibles Derek DeSantis [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/mathstudent Part of the Analysis Commons, Harmonic Analysis and Representation Commons, and the Other Mathematics Commons DeSantis, Derek, "Operator algebras generated by left invertibles" (2019). Dissertations, Theses, and Student Research Papers in Mathematics. 93. https://digitalcommons.unl.edu/mathstudent/93 This Article is brought to you for free and open access by the Mathematics, Department of at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Dissertations, Theses, and Student Research Papers in Mathematics by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. OPERATOR ALGEBRAS GENERATED BY LEFT INVERTIBLES by Derek DeSantis A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfilment of Requirements For the Degree of Doctor of Philosophy Major: Mathematics Under the Supervision of Professor David Pitts Lincoln, Nebraska March, 2019 OPERATOR ALGEBRAS GENERATED BY LEFT INVERTIBLES Derek DeSantis, Ph.D. University of Nebraska, 2019 Adviser: David Pitts Operator algebras generated by partial isometries and their adjoints form the basis for some of the most well studied classes of C*-algebras. Representations of such algebras encode the dynamics of orthonormal sets in a Hilbert space. We instigate a research program on concrete operator algebras that model the dynamics of Hilbert space frames. The primary object of this thesis is the norm-closed operator algebra generated by a left y invertible T together with its Moore-Penrose inverse T . -
Lower Bounds for the Ruelle Spectrum of Analytic Expanding Circle Maps
LOWER BOUNDS FOR THE RUELLE SPECTRUM OF ANALYTIC EXPANDING CIRCLE MAPS OSCAR F. BANDTLOW AND FRED´ ERIC´ NAUD Abstract. We prove that there exists a dense set of analytic expanding maps of the circle for which the Ruelle eigenvalues enjoy exponential lower bounds. The proof combines potential theoretic techniques and explicit calculations for the spectrum of expanding Blaschke products. 1. Introduction and statement One of the basic problems of smooth ergodic theory is to investigate the asymptotic behaviour of mixing systems. In particular, there is interest in precise quantitative results on the rate of decay of correlations for smooth observables. In his seminal paper [17], Ruelle showed that the long time asymptotic behaviour of analytic hyperbolic systems can be understood in terms of the Ruelle spectrum, that is, the spectrum of certain operators, known as transfer operators in this context, acting on suitable Banach spaces of holomorphic functions. Surprisingly, even for the simplest systems like analytic expanding maps of the circle, very few quantitative results on the Ruelle spectrum are known so far. Let us be more precise. With T denoting the unit circle in the complex plane C, a map τ : T ! T is said to be analytic expanding if τ has a holomorphic extension to a neighbourhood of T and we have inf jτ 0(z)j > 1 : z2T The Ruelle spectrum is commonly defined as the spectrum of the Perron-Frobenius or transfer operator (on a suitable space of holomorphic functions) given locally by d X 0 (Lτ f)(z) := !(τ) φk(z)f(φk(z)) ; (1) k=1 where φk denotes the k-th local inverse branch of the covering map τ : T ! T, and ( +1 if τ is orientation preserving; arXiv:1605.06247v1 [math.DS] 20 May 2016 !(τ) = (2) −1 if τ is orientation reversing. -
An Introduction to Basic Functional Analysis
AN INTRODUCTION TO BASIC FUNCTIONAL ANALYSIS T. S. S. R. K. RAO . 1. Introduction This is a write-up of the lectures given at the Advanced Instructional School (AIS) on ` Functional Analysis and Harmonic Analysis ' during the ¯rst week of July 2006. All of the material is standard and can be found in many basic text books on Functional Analysis. As the prerequisites, I take the liberty of assuming 1) Basic Metric space theory, including completeness and the Baire category theorem. 2) Basic Lebesgue measure theory and some abstract (σ-¯nite) measure theory in- cluding the Radon-Nikodym theorem and the completeness of Lp-spaces. I thank Professor A. Mangasuli for carefully proof reading these notes. 2. Banach spaces and Examples. Let X be a vector space over the real or complex scalar ¯eld. Let k:k : X ! R+ be a function such that (1) kxk = 0 , x = 0 (2) k®xk = j®jkxk (3) kx + yk · kxk + kyk for all x; y 2 X and scalars ®. Such a function is called a norm on X and (X; k:k) is called a normed linear space. Most often we will be working with only one speci¯c k:k on any given vector space X thus we omit writing k:k and simply say that X is a normed linear space. It is an easy exercise to show that d(x; y) = kx ¡ yk de¯nes a metric on X and thus there is an associated notion of topology and convergence. If X is a normed linear space and Y ½ X is a subspace then by restricting the norm to Y , we can consider Y as a normed linear space. -
