On the Duality of Operator Spaces
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Differentiation of Operator Functions in Non-Commutative Lp-Spaces B
ARTICLE IN PRESS Journal of Functional Analysis 212 (2004) 28–75 Differentiation of operator functions in non-commutative Lp-spaces B. de Pagtera and F.A. Sukochevb,Ã a Department of Mathematics, Faculty ITS, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands b School of Informatics and Engineering, Flinders University of South Australia, Bedford Park, 5042 SA, Australia Received 14 May 2003; revised 18 September 2003; accepted 15 October 2003 Communicated by G. Pisier Abstract The principal results in this paper are concerned with the description of differentiable operator functions in the non-commutative Lp-spaces, 1ppoN; associated with semifinite von Neumann algebras. For example, it is established that if f : R-R is a Lipschitz function, then the operator function f is Gaˆ teaux differentiable in L2ðM; tÞ for any semifinite von Neumann algebra M if and only if it has a continuous derivative. Furthermore, if f : R-R has a continuous derivative which is of bounded variation, then the operator function f is Gaˆ teaux differentiable in any LpðM; tÞ; 1opoN: r 2003 Elsevier Inc. All rights reserved. 1. Introduction Given an arbitrary semifinite von Neumann algebra M on the Hilbert space H; we associate with any Borel function f : R-R; via the usual functional calculus, the corresponding operator function a/f ðaÞ having as domain the set of all (possibly unbounded) self-adjoint operators a on H which are affiliated with M: This paper is concerned with the study of the differentiability properties of such operator functions f : Let LpðM; tÞ; with 1pppN; be the non-commutative Lp-space associated with ðM; tÞ; where t is a faithful normal semifinite trace on the von f Neumann algebra M and let M be the space of all t-measurable operators affiliated ÃCorresponding author. -
The Uniform Boundedness Theorem in Asymmetric Normed Spaces
Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2012, Article ID 809626, 8 pages doi:10.1155/2012/809626 Research Article The Uniform Boundedness Theorem in Asymmetric Normed Spaces C. Alegre,1 S. Romaguera,1 and P. Veeramani2 1 Instituto Universitario de Matematica´ Pura y Aplicada, Universitat Politecnica` de Valencia,` 46022 Valencia, Spain 2 Department of Mathematics, Indian Institute of Technology Madras, Chennai 6000 36, India Correspondence should be addressed to C. Alegre, [email protected] Received 5 July 2012; Accepted 27 August 2012 Academic Editor: Yong Zhou Copyright q 2012 C. Alegre et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We obtain a uniform boundedness type theorem in the frame of asymmetric normed spaces. The classical result for normed spaces follows as a particular case. 1. Introduction Throughout this paper the letters R and R will denote the set of real numbers and the set of nonnegative real numbers, respectively. The book of Aliprantis and Border 1 provides a good basic reference for functional analysis in our context. Let X be a real vector space. A function p : X → R is said to be an asymmetric norm on X 2, 3 if for all x, y ∈ X,andr ∈ R , i pxp−x0 if and only if x 0; ii prxrpx; iii px y ≤ pxpy. The pair X, p is called an asymmetric normed space. Asymmetric norms are also called quasinorms in 4–6, and nonsymmetric norms in 7. -
On the Second Dual of the Space of Continuous Functions
