Exponentiation of Lie Algebras of Linear Operators on Locally Convex Spaces
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Adjoint of Unbounded Operators on Banach Spaces
November 5, 2013 ADJOINT OF UNBOUNDED OPERATORS ON BANACH SPACES M.T. NAIR Banach spaces considered below are over the field K which is either R or C. Let X be a Banach space. following Kato [2], X∗ denotes the linear space of all continuous conjugate linear functionals on X. We shall denote hf; xi := f(x); x 2 X; f 2 X∗: On X∗, f 7! kfk := sup jhf; xij kxk=1 defines a norm on X∗. Definition 1. The space X∗ is called the adjoint space of X. Note that if K = R, then X∗ coincides with the dual space X0. It can be shown, analogues to the case of X0, that X∗ is a Banach space. Let X and Y be Banach spaces, and A : D(A) ⊆ X ! Y be a densely defined linear operator. Now, we st out to define adjoint of A as in Kato [2]. Theorem 2. There exists a linear operator A∗ : D(A∗) ⊆ Y ∗ ! X∗ such that hf; Axi = hA∗f; xi 8 x 2 D(A); f 2 D(A∗) and for any other linear operator B : D(B) ⊆ Y ∗ ! X∗ satisfying hf; Axi = hBf; xi 8 x 2 D(A); f 2 D(B); D(B) ⊆ D(A∗) and B is a restriction of A∗. Proof. Suppose D(A) is dense in X. Let S := ff 2 Y ∗ : x 7! hf; Axi continuous on D(A)g: For f 2 S, define gf : D(A) ! K by (gf )(x) = hf; Axi 8 x 2 D(A): Since D(A) is dense in X, gf has a unique continuous conjugate linear extension to all ∗ of X, preserving the norm. -
Quantum Duality, Unbounded Operators, And
JOURNAL OF MATHEMATICAL PHYSICS 51, 063511 ͑2010͒ Quantum duality, unbounded operators, and inductive limits ͒ Anar Dosia Middle East Technical University Northern Cyprus Campus, Guzelyurt, KKTC, Mersin 10, Turkey ͑Received 20 October 2009; accepted 7 April 2010; published online 14 June 2010͒ In this paper, we investigate the inductive limits of quantum normed ͑or operator͒ spaces. This construction allows us to treat the space of all noncommutative con- tinuous functions over a quantum domain as a quantum ͑or local operator͒ space of all matrix continuous linear operators equipped with S-quantum topology. In par- ticular, we classify all quantizations of the polynormed topologies compatible with the given duality proposing a noncommutative Arens–Mackey theorem. Further, the inductive limits of operator spaces are used to introduce locally compact and lo- cally trace class unbounded operators on a quantum domain and prove the dual realization theorem for an abstract quantum space. © 2010 American Institute of Physics. ͓doi:10.1063/1.3419771͔ I. INTRODUCTION The operator analogs of locally convex spaces have been started to develop in Ref. 14 by Effros and Webster. The central goal of this direction is to create a theory of quantum polynormed spaces or quantum spaces, which should reflect the “locally convex space chapters” of quantum functional analysis. This theory has been created as the basic language of quantum physics. To have a comprehensive mathematical model of quantum physics, it is necessary to consider the linear spaces of unbounded Hilbert space operators or “noncommutative variable spaces.”19 This is the mathematical background of Heisenberg’s “matrix mechanics.” The quantizations of a variable space have provided functional analysis with the new constructions, methods, and problems.22 Being a new and modern branch of functional analysis, quantum functional analysis can be divided into the normed13,20 and polynormed ͑or locally convex͒ topics,14,25 as in the classical theory. -
Topology of Differentiable Manifolds