Spectral Theory of Composition Operators on Hardy Spaces of the Unit Disc and of the Upper Half-Plane
SPECTRAL THEORY OF COMPOSITION OPERATORS ON HARDY SPACES OF THE UNIT DISC AND OF THE UPPER HALF-PLANE UGURˇ GUL¨ FEBRUARY 2007 SPECTRAL THEORY OF COMPOSITION OPERATORS ON HARDY SPACES OF THE UNIT DISC AND OF THE UPPER HALF-PLANE A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY BY UGURˇ GUL¨ IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS FEBRUARY 2007 Approval of the Graduate School of Natural and Applied Sciences Prof. Dr. Canan OZGEN¨ Director I certify that this thesis satisfies all the requirements as a thesis for the degree of Doctor of Philosophy. Prof. Dr. Zafer NURLU Head of Department This is to certify that we have read this thesis and that in our opinion it is fully adequate, in scope and quality, as a thesis for the degree of Doctor of Philosophy. Prof. Dr. Aydın AYTUNA Prof. Dr. S¸afak ALPAY Co-Supervisor Supervisor Examining Committee Members Prof. Dr. Aydın AYTUNA (Sabancı University) Prof. Dr. S¸afak ALPAY (METU MATH) Prof. Dr. Zafer NURLU (METU MATH) Prof. Dr. Eduard EMELYANOV (METU MATH) Prof. Dr. Murat YURDAKUL (METU MATH) I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work. Name, Last name : U˘gurG¨ul. Signature : abstract SPECTRAL THEORY OF COMPOSITION OPERATORS ON HARDY SPACES OF THE UNIT DISC AND OF THE UPPER HALF-PLANE G¨ul,Uˇgur Ph.D., Department of Mathematics Supervisor: Prof. -
Frames and Representing Systems in Fréchet Spaces and Their Duals
Frames and representing systems in Fr´echet spaces and their duals J. Bonet, C. Fern´andez,A. Galbis, J.M. Ribera Abstract Frames and Bessel sequences in Fr´echet spaces and their duals are defined and studied. Their relation with Schauder frames and representing systems is analyzed. The abstract results presented here, when applied to concrete spaces of analytic functions, give many examples and consequences about sampling sets and Dirichlet series expansions. 1 Introduction and preliminaries The purpose of this paper is twofold. On the one hand we study Λ-Bessel sequences 0 (gi)i ⊂ E , Λ-frames and frames with respect to Λ in the dual of a Hausdorff locally convex space E, in particular for Fr´echet spaces and complete (LB)-spaces E, with Λ a sequence space. We investigate the relation of these concepts with representing systems in the sense of Kadets and Korobeinik [12] and with the Schauder frames, that were investigated by the authors in [7]. On the other hand our article emphasizes the deep connection of frames for Fr´echet and (LB)-spaces with the sufficient and weakly sufficient sets for weighted Fr´echet and (LB)-spaces of holomorphic functions. These concepts correspond to sampling sets in the case of Banach spaces of holomorphic functions. Our general results in Sections 2 and 3 permit us to obtain as a consequence many examples and results in the literature in a unified way in Section 4. Section 2 of our article is inspired by the work of Casazza, Christensen and Stoeva [9] in the context of Banach spaces. -
Methods of Applied Mathematics
Methods of Applied Mathematics Todd Arbogast and Jerry L. Bona Department of Mathematics, and Institute for Computational Engineering and Sciences The University of Texas at Austin Copyright 1999{2001, 2004{2005, 2007{2008 (corrected version) by T. Arbogast and J. Bona. Contents Chapter 1. Preliminaries 5 1.1. Elementary Topology 5 1.2. Lebesgue Measure and Integration 13 1.3. Exercises 23 Chapter 2. Normed Linear Spaces and Banach Spaces 27 2.1. Basic Concepts and Definitions. 