ON THE SECOND DUAL OF THE SPACE OF CONTINUOUS FUNCTIONS BY SAMUEL KAPLAN Introduction. By "the space of continuous functions" we mean the Banach space C of real continuous functions on a compact Hausdorff space X. C, its first dual, and its second dual are not only Banach spaces, but also vector lattices, and this aspect of them plays a central role in our treatment. In §§1-3, we collect the properties of vector lattices which we need in the paper. In §4 we add some properties of the first dual of C—the space of Radon measures on X—denoted by L in the present paper. In §5 we imbed C in its second dual, denoted by M, and then identify the space of all bounded real functions on X with a quotient space of M, in fact with a topological direct summand. Thus each bounded function on X represents an entire class of elements of M. The value of an element/ of M on an element p. of L is denoted as usual by/(p.)- If/lies in C then of course f(u) =p(f) (usually written J fdp), and thus the multiplication between Afand P is an extension of the multiplication between L and C. Now in integration theory, for each pEL, there is a stand- ard "extension" of ffdu to certain of the bounded functions on X (we confine ourselves to bounded functions in this paper), which are consequently called u-integrable. The question arises: how is this "extension" related to the multi- plication between M and L. -
Introduction to Operator Spaces
Lecture 1: Introduction to Operator Spaces Zhong-Jin Ruan at Leeds, Monday, 17 May , 2010 1 Operator Spaces A Natural Quantization of Banach Spaces 2 Banach Spaces A Banach space is a complete normed space (V/C, k · k). In Banach spaces, we consider Norms and Bounded Linear Maps. Classical Examples: ∗ C0(Ω),M(Ω) = C0(Ω) , `p(I),Lp(X, µ), 1 ≤ p ≤ ∞. 3 Hahn-Banach Theorem: Let V ⊆ W be Banach spaces. We have W ↑ & ϕ˜ ϕ V −−−→ C with kϕ˜k = kϕk. It follows from the Hahn-Banach theorem that for every Banach space (V, k · k) we can obtain an isometric inclusion (V, k · k) ,→ (`∞(I), k · k∞) ∗ ∗ where we may choose I = V1 to be the closed unit ball of V . So we can regard `∞(I) as the home space of Banach spaces. 4 Classical Theory Noncommutative Theory `∞(I) B(H) Banach Spaces Operator Spaces (V, k · k) ,→ `∞(I)(V, ??) ,→ B(H) norm closed subspaces of B(H)? 5 Matrix Norm and Concrete Operator Spaces [Arveson 1969] Let B(H) denote the space of all bounded linear operators on H. For each n ∈ N, n H = H ⊕ · · · ⊕ H = {[ξj]: ξj ∈ H} is again a Hilbert space. We may identify ∼ Mn(B(H)) = B(H ⊕ ... ⊕ H) by letting h i h i X Tij ξj = Ti,jξj , j and thus obtain an operator norm k · kn on Mn(B(H)). A concrete operator space is norm closed subspace V of B(H) together with the canonical operator matrix norm k · kn on each matrix space Mn(V ). -
Notes on Linear Functional Analysis by M.A Sofi
Linear Functional Analysis Prof. M. A. Sofi Department of Mathematics University of Kashmir Srinagar-190006 1 Bounded linear operators In this section, we shall characterize continuity of linear operators acting between normed spaces. It turns out that a linear operator is continuous on a normed linear space as soon as it is continuous at the origin or for that matter, at any point of the domain of its definition. Theorem 1.1. Given normed spaces X and Y and a linear map T : X ! Y; then T is continuous on X if and only if 9 c > 0 such that kT (x)k ≤ ckxk; 8 x 2 X: (1) Proof. Assume that T is continuous. In particular, T is continuous at the origin. By the definition of continuity, since T (0) = 0; there exits a neigh- bourhood U at the origin in X such that T (U) ⊆ B(Y ): But then 9r > 0 such that Sr(0) ⊂ U: This gives T (Sr(0)) ⊆ T (U) ⊆ B(Y ): (2) x Let x(6= 0) 2 X; for otherwise, (1) is trivially satisfied. Then kxk 2 Sr(0); so rx that by (2), T 2kxk 2 B(Y ): In other words, 2 kT (x)k ≤ ckxk where c = r which gives (1). Conversely, assume that (1) is true. To show that T is continuous on X; let 1 n n x 2 X and assume that xn −! x in X: Then xn − x −! 0 in X: Thus, 8 > 0; 9 N 3: 8 n > N; kT (xn) − T (x)k = kT (xn − x)k ≤ ckxn − xk ≤ , n which holds for all n > N: In other words, T (xn) −! T (x) in Y and therefore, T is continuous at x 2 X: Since x 2 X was arbitrarily chosen, it follows that T is continuous on X: Definition 1.2. -
Duality, Part 1: Dual Bases and Dual Maps Notation
