Numerical Ranges And Commutation Properties Of Hilbert Space Operators
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Commutation Properties Of Hilbert Space Operators And Related Topics
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Author : Calvin R. Putnam
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Commutation Properties Of Hilbert Space Operators And Related Topics written by Calvin R. Putnam and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
What could be regarded as the beginning of a theory of commutators AB - BA of operators A and B on a Hilbert space, considered as a dis cipline in itself, goes back at least to the two papers of Weyl [3] {1928} and von Neumann [2] {1931} on quantum mechanics and the commuta tion relations occurring there. Here A and B were unbounded self-adjoint operators satisfying the relation AB - BA = iI, in some appropriate sense, and the problem was that of establishing the essential uniqueness of the pair A and B. The study of commutators of bounded operators on a Hilbert space has a more recent origin, which can probably be pinpointed as the paper of Wintner [6] {1947}. An investigation of a few related topics in the subject is the main concern of this brief monograph. The ensuing work considers commuting or "almost" commuting quantities A and B, usually bounded or unbounded operators on a Hilbert space, but occasionally regarded as elements of some normed space. An attempt is made to stress the role of the commutator AB - BA, and to investigate its properties, as well as those of its components A and B when the latter are subject to various restrictions. Some applica tions of the results obtained are made to quantum mechanics, perturba tion theory, Laurent and Toeplitz operators, singular integral trans formations, and Jacobi matrices.
Numerical Ranges And Commutation Properties Of Hilbert Space Operators
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Author : Bruce Kent Richard
language : en
Publisher:
Release Date : 1974
Numerical Ranges And Commutation Properties Of Hilbert Space Operators written by Bruce Kent Richard and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1974 with categories.
Product And Commutation Properties Of Hilbert Space Operators
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Author : Michael John Hoffman
language : en
Publisher:
Release Date : 1979
Product And Commutation Properties Of Hilbert Space Operators written by Michael John Hoffman and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1979 with categories.
Operator Commutation Relations
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Author : P.E.T. Jørgensen
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Operator Commutation Relations written by P.E.T. Jørgensen and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
In his Retiring Presidential address, delivered before the Annual Meeting of The American Mathematical Society on December, 1948, the late Professor Einar Hille spoke on his recent results on the Lie theory of semigroups of linear transformations, . . • "So far only commutative operators have been considered and the product law . . . is the simplest possible. The non-commutative case has resisted numerous attacks in the past and it is only a few months ago that any headway was made with this problem. I shall have the pleasure of outlining the new theory here; it is a blend of the classical theory of Lie groups with the recent theory of one-parameter semigroups. " The list of references in the subsequent publication of Hille's address (Bull. Amer. Math •. Soc. 56 (1950)) includes pioneering papers of I. E. Segal, I. M. Gelfand, and K. Yosida. In the following three decades the subject grew tremendously in vitality, incorporating a number of different fields of mathematical analysis. Early papers of V. Bargmann, I. E. Segal, L. G~ding, Harish-Chandra, I. M. Singer, R. Langlands, B. Konstant, and E. Nelson developed the theoretical basis for later work in a variety of different applications: Mathematical physics, astronomy, partial differential equations, operator algebras, dynamical systems, geometry, and, most recently, stochastic filtering theory. As it turned out, of course, the Lie groups, rather than the semigroups, provided the focus of attention.
Inequalities For The Numerical Radius Of Linear Operators In Hilbert Spaces
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Author : Silvestru Sever Dragomir
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-09-14
Inequalities For The Numerical Radius Of Linear Operators In Hilbert Spaces written by Silvestru Sever Dragomir and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-09-14 with Mathematics categories.
Aimed toward researchers, postgraduate students, and scientists in linear operator theory and mathematical inequalities, this self-contained monograph focuses on numerical radius inequalities for bounded linear operators on complex Hilbert spaces for the case of one and two operators. Students at the graduate level will learn some essentials that may be useful for reference in courses in functional analysis, operator theory, differential equations, and quantum computation, to name several. Chapter 1 presents fundamental facts about the numerical range and the numerical radius of bounded linear operators in Hilbert spaces. Chapter 2 illustrates recent results obtained concerning numerical radius and norm inequalities for one operator on a complex Hilbert space, as well as some special vector inequalities in inner product spaces due to Buzano, Goldstein, Ryff and Clarke as well as some reverse Schwarz inequalities and Grüss type inequalities obtained by the author. Chapter 3 presents recent results regarding the norms and the numerical radii of two bounded linear operators. The techniques shown in this chapter are elementary but elegant and may be accessible to undergraduate students with a working knowledge of operator theory. A number of vector inequalities in inner product spaces as well as inequalities for means of nonnegative real numbers are also employed in this chapter. All the results presented are completely proved and the original references are mentioned.
Matrix And Operator Equations And Applications
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Author : Mohammad Sal Moslehian
language : en
Publisher: Springer Nature
Release Date : 2023-07-29
Matrix And Operator Equations And Applications written by Mohammad Sal Moslehian and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-07-29 with Mathematics categories.