27 2.2. Some Important Examples 34 2.3. Hahn-Banach Theorems 43 2.4. Applications of Hahn-Banach 48 2.5. The Embedding of X into its Double Dual X∗∗ 52 2.6. The Open Mapping Theorem 53 2.7. Uniform Boundedness Principle 57 2.8. Compactness and Weak Convergence in a NLS 58 2.9. The Dual of an Operator 63 2.10. Exercises 66 Chapter 3. Hilbert Spaces 73 3.1. Basic Properties of Inner-Products 73 3.2. Best Approximation and Orthogonal Projections 75 3.3. The Dual Space 78 3.4. Orthonormal Subsets 79 3.5. Weak Convergence in a Hilbert Space 86 3.6. Exercises 87 Chapter 4. Spectral Theory and Compact Operators 89 4.1. Definitions of the Resolvent and Spectrum 90 4.2. Basic Spectral Theory in Banach Spaces 91 4.3. Compact Operators on a Banach Space 93 4.4. Bounded Self-Adjoint Linear Operators on a Hilbert Space 99 4.5. Compact Self-Adjoint Operators on a Hilbert Space 104 4.6. The Ascoli-Arzel`aTheorem 107 4.7. Sturm Liouville Theory 109 4.8. -
On Linear Operators with Closed Range 1 Introduction
Journal of Applied Mathematics & Bioinformatics, vol.1, no.2, 2011, 175-182 ISSN: 1792-6602 (print), 1792-6939 (online) International Scientific Press, 2011 On Linear Operators with Closed Range P. Sam Johnson1 and S. Balaji2 Abstract Linear operators between Fr´echet spaces such that the closedness of range of an operator implies the closedness of range of another operator are discussed when the operators are topologically dominated by each other. Mathematics Subject Classification (2010): 47A05, 47A30. Keywords: Fr´echet spaces, closed range operators 1 Introduction Let X and Y be normed spaces, B(X,Y ) will denote the normed space of all continuous linear operators from X to Y . R(T ) and N(T ) will denote the range and null spaces of a linear operator T respectively. The Banach’s closed range theorem [5] reads as follows : If X and Y are Banach spaces and if T ∈B(X,Y ), then R(T ) is closed in Y if and only if R(T ∗) is closed in X∗. Given a differential operator T defined on some subspace of Lp(Ω), one may be interested in determining the family of functions y ∈ Lp(Ω) for which Tf = y has a solution. It is well known that if T ∈ B(X,Y ) has closed range, then the space of such y is the orthogonal complement of the solutions to the homogeneous equation T ∗g = 0. There are many important 1 National Institute of Technology Karnataka, Mangalore, India, e-mail: [email protected] 2 National Institute of Technology Karnataka, Mangalore, India, e-mail: [email protected] Article Info: Revised : August 24, 2011. -
Unbounded Linear Operators Jan Derezinski
Unbounded linear operators Jan Derezi´nski Department of Mathematical Methods in Physics Faculty of Physics University of Warsaw Ho_za74, 00-682, Warszawa, Poland Lecture notes, version of June 2013 Contents 3 1 Unbounded operators on Banach spaces 13 1.1 Relations . 13 1.2 Linear partial operators . 17 1.3 Closed operators . 18 1.4 Bounded operators . 20 1.5 Closable operators . 21 1.6 Essential domains . 23 1.7 Perturbations of closed operators . 24 1.8 Invertible unbounded operators . 28 1.9 Spectrum of unbounded operators . 32 1.10 Functional calculus . 36 1.11 Spectral idempotents . 41 1.12 Examples of unbounded operators . 43 1.13 Pseudoresolvents . 45 2 One-parameter semigroups on Banach spaces 47 2.1 (M; β)-type semigroups . 47 2.2 Generator of a semigroup . 49 2.3 One-parameter groups . 54 2.4 Norm continuous semigroups . 55 2.5 Essential domains of generators . 56 2.6 Operators of (M; β)-type ............................ 58 2.7 The Hille-Philips-Yosida theorem . 60 2.8 Semigroups of contractions and their generators . 65 3 Unbounded operators on Hilbert spaces 67 3.1 Graph scalar product . 67 3.2 The adjoint of an operator . 68 3.3 Inverse of the adjoint operator . 72 3.4 Numerical range and maximal operators . 74 3.5 Dissipative operators . 78 3.6 Hermitian operators . 80 3.7 Self-adjoint operators . 82 3.8 Spectral theorem . 85 3.9 Essentially self-adjoint operators . 89 3.10 Rigged Hilbert space . 90 3.11 Polar decomposition . 94 3.12 Scale of Hilbert spaces I . 98 3.13 Scale of Hilbert spaces II . -
Gradient Systems