Duality, part 1: Dual Bases and Dual Maps Notation F denotes either R or C. V and W denote vector spaces over F. Define ': R3 ! R by '(x; y; z) = 4x − 5y + 2z. Then ' is a linear functional on R3. n n Fix (b1;:::; bn) 2 C . Define ': C ! C by '(z1;:::; zn) = b1z1 + ··· + bnzn: Then ' is a linear functional on Cn. Define ': P(R) ! R by '(p) = 3p00(5) + 7p(4). Then ' is a linear functional on P(R). R 1 Define ': P(R) ! R by '(p) = 0 p(x) dx. Then ' is a linear functional on P(R). Examples: Linear Functionals Definition: linear functional A linear functional on V is a linear map from V to F. In other words, a linear functional is an element of L(V; F). n n Fix (b1;:::; bn) 2 C . Define ': C ! C by '(z1;:::; zn) = b1z1 + ··· + bnzn: Then ' is a linear functional on Cn. Define ': P(R) ! R by '(p) = 3p00(5) + 7p(4). Then ' is a linear functional on P(R). R 1 Define ': P(R) ! R by '(p) = 0 p(x) dx. Then ' is a linear functional on P(R). Linear Functionals Definition: linear functional A linear functional on V is a linear map from V to F. In other words, a linear functional is an element of L(V; F). Examples: Define ': R3 ! R by '(x; y; z) = 4x − 5y + 2z. Then ' is a linear functional on R3. Define ': P(R) ! R by '(p) = 3p00(5) + 7p(4). Then ' is a linear functional on P(R). R 1 Define ': P(R) ! R by '(p) = 0 p(x) dx. -
Eberlein-Šmulian Theorem and Some of Its Applications
Eberlein-Šmulian theorem and some of its applications Kristina Qarri Supervisors Trond Abrahamsen Associate professor, PhD University of Agder Norway Olav Nygaard Professor, PhD University of Agder Norway This master’s thesis is carried out as a part of the education at the University of Agder and is therefore approved as a part of this education. However, this does not imply that the University answers for the methods that are used or the conclusions that are drawn. University of Agder, 2014 Faculty of Engineering and Science Department of Mathematics Contents Abstract 1 1 Introduction 2 1.1 Notation and terminology . 4 1.2 Cornerstones in Functional Analysis . 4 2 Basics of weak and weak* topologies 6 2.1 The weak topology . 7 2.2 Weak* topology . 16 3 Schauder Basis Theory 21 3.1 First Properties . 21 3.2 Constructing basic sequences . 37 4 Proof of the Eberlein Šmulian theorem due to Whitley 50 5 The weak topology and the topology of pointwise convergence on C(K) 58 6 A generalization of the Ebrlein-Šmulian theorem 64 7 Some applications to Tauberian operator theory 69 Summary 73 i Abstract The thesis is about Eberlein-Šmulian and some its applications. The goal is to investigate and explain different proofs of the Eberlein-Šmulian theorem. First we introduce the general theory of weak and weak* topology defined on a normed space X. Next we present the definition of a basis and a Schauder basis of a given Banach space. We give some examples and prove the main theorems which are needed to enjoy the proof of the Eberlein-Šmulian theorem given by Pelchynski in 1964. -
A Novel Dual Approach to Nonlinear Semigroups of Lipschitz Operators
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 357, Number 1, Pages 409{424 S 0002-9947(04)03635-9 Article electronically published on August 11, 2004 A NOVEL DUAL APPROACH TO NONLINEAR SEMIGROUPS OF LIPSCHITZ OPERATORS JIGEN PENG AND ZONGBEN XU Abstract. Lipschitzian semigroup refers to a one-parameter semigroup of Lipschitz operators that is strongly continuous in the parameter. It con- tains C0-semigroup, nonlinear semigroup of contractions and uniformly k- Lipschitzian semigroup as special cases. In this paper, through developing a series of Lipschitz dual notions, we establish an analysis approach to Lips- chitzian semigroup. It is mainly proved that a (nonlinear) Lipschitzian semi- group can be isometrically embedded into a certain C0-semigroup. As ap- plication results, two representation formulas of Lipschitzian semigroup are established, and many asymptotic properties of C0-semigroup are generalized to Lipschitzian semigroup. 