TOPOLOGY OF DIFFERENTIABLE MANIFOLDS D. MART´INEZ TORRES Contents 1. Introduction 1 1.1. Topology 2 1.2. Manifolds 3 2. More definitions and basic results 5 2.1. Submanifold vs. embedding 7 2.2. The tangent bundle of a Cr-manifold, r ≥ 1. 7 2.3. Transversality and submanifolds 9 2.4. Topology with Cr-functions. 9 2.5. Manifolds with boundary 13 2.6. 1-dimensional manifolds 16 3. Function spaces 19 4. Approximations 27 5. Sard's theorem and transversality 32 5.1. Transversality 35 6. Tubular neighborhoods, homotopies and isotopies 36 6.1. Homotopies, isotopies and linearizations 38 6.2. Linearizations 39 7. Degree, intersection number and Euler characteristic 42 7.1. Orientations 42 7.2. The degree of a map 43 7.3. Intersection number and Euler characteristic 45 7.4. Vector fields 46 8. Isotopies and gluings and Morse theory 47 8.1. Gluings 48 8.2. Morse functions 49 8.3. More on k-handles and smoothings 57 9. 2 and 3 dimensional compact oriented manifolds 60 9.1. Compact, oriented surfaces 60 9.2. Compact, oriented three manifolds 64 9.3. Heegard decompositions 64 9.4. Lens spaces 65 9.5. Higher genus 66 10. Exercises 66 References 67 1. Introduction Let us say a few words about the two key concepts in the title of the course, topology and differentiable manifolds. 1 2 D. MART´INEZ TORRES 1.1. Topology. It studies topological spaces and continuous maps among them, i.e. the category TOP with objects topological spaces and arrows continuous maps. -
Spectral Measures in Locally Convex Algebras by Helmut H
SPECTRAL MEASURES IN LOCALLY CONVEX ALGEBRAS BY HELMUT H. SCHAEFER The University of Michigan, U.S.A.Q) Introduction The central subject of classical spectral theory has been the spectral representation of self-adjoint and normal operators in Hilbert space; after Hilbert had given a complete treatment of the bounded case, the spectra] theory of unbounded self-adjoint and normal operators was developed by yon Neumann and others. (An excellent account can be found in [23].) Abstractions of algebras of Hermitian and normal operators were considered by various authors, notably Stone [22] who characterized such algebras as algebras of conti- nuous functions on a compact (Hausdorff) space. Investigations by Freudenthal [9] and Nakano [15] (especially papers 1 and 2), also leading to spectral theories, went in a different direction. Generalizations of unitary operators to reflexive Banach spaces were considered by Lorch [13]; Lorch also developed an operational calculus for those operators in reflexive Banach spaces that can be represented by a spectral measure [12]. (Taylor [24] developed such a calculus for closed operators on a Banach space whose spectrum does not cover the plane, but necessarily restricted to functions locally holomorphic on the spectrum. There are some recent results in this direction for operators on a locally convex space [26], [16].) The most extensive work on bounded and unbounded operators in a Banach space is due to Dunford, Schwartz, Bade and others (for a detailed bibliography, see [6] and [8]). Dunford considers operators that have a countably additive resolution of the identity, but may differ from a spectral operator (in the sense of Definition 4, Section 4 of this paper) by a quasi-nilpotent. -
Multiple Operator Integrals: Development and Applications
Multiple Operator Integrals: Development and Applications by Anna Tomskova A thesis submitted for the degree of Doctor of Philosophy at the University of New South Wales. School of Mathematics and Statistics Faculty of Science May 2017 PLEASE TYPE THE UNIVERSITY OF NEW SOUTH WALES Thesis/Dissertation Sheet Surname or Family name: Tomskova First name: Anna Other name/s: Abbreviation for degree as given in the University calendar: PhD School: School of Mathematics and Statistics Faculty: Faculty of Science Title: Multiple Operator Integrals: Development and Applications Abstract 350 words maximum: (PLEASE TYPE) Double operator integrals, originally introduced by Y.L. Daletskii and S.G. Krein in 1956, have become an indispensable tool in perturbation and scattering theory. Such an operator integral is a special mapping defined on the space of all bounded linear operators on a Hilbert space or, when it makes sense, on some operator ideal. Throughout the last 60 years the double and multiple operator integration theory has been greatly expanded in different directions and several definitions of operator integrals have been introduced reflecting the nature of a particular problem under investigation. The present thesis develops multiple operator integration theory and demonstrates how this theory applies to solving of several deep problems in Noncommutative Analysis. The first part of the thesis considers double operator integrals. Here we present the key definitions and prove several important properties of this mapping. In addition, we give a solution of the Arazy conjecture, which was made by J. Arazy in 1982. In this part we also discuss the theory in the setting of Banach spaces and, as an application, we study the operator Lipschitz estimate problem in the space of all bounded linear operators on classical Lp-spaces of scalar sequences. -
Class Notes, Functional Analysis 7212
Class notes, Functional Analysis 7212 Ovidiu Costin Contents 1 Banach Algebras 2 1.1 The exponential map.....................................5 1.2 The index group of B = C(X) ...............................6 1.2.1 p1(X) .........................................7 1.3 Multiplicative functionals..................................7 1.3.1 Multiplicative functionals on C(X) .........................8 1.4 Spectrum of an element relative to a Banach algebra.................. 10 1.5 Examples............................................ 19 1.5.1 Trigonometric polynomials............................. 19 1.6 The Shilov boundary theorem................................ 21 1.7 Further examples....................................... 21 1.7.1 The convolution algebra `1(Z) ........................... 21 1.7.2 The return of Real Analysis: the case of L¥ ................... 23 2 Bounded operators on Hilbert spaces 24 2.1 Adjoints............................................ 24 2.2 Example: a space of “diagonal” operators......................... 30 2.3 The shift operator on `2(Z) ................................. 32 2.3.1 Example: the shift operators on H = `2(N) ................... 38 3 W∗-algebras and measurable functional calculus 41 3.1 The strong and weak topologies of operators....................... 42 4 Spectral theorems 46 4.1 Integration of normal operators............................... 51 4.2 Spectral projections...................................... 51 5 Bounded and unbounded operators 54 5.1 Operations.......................................... -
Some Elements of Functional Analysis
Some elements of functional analysis J´erˆome Le Rousseau January 8, 2019 Contents 1 Linear operators in Banach spaces 1 2 Continuous and bounded operators 2 3 Spectrum of a linear operator in a Banach space 3 4 Adjoint operator 4 5 Fredholm operators 4 5.1 Characterization of bounded Fredholm operators ............ 5 6 Linear operators in Hilbert spaces 10 Here, X and Y will denote Banach spaces with their norms denoted by k.kX , k.kY , or simply k.k when there is no ambiguity. 1 Linear operators in Banach spaces An operator A from X to Y is a linear map on its domain, a linear subspace of X, to Y . One denotes by D(A) the domain of this operator. An operator from X to Y is thus characterized by its domain and how it acts on this domain. Operators defined this way are usually referred to as unbounded operators. One writes (A, D(A)) to denote the operator along with its domain. The set of linear operators from X to Y is denoted by L (X,Y ). If D(A) is dense in X the operator is said to be densely defined. If D(A) = X one says that the operator A is on X to Y . The range of the operator is denoted by Ran(A), that is, Ran(A)= {Ax; x ∈ D(A)} ⊂ Y, 1 and its kernel, ker(A), is the set of all x ∈ D(A) such that Ax = 0. The graph of A, G(A), is given by G(A)= {(x, Ax); x ∈ D(A)} ⊂ X × Y. -
On Lifting of Operators to Hilbert Spaces Induced by Positive Selfadjoint Operators
View metadata, citation and similar papers at core.ac.uk brought to you by CORE J. Math. Anal. Appl. 304 (2005) 584–598 provided by Elsevier - Publisher Connector www.elsevier.com/locate/jmaa On lifting of operators to Hilbert spaces induced by positive selfadjoint operators Petru Cojuhari a, Aurelian Gheondea b,c,∗ a Department of Applied Mathematics, AGH University of Science and Technology, Al. Mickievicza 30, 30-059 Cracow, Poland b Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey c Institutul de Matematic˘a al Academiei Române, C.P. 1-764, 70700 Bucure¸sti, Romania Received 10 May 2004 Available online 29 January 2005 Submitted by F. Gesztesy Abstract We introduce the notion of induced Hilbert spaces for positive unbounded operators and show that the energy spaces associated to several classical boundary value problems for partial differential operators are relevant examples of this type. The main result is a generalization of the Krein–Reid lifting theorem to this unbounded case and we indicate how it provides estimates of the spectra of operators with respect to energy spaces. 2004 Elsevier Inc. All rights reserved. Keywords: Energy space; Induced Hilbert space; Lifting of operators; Boundary value problems; Spectrum 1. Introduction One of the central problem in spectral theory refers to the estimation of the spectra of linear operators associated to different partial differential equations. Depending on the specific problem that is considered, we have to choose a certain space of functions, among * Corresponding author. E-mail addresses: [email protected] (P. Cojuhari), [email protected], [email protected] (A. -
Subspaces and Quotients of Topological and Ordered Vector Spaces
Zoran Kadelburg Stojan Radenovi´c SUBSPACES AND QUOTIENTS OF TOPOLOGICAL AND ORDERED VECTOR SPACES Novi Sad, 1997. CONTENTS INTRODUCTION::::::::::::::::::::::::::::::::::::::::::::::::::::: 1 I: TOPOLOGICAL VECTOR SPACES::::::::::::::::::::::::::::::: 3 1.1. Some properties of subsets of vector spaces ::::::::::::::::::::::: 3 1.2. Topological vector spaces::::::::::::::::::::::::::::::::::::::::: 6 1.3. Locally convex spaces :::::::::::::::::::::::::::::::::::::::::::: 12 1.4. Inductive and projective topologies ::::::::::::::::::::::::::::::: 15 1.5. Topologies of uniform convergence. The Banach-Steinhaus theorem 21 1.6. Duality theory ::::::::::::::::::::::::::::::::::::::::::::::::::: 28 II: SUBSPACES AND QUOTIENTS OF TOPOLOGICAL VECTOR SPACES ::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 39 2.1. Subspaces of lcs’s belonging to the basic classes ::::::::::::::::::: 39 2.2. Subspaces of lcs’s from some other classes :::::::::::::::::::::::: 47 2.3. Subspaces of topological vector spaces :::::::::::::::::::::::::::: 56 2.4. Three-space-problem for topological vector spaces::::::::::::::::: 60 2.5. Three-space-problem in Fr´echet spaces:::::::::::::::::::::::::::: 65 III: ORDERED TOPOLOGICAL VECTOR SPACES :::::::::::::::: 72 3.1. Basics of the theory of Riesz spaces::::::::::::::::::::::::::::::: 72 3.2. Topological vector Riesz spaces ::::::::::::::::::::::::::::::::::: 79 3.3. The basic classes of locally convex Riesz spaces ::::::::::::::::::: 82 3.4. l-ideals of topological vector Riesz spaces ::::::::::::::::::::::::: -
Positive Forms on Banach Spaces
POSITIVE FORMS ON BANACH SPACES B´alint Farkas, M´at´eMatolcsi 25th June 2001 Abstract The first representation theorem establishes a correspondence between positive, self-adjoint operators and closed, positive forms on Hilbert spaces. The aim of this paper is to show that some of the results remain true if the underlying space is a reflexive Banach space. In particular, the construc- tion of the Friedrichs extension and the form sum of positive operators can be carried over to this case. 1 Introduction Let X denote a reflexive complex Banach space, and X∗ its conjugate dual space (i.e. the space of all continuous, conjugate linear functionals over X). We will use the notation (v, x) := v(x) for v X∗ , x X, and (x, v) := v(x). Let A be a densely defined linear operator from∈ X to ∈X∗. Notice that in this context it makes sense to speak about positivity and self-adjointness of A. Indeed, A defines a sesquilinear form on Dom A Dom A via t (x, y) = (Ax)(y) = (Ax, y) × A and A is called positive if tA is positive, i.e. if (Ax, x) 0 for all x Dom A. Also, the adjoint A∗ of A is defined (because A is densely≥ defined)∈ and is a mapping from X∗∗ to X∗, i.e. from X to X∗. Thus, A is called self-adjoint ∗ if A = A . Similarly, the operator A is called symmetric if the form tA is symmetric. In Section 2 we deal with closed, positive forms and associated operators, and arXiv:math/0612015v1 [math.FA] 1 Dec 2006 we establish a generalized version of the first representation theorem. -