Ralph Chill, Eva Fasangovˇ a´ Gradient Systems – 13th International Internet Seminar – February 10, 2010 c by R. Chill, E. Faˇsangov´a Contents 0 Introduction ................................................... 1 1 Gradient systems in euclidean space .............................. 5 1.1 Thespace Rd .............................................. 5 1.2 Fr´echetderivative .............................. ............ 7 1.3 Euclideangradient............................... ........... 8 1.4 Euclideangradientsystems ........................ .......... 9 1.5 Algebraicequationsand steepest descent ............ ........... 12 1.6 Exercises ....................................... .......... 13 2 Gradient systems in finite dimensional space ...................... 15 2.1 Definitionofgradient ............................. .......... 17 2.2 Definitionofgradientsystem....................... .......... 18 2.3 Local existence for ordinary differential equations . ............. 18 2.4 Local and global existence for gradientsystems .. ..... .......... 21 2.5 Newton’smethod.................................. ......... 23 2.6 Exercises ....................................... .......... 25 3 Gradients in infinite dimensional space: the Laplace operator on a bounded interval ....................... 27 3.1 Definitionofgradient ............................. .......... 28 3.2 Gradientsofquadraticforms ....................... .......... 30 3.3 Sobolevspacesinonespacedimension ................ ........ 31 3.4 The Dirichlet-Laplace operator on a bounded interval . ......... -
On the Distributional Divergence of Vector Fields Vanishing at Infinity
Proceedings of the Royal Society of Edinburgh, 141A, 65–76, 2011 On the distributional divergence of vector fields vanishing at infinity Thierry De Pauw Institut de Recherches en Math´ematiques et Physique, Universit´e Catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium ([email protected]) Monica Torres Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, USA ([email protected]) (MS received 31 August 2009; accepted 1 April 2010) The equation div v = F has a solution v in the space of continuous vector fields m vanishing at infinity if and only if F acts linearly on BVm/(m−1)(R ) (the space of functions in Lm/(m−1)(Rm) whose distributional gradient is a vector-valued measure) and satisfies the following continuity condition: F (uj ) converges to zero for each sequence {uj } such that the measure norms of ∇uj are uniformly bounded and m/(m−1) m uj 0 weakly in L (R ). 1. Introduction The equation ∆u = f ∈ Lm(Rm) need not have a solution u ∈ C1(Rm). In this paper we prove that, to each f ∈ Lm(Rm), there corresponds a continuous vector m m field, vanishing at infinity, v ∈ C0(R ; R ) such that div v = f weakly. In fact, we characterize those distributions F on Rm such that the equation div v = F admits m m a weak solution v ∈ C0(R ; R ). Related results have been obtained in [1–4, 6]. Our first proof, contained in §§ 3–6, follows the same pattern as [2]. -
Functional Analysis
Principles of Functional Analysis SECOND EDITION Martin Schechter Graduate Studies in Mathematics Volume 36 American Mathematical Society http://dx.doi.org/10.1090/gsm/036 Selected Titles in This Series 36 Martin Schechter, Principles of functional analysis, second edition, 2002 35 JamesF.DavisandPaulKirk, Lecture notes in algebraic topology, 2001 34 Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, 2001 33 Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, 2001 32 Robert G. Bartle, A modern theory of integration, 2001 31 Ralf Korn and Elke Korn, Option pricing and portfolio optimization: Modern methods of financial mathematics, 2001 30 J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, 2001 29 Javier Duoandikoetxea, Fourier analysis, 2001 28 Liviu I. Nicolaescu, Notes on Seiberg-Witten theory, 2000 27 Thierry Aubin, A course in differential geometry, 2001 26 Rolf Berndt, An introduction to symplectic geometry, 2001 25 Thomas Friedrich, Dirac operators in Riemannian geometry, 2000 24 Helmut Koch, Number theory: Algebraic numbers and functions, 2000 23 Alberto Candel and Lawrence Conlon, Foliations I, 2000 22 G¨unter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, 2000 21 John B. Conway, A course in operator theory, 2000 20 Robert E. Gompf and Andr´as I. Stipsicz, 4-manifolds and Kirby calculus, 1999 19 Lawrence C. Evans, Partial differential equations, 1998 18 Winfried Just and Martin Weese, Discovering modern set theory. II: Set-theoretic tools for every mathematician, 1997 17 Henryk Iwaniec, Topics in classical automorphic forms, 1997 16 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras.