Numerical Ranges of Hilbert Space Operators

Numerical Ranges of Hilbert Space Operators PDF

Author: Hwa-Long Gau

Publisher: Cambridge University Press

Published: 2021-08-05

Total Pages: 556

ISBN-13: 1108787606

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Starting with elementary operator theory and matrix analysis, this book introduces the basic properties of the numerical range and gradually builds up the whole numerical range theory. Over 400 assorted problems, ranging from routine exercises to published research results, give you the chance to put the theory into practice and test your understanding. Interspersed throughout the text are numerous comments and references, allowing you to discover related developments and to pursue areas of interest in the literature. Also included is an appendix on basic convexity properties on the Euclidean space. Targeted at graduate students as well as researchers interested in functional analysis, this book provides a comprehensive coverage of classic and recent works on the numerical range theory. It serves as an accessible entry point into this lively and exciting research area.

Numerical Range

Numerical Range PDF

Author: Karl E. Gustafson

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 202

ISBN-13: 1461384982

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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.

Inequalities for the Numerical Radius of Linear Operators in Hilbert Spaces

Inequalities for the Numerical Radius of Linear Operators in Hilbert Spaces PDF

Author: Silvestru Sever Dragomir

Publisher: Springer Science & Business Media

Published: 2013-09-14

Total Pages: 130

ISBN-13: 331901448X

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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.

The Numerical Range and the Core of Hilbert-space Operators [microform]

The Numerical Range and the Core of Hilbert-space Operators [microform] PDF

Author: Ching-Nam Hung

Publisher: Library and Archives Canada = Bibliothèque et Archives Canada

Published: 2004

Total Pages: 160

ISBN-13: 9780612944039

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The main object of this thesis is to study the numerical range of Hilbert-space operators. In 1973, T. Ando examined the geometric and algebraic properties of operators and developed a structure theory. In continuation of his work, there has been much progress, especially in the study of the core of a numerical contraction in terms of dilation theory and representation theory. In the first half of this thesis, explicit expressions for the minimum and the maximum of the core of a numerical contraction are studied. The expressions for these extremals are given as strongly convergent non-commutative operator series in terms of the given numerical contraction and its adjoint. This part of the thesis serves as a complement to T. Ando's theorem, in which we find that the operator series provides an efficient mechanism for writing a numerical contraction in terms of dilations and representations. The main tool employed is the theory of Schur complements of positive semi-definite operator matrices. Further discussions on the classical Catalan problem and another related combinatorial problem are also presented. In the second half of this thesis, matrices whose numerical ranges are the closed unit disc are investigated, and the structural expressions of those matrices are studied. As a result, matrices having elliptical discs as numerical range are found to possess the property that the foci of the disc are their eigenvalues. The structure theory obtained by T. Ando, especially the representation of numerical contractions, is essential in proving these results. Finally, the structural expressions of matrices with numerical range equal to the closed unit disc are used to provide an alternative proof for P.Y. Wu's theorem concerning the norms of matrices.

Operator Theory

Operator Theory PDF

Author: Samuel Kisengo

Publisher: LAP Lambert Academic Publishing

Published: 2011-10

Total Pages: 84

ISBN-13: 9783846532522

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This book is about numerical ranges and spectral properties of Hilbert space operators which have been of great interest to many mathematicians in the past decades. Nice properties with examples are explored. The properties of numerical range, for example, convexity and closedness are well known as proved in the classic Toeplitz - Hausdorff Theorem. In this book, we embark on the relationship between the spectrum and the numerical range of an operator, in particular, when the operator is normal. It is known that for a bounded linear operator on a Hilbert space, the spectrum is contained in the closure of its numerical range. For a normal operator, the numerical radius and the spectral radius coincides with the norm of the operator. These results are actually a contribution to the field of numerical ranges and spectra. For the reader to understand this book, it is paramount that a deep understanding of the theory of operators, especially on Hilbert spaces, General Topology, Functional Analysis and Abstract Algebra be put in place. This book is useful to both undergraduate students and postgraduate students.

Numerical Ranges II

Numerical Ranges II PDF

Author: F. F. Bonsall

Publisher: Cambridge University Press

Published: 1973-08-02

Total Pages: 189

ISBN-13: 0521202272

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