Grothendieck-Topological Group Objects
GROTHENDIECK-TOPOLOGICAL GROUP OBJECTS JOAQU´IN LUNA-TORRES In memory of Carlos J. Ruiz Salguero Abstract. In analogy with the classical theory of topological groups, for finitely complete categories enriched with Grothendieck topologies, we provide the concepts of localized G-topological space, initial Grothendieck topologies and continuous morphisms, in order to obtain the concepts of G-topological monoid and G-topological group objects. 1. Introduction Topological groups (monoids) are very important mathematical objects with not only applications in mathematics, for example in Lie group theory, but also in physics. Topological groups are defined on topological spaces that involve points as fundamental objects of such topological spaces. The aim of this paper is to introduce a topological theory of groups within any category, finitely complete, enriched with Grothendieck topologies. The paper is organized as follows: We describe, in section 2, the notion of Grothendieck topology as in S. MacLane and I. Moerdijk [8] . In section 3, we present the concepts of localized G-topological space, initial Grothendieck topologies and continuous morphisms, and we study the lattice structure of all Grothendieck topologies on a category C ; after that, in section 4, as in M. Forrester-Baker [3], we present the concepts of monoid and group objects. In section 5 we present the concepts of G-topological monoid and G-topological group objects; finally, in section 6 we give some examples of arXiv:1909.11777v1 [math.CT] 25 Sep 2019 these ideas. 2. Theoretical Considerations Throughout this paper, we will work within an ambient category C which is finitely complete. From S. MacLane and I. -
Closed Linear Operators with Domain Containing Their Range
Proceedings of the Edinburgh Mathematical Society (1984) 27, 229-233 © CLOSED LINEAR OPERATORS WITH DOMAIN CONTAINING THEIR RANGE by SCHOICHI OTA (Received 8th February 1984) 1. Introduction In connection with algebras of unbounded operators, Lassner showed in [4] that, if T is a densely defined, closed linear operator in a Hilbert space such that its domain is contained in the domain of its adjoint T* and is globally invariant under T and T*, then T is bounded. In the case of a Banach space (in particular, a C*-algebra) we showed in [6] that a densely defined closed derivation in a C*-algebra with domain containing its range is automatically bounded (see the references in [6] and [7] for the theory of derivations in C*-algebras). In general there exists a densely defined, unbounded closed linear operator with domain containing its range (see Example 3.1). Therefore it is of great interest to study the boundedness and properties of such an operator. We show in Section 2 that a dissipative closed linear operator in a Banach space with domain containing its range is automatically bounded. In Section 3, we deal with a densely defined, closed linear operator in a Hilbert space. Using the result in Section 2, we first show that a closed operator which maps its domain into the domain of its adjoint is bounded and, as a corollary, that a closed symmetric operator with domain containing its range is automatically bounded. Furthermore we study some properties of an unbounded closed operator with domain containing its range and show that the numerical range of such an operator is the whole complex plane.