1. Introduction Let X and Y be Banach spaces over the same coefficient field K (= R or C), and let C ⊂ X and D ⊂ Y be their subsets. A mapping T from C into D is called Lipschitz operator if there exists a real constant M>0 such that (1.1) k Tx− Ty k≤ M k x − y k; 8x; y 2 C; where the constant M is commonly referred to as a Lipschitz constant of T . Clearly, if the mapping T is a constant (that is, there exists a vector y0 2 D such that Tx= y0 for all x 2 C), then T is a Lipschitz operator. Let Lip(C; D) denote the space of Lipschitz operators from C into D,andfor every T 2 Lip(C; D)letL(T ) denote the minimum Lipschitz constant of T , i.e., k Tx− Ty k (1.2) L(T )= sup : x;y2C;x=6 y k x − y k Then, it can be shown that the nonnegative functional L(·)isaseminormof Lip(C; D) and hence (Lip(C; D);L(·)) is a seminormed linear space. -
Multilinear Algebra
Appendix A Multilinear Algebra This chapter presents concepts from multilinear algebra based on the basic properties of finite dimensional vector spaces and linear maps. The primary aim of the chapter is to give a concise introduction to alternating tensors which are necessary to define differential forms on manifolds. Many of the stated definitions and propositions can be found in Lee [1], Chaps. 11, 12 and 14. Some definitions and propositions are complemented by short and simple examples. First, in Sect. A.1 dual and bidual vector spaces are discussed. Subsequently, in Sects. A.2–A.4, tensors and alternating tensors together with operations such as the tensor and wedge product are introduced. Lastly, in Sect. A.5, the concepts which are necessary to introduce the wedge product are summarized in eight steps. A.1 The Dual Space Let V be a real vector space of finite dimension dim V = n.Let(e1,...,en) be a basis of V . Then every v ∈ V can be uniquely represented as a linear combination i v = v ei , (A.1) where summation convention over repeated indices is applied. The coefficients vi ∈ R arereferredtoascomponents of the vector v. Throughout the whole chapter, only finite dimensional real vector spaces, typically denoted by V , are treated. When not stated differently, summation convention is applied. Definition A.1 (Dual Space)Thedual space of V is the set of real-valued linear functionals ∗ V := {ω : V → R : ω linear} . (A.2) The elements of the dual space V ∗ are called linear forms on V . © Springer International Publishing Switzerland 2015 123 S.R. -
Spectral Properties of the Koopman Operator in the Analysis of Nonstationary Dynamical Systems
UNIVERSITY of CALIFORNIA Santa Barbara Spectral Properties of the Koopman Operator in the Analysis of Nonstationary Dynamical Systems A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering by Ryan M. Mohr Committee in charge: Professor Igor Mezi´c,Chair Professor Bassam Bamieh Professor Jeffrey Moehlis Professor Mihai Putinar June 2014 The dissertation of Ryan M. Mohr is approved: Bassam Bamieh Jeffrey Moehlis Mihai Putinar Igor Mezi´c,Committee Chair May 2014 Spectral Properties of the Koopman Operator in the Analysis of Nonstationary Dynamical Systems Copyright c 2014 by Ryan M. Mohr iii To my family and dear friends iv Acknowledgements Above all, I would like to thank my family, my parents Beth and Jim, and brothers Brennan and Gralan. Without their support, encouragement, confidence and love, I would not be the person I am today. I am also grateful to my advisor, Professor Igor Mezi´c,who introduced and guided me through the field of dynamical systems and encouraged my mathematical pursuits, even when I fell down the (many) proverbial (mathematical) rabbit holes. I learned many fascinating things from these wandering forays on a wide range of topics, both contributing to my dissertation and not. Our many interesting discussions on math- ematics, research, and philosophy are among the highlights of my graduate school career. His confidence in my abilities and research has been a constant source of encouragement, especially when I felt uncertain in them myself. His enthusiasm for science and deep, meaningful work has set the tone for the rest of my career and taught me the level of research problems I should consider and the quality I should strive for. -
