Topic

# Spectral theorem

About: Spectral theorem is a research topic. Over the lifetime, 5454 publications have been published within this topic receiving 162512 citations.

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01 Jan 1966

TL;DR: The monograph by T Kato as discussed by the authors is an excellent reference work in the theory of linear operators in Banach and Hilbert spaces and is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory.

Abstract: "The monograph by T Kato is an excellent textbook in the theory of linear operators in Banach and Hilbert spaces It is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory
In chapters 1, 3, 5 operators in finite-dimensional vector spaces, Banach spaces and Hilbert spaces are introduced Stability and perturbation theory are studied in finite-dimensional spaces (chapter 2) and in Banach spaces (chapter 4) Sesquilinear forms in Hilbert spaces are considered in detail (chapter 6), analytic and asymptotic perturbation theory is described (chapter 7 and 8) The fundamentals of semigroup theory are given in chapter 9 The supplementary notes appearing in the second edition of the book gave mainly additional information concerning scattering theory described in chapter 10
The first edition is now 30 years old The revised edition is 20 years old Nevertheless it is a standard textbook for the theory of linear operators It is user-friendly in the sense that any sought after definitions, theorems or proofs may be easily located In the last two decades much progress has been made in understanding some of the topics dealt with in the book, for instance in semigroup and scattering theory However the book has such a high didactical and scientific standard that I can recomment it for any mathematician or physicist interested in this field
Zentralblatt MATH, 836

19,846 citations

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TL;DR: Dunford and Schwartz as discussed by the authors provided a comprehensive survey of the general theory of linear operations, together with applications to the diverse fields of more classical analysis, and emphasized the significance of the relationships between the abstract theory and its applications.

Abstract: This classic text, written by two notable mathematicians, constitutes a comprehensive survey of the general theory of linear operations, together with applications to the diverse fields of more classical analysis. Dunford and Schwartz emphasize the significance of the relationships between the abstract theory and its applications. This text has been written for the student as well as for the mathematician—treatment is relatively self-contained. This is a paperback edition of the original work, unabridged, in three volumes.

2,890 citations

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Lund University

^{1}TL;DR: In this paper, a more general class of pseudo-differential operators for non-elliptic problems is discussed. But their value is rather limited in genuinely nonelliptical problems.

Abstract: Pseudo-differential operators have been developed as a tool for the study of elliptic differential equations. Suitably extended versions are also applicable to hypoelliptic equations, but their value is rather limited in genuinely non-elliptic problems. In this paper we shall therefore discuss some more general classes of operators which are adapted to such applications. For these operators we shall develop a calculus which is almost as smooth as that of pseudo-differential operators. It also seems that one gains some more insight into the theory of pseudo-differential operators by considering them from the point of view of the wider classes of operators to be discussed here so we shall take the opportunity to include a short exposition.

2,450 citations