Spectra and Pseudospectra

Spectra and Pseudospectra PDF

Author: Lloyd N. Trefethen

Publisher: Princeton University Press

Published: 2020-05-26

Total Pages:

ISBN-13: 0691213100

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Pure and applied mathematicians, physicists, scientists, and engineers use matrices and operators and their eigenvalues in quantum mechanics, fluid mechanics, structural analysis, acoustics, ecology, numerical analysis, and many other areas. However, in some applications the usual analysis based on eigenvalues fails. For example, eigenvalues are often ineffective for analyzing dynamical systems such as fluid flow, Markov chains, ecological models, and matrix iterations. That's where this book comes in. This is the authoritative work on nonnormal matrices and operators, written by the authorities who made them famous. Each of the sixty sections is written as a self-contained essay. Each document is a lavishly illustrated introductory survey of its topic, complete with beautiful numerical experiments and all the right references. The breadth of included topics and the numerous applications that provide links between fields will make this an essential reference in mathematics and related sciences.

Linear Operators and their Spectra

Linear Operators and their Spectra PDF

Author: E. Brian Davies

Publisher: Cambridge University Press

Published: 2007-04-26

Total Pages: 436

ISBN-13: 1139464337

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This wide ranging but self-contained account of the spectral theory of non-self-adjoint linear operators is ideal for postgraduate students and researchers, and contains many illustrative examples and exercises. Fredholm theory, Hilbert-Schmidt and trace class operators are discussed, as are one-parameter semigroups and perturbations of their generators. Two chapters are devoted to using these tools to analyze Markov semigroups. The text also provides a thorough account of the new theory of pseudospectra, and presents the recent analysis by the author and Barry Simon of the form of the pseudospectra at the boundary of the numerical range. This was a key ingredient in the determination of properties of the zeros of certain orthogonal polynomials on the unit circle. Finally, two methods, both very recent, for obtaining bounds on the eigenvalues of non-self-adjoint Schrodinger operators are described. The text concludes with a description of the surprising spectral properties of the non-self-adjoint harmonic oscillator.

Spectral Theory of Multivalued Linear Operators

Spectral Theory of Multivalued Linear Operators PDF

Author: Aymen Ammar

Publisher: CRC Press

Published: 2021-09-14

Total Pages: 314

ISBN-13: 1000293092

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The concept of multivalued linear operators—or linear relations—is the one of the most exciting and influential fields of research in modern mathematics. Applications of this theory can be found in economic theory, noncooperative games, artificial intelligence, medicine, and more. This new book focuses on the theory of linear relations, responding to the lack of resources exclusively dealing with the spectral theory of multivalued linear operators. The subject of this book is the study of linear relations over real or complex Banach spaces. The main purposes are the definitions and characterization of different kinds of spectra and extending the notions of spectra that are considered for the usual one single-valued operator bounded or not bounded. The volume introduces the theory of pseudospectra of multivalued linear operators. The main topics include demicompact linear relations, essential spectra of linear relation, pseudospectra, and essential pseudospectra of linear relations. The volume will be very useful for researchers since it represents not only a collection of a previously heterogeneous material but is also an innovation through several extensions. Beginning graduate students who wish to enter the field of spectral theory of multivalued linear operators will benefit from the material covered, and expert readers will also find sources of inspiration.

S-Essential Spectra and the Weyl Pseudospectra of Linear Operators

S-Essential Spectra and the Weyl Pseudospectra of Linear Operators PDF

Author: Ammar Aymen

Publisher: LAP Lambert Academic Publishing

Published: 2015-10-22

Total Pages: 132

ISBN-13: 9783659681127

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In this work we present a spectral study of semi regular operator. The concept of semi regularity and essentially semi regularity amongst the various concepts of regularity originating from Fredholm theory, and we introduce new notions and unprecedented denition to develop pseudospectral theory. We built our denition on the basis of essential spectra and tried to comply it with pseudospectral theory. So, we intend to obtain what we call "Weyl pseudospectra."

Linear Operators and Their Essential Pseudospectra

Linear Operators and Their Essential Pseudospectra PDF

Author: Aref Jeribi

Publisher: CRC Press

Published: 2018-04-17

Total Pages: 352

ISBN-13: 1351046268

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Linear Operators and Their Essential Pseudospectra provides a comprehensive study of spectral theory of linear operators defined on Banach spaces. The central items of interest in the volume include various essential spectra, but the author also considers some of the generalizations that have been studied. In recent years, spectral theory has witnessed an explosive development. This volume presents a survey of results concerning various types of essential spectra and pseudospectra in a unified, axiomatic way and also discusses several topics that are new but which relate to the concepts and methods emanating from the book. The main topics include essential spectra, essential pseudospectra, structured essential pseudospectra, and their relative sets. This volume will be very useful for several researchers since it represents not only a collection of previously heterogeneous material but also includes discussions of innovation through several extensions. As the spectral theory of operators is an important part of functional analysis and has numerous applications in many areas of mathematics, the author suggests that some modest prerequisites from functional analysis and operator theory should be in place to be accessible to newcomers and graduate students of mathematics.

Pseudospectra and Structured Pseudospectra

Pseudospectra and Structured Pseudospectra PDF

Author: John D. Hardee

Publisher:

Published: 2012

Total Pages: 54

ISBN-13:

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"Pseudospectra and structured pseudospectra are subsets of the complex plane which give a geometric representation, via eigenvalues, of the effects of perturbations to a matrix. We survey the historical development of the subject, and the definitions and characterizations of the various sets of pseudospectra. Motivated by the fact that a nonnormal matrix in the 2-norm can become normal in a different norm, we describe a numerical investigation into the question of characterizing which perturbations have the greatest effect on the eigenvalues of the matrix."--Abstract from author supplied metadata.

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