Recent Developments in the Theory of Duality in Locally Convex Vector Spaces
[ VOLUME 6 I ISSUE 2 I APRIL– JUNE 2019] E ISSN 2348 –1269, PRINT ISSN 2349-5138 RECENT DEVELOPMENTS IN THE THEORY OF DUALITY IN LOCALLY CONVEX VECTOR SPACES CHETNA KUMARI1 & RABISH KUMAR2* 1Research Scholar, University Department of Mathematics, B. R. A. Bihar University, Muzaffarpur 2*Research Scholar, University Department of Mathematics T. M. B. University, Bhagalpur Received: February 19, 2019 Accepted: April 01, 2019 ABSTRACT: : The present paper concerned with vector spaces over the real field: the passage to complex spaces offers no difficulty. We shall assume that the definition and properties of convex sets are known. A locally convex space is a topological vector space in which there is a fundamental system of neighborhoods of 0 which are convex; these neighborhoods can always be supposed to be symmetric and absorbing. Key Words: LOCALLY CONVEX SPACES We shall be exclusively concerned with vector spaces over the real field: the passage to complex spaces offers no difficulty. We shall assume that the definition and properties of convex sets are known. A convex set A in a vector space E is symmetric if —A=A; then 0ЄA if A is not empty. A convex set A is absorbing if for every X≠0 in E), there exists a number α≠0 such that λxЄA for |λ| ≤ α ; this implies that A generates E. A locally convex space is a topological vector space in which there is a fundamental system of neighborhoods of 0 which are convex; these neighborhoods can always be supposed to be symmetric and absorbing. Conversely, if any filter base is given on a vector space E, and consists of convex, symmetric, and absorbing sets, then it defines one and only one topology on E for which x+y and λx are continuous functions of both their arguments. -
Norm Attaining Functionals on C(T)
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 1, January 1998, Pages 153{157 S 0002-9939(98)04008-8 NORM ATTAINING FUNCTIONALS ON C(T ) P. S. KENDEROV, W. B. MOORS, AND SCOTT SCIFFER (Communicated by Dale Alspach) Abstract. It is shown that for any infinite compact Hausdorff space T ,the Bishop-Phelps set in C(T )∗ is of the first Baire category when C(T )hasthe supremum norm. For a Banach space X with closed unit ball B(X), the Bishop-Phelps set is the set of all functionals in the dual X∗ which attain their supremum on B(X); that is, the set µ X∗:thereexistsx B(X) with µ(x)= µ .Itiswellknownthat { ∈ ∈ k k} the Bishop-Phelps set is always dense in the dual X∗. It is more difficult, however, to decide whether the Bishop-Phelps set is residual as this would require a concrete characterisation of those linear functionals in X∗ which attain their supremum on B(X). In general it is difficult to characterise such functionals, however it is possible when X = C(T ), for some compact T , with the supremum norm. The aim of this note is to show that in this case the Bishop-Phelps set is a first Baire category subset of C(T )∗. A problem of considerable interest in differentiability theory is the following. If the Bishop-Phelps set is a residual subset of the dual X ∗, is the dual norm necessarily densely Fr´echet differentiable? It is known, for example, that this is the case if the dual X∗ is weak Asplund, [1, Corollary 1.6(i)], or if X has an equivalent weakly mid-point locally uniformly rotund norm and the weak topology on X is σ-fragmented by the norm, [2, Theo- rems 3.3 and 4.4].