## Linear Operators: Spectral theory |

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Page 1145

The Peter-Weyl Theorem 1.4 is basic to the theory of

groups. The principal definitions and theorems of this theory are as follows.

DEFINITION: Let G be a topological group, and 3: a B-space. Then a

The Peter-Weyl Theorem 1.4 is basic to the theory of

**representations**of compactgroups. The principal definitions and theorems of this theory are as follows.

DEFINITION: Let G be a topological group, and 3: a B-space. Then a

**representation**R ...Page 1146

M. Any finite dimensional

irreducible

dimensional

generality, ...

M. Any finite dimensional

**representation**of a compact group G is a direct sum ofirreducible

**representations**. This theorem shows that in studying finitedimensional

**representations**of a compact group G we may, without loss ofgenerality, ...

Page 1217

A spectral

adjoint operator T in S) is said to be an ordered

The measure u