This book concerns matrix and operator equations that are widely applied in various disciplines of science to formulate challenging problems and solve them in a faithful way. The main aim of this contributed book is to study several important matrix and operator equalities and equations in a systematic and self-contained fashion. Some powerful methods have been used to investigate some significant equations in functional analysis, operator theory, matrix analysis, and numerous subjects in the last decades. The book is divided into two parts: (I) Matrix Equations and (II) Operator Equations. In the first part, the state-of-the-art of systems of matrix equations is given and generalized inverses are used to find their solutions. The semi-tensor product of matrices is used to solve quaternion matrix equations. The contents of some chapters are related to the relationship between matrix inequalities, matrix means, numerical range, and matrix equations. In addition, quaternion algebras and their applications are employed in solving some famous matrix equations like Sylvester, Stein, and Lyapunov equations. A chapter devoted to studying Hermitian polynomial matrix equations, which frequently arise from linear-quadratic control problems. Moreover, some classical and recently discovered inequalities for matrix exponentials are reviewed. In the second part, the latest developments in solving several equations appearing in modern operator theory are demonstrated. These are of interest to a wide audience of pure and applied mathematicians. For example, the Daugavet equation in the linear and nonlinear setting, iterative processes and Volterra-Fredholm integral equations, semicircular elements induced by connected finite graphs, free probability, singular integral operators with shifts, and operator differential equations closely related to the properties of the coefficient operators in some equations are discussed. The chapters give a comprehensive account of their subjects. The exhibited chapters are written in a reader-friendly style and can be read independently. Each chapter contains a rich bibliography. This book is intended for use by both researchers and graduate students of mathematics, physics, and engineering.
Numerical Range
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Author : Karl E. Gustafson
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Numerical Range written by Karl E. Gustafson and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
The theories of quadratic forms and their applications appear in many parts of mathematics and the sciences. All students of mathematics have the opportunity to encounter such concepts and applications in their first course in linear algebra. This subject and its extensions to infinite dimen sions comprise the theory of the numerical range W(T). There are two competing names for W(T), namely, the numerical range of T and the field of values for T. The former has been favored historically by the func tional analysis community, the latter by the matrix analysis community. It is a toss-up to decide which is preferable, and we have finally chosen the former because it is our habit, it is a more efficient expression, and because in recent conferences dedicated to W(T), even the linear algebra commu nity has adopted it. Also, one universally refers to the numerical radius, and not to the field of values radius. Originally, Toeplitz and Hausdorff called it the Wertvorrat of a bilinear form, so other good names would be value field or form values. The Russian community has referred to it as the Hausdorff domain. Murnaghan in his early paper first called it the region of the complex plane covered by those values for an n x n matrix T, then the range of values of a Hermitian matrix, then the field of values when he analyzed what he called the sought-for region.
Numerical Ranges Of Operators On Normed Spaces And Of Elements Of Normed Algebras
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Author : F. F. Bonsall
language : en
Publisher: CUP Archive
Release Date : 1971-03-02
Numerical Ranges Of Operators On Normed Spaces And Of Elements Of Normed Algebras written by F. F. Bonsall and has been published by CUP Archive this book supported file pdf, txt, epub, kindle and other format this book has been release on 1971-03-02 with Mathematics categories.
The authors develop various applications, in particular to the study of Banach algebras where the numerical range provides an important link between the algebraic and metric structures.
Operators On Hilbert Space
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Author : V. S. Sunder
language : en
Publisher: Springer
Release Date : 2016-08-05
Operators On Hilbert Space written by V. S. Sunder and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-08-05 with Mathematics categories.
The primarily objective of the book is to serve as a primer on the theory of bounded linear operators on separable Hilbert space. The book presents the spectral theorem as a statement on the existence of a unique continuous and measurable functional calculus. It discusses a proof without digressing into a course on the Gelfand theory of commutative Banach algebras. The book also introduces the reader to the basic facts concerning the various von Neumann–Schatten ideals, the compact operators, the trace-class operators and all bounded operators.
Unitary Invariants In Multivariable Operator Theory
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Author : Gelu Popescu
language : en
Publisher: American Mathematical Soc.
Release Date : 2009-06-05
Unitary Invariants In Multivariable Operator Theory written by Gelu Popescu and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-06-05 with Mathematics categories.
This paper concerns unitary invariants for $n$-tuples $T:=(T_1,\ldots, T_n)$ of (not necessarily commuting) bounded linear operators on Hilbert spaces. The author introduces a notion of joint numerical radius and works out its basic properties. Multivariable versions of Berger's dilation theorem, Berger-Kato-Stampfli mapping theorem, and Schwarz's lemma from complex analysis are obtained. The author studies the joint (spatial) numerical range of $T$ in connection with several unitary invariants for $n$-tuples of operators such as: right joint spectrum, joint numerical radius, euclidean operator radius, and joint spectral radius. He also proves an analogue of Toeplitz-Hausdorff theorem on the convexity of the spatial numerical range of an operator on a Hilbert space, for the joint numerical range of operators in the noncommutative analytic Toeplitz algebra $F_n^\infty$.