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Published in:
Spectrum of an Operator Read chapter Read first chapter

2018 | OriginalPaper | Chapter

5. Compact Operators

Author : Piotr Sołtan

Published in: A Primer on Hilbert Space Operators

Publisher: Springer International Publishing

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Abstract

In this chapter we will discuss the most fundamental results on compact operators on Hilbert spaces. It is important to stress that analogous theory for operators on Banach spaces is somewhat more complicated and, in particular, requires more considerations of topological nature. The very definition of a compact operator between Banach spaces is significantly different from the one we will propose in the case of operators on a Hilbert space. In the more general framework of Banach spaces the definition of a compact operator amounts to the conclusion of Proposition 5.1. An interested reader will find more information on compact operators in any functional analysis textbook, e.g. in [Rud2, Chapter 4].

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previous chapter Spectral Theorems and Functional Calculus
next chapter The Trace
Footnotes
1
A subset of a topological space is pre-compact if its closure is compact. A subset S of a complete metric space is pre-compact if and only if for any ε > 0 there exists a finite family of balls with radius ε covering S. The collection of centers of such balls is called an ε-net for S.
 
Literature
[Hal]
[Mau]
K. Maurin, Methods of Hilbert Spaces (Polish Scientific Publishers, Warsaw, 1972)MATH
[Ped]
[ReSi1]
M. Reed, B. Simon, Methods of Modern Mathematical Physics I. Functional Analysis (Academic Press, London, 1980)
[Rud2]
W. Rudin, Functional Analysis (McGraw-Hill, New York, 1991)MATH
Metadata
Title
Compact Operators
Author
Piotr Sołtan
Copyright Year
2018
Book
A Primer on Hilbert Space Operators

Print ISBN: 978-3-319-92060-3

Electronic ISBN: 978-3-319-92061-0

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-92061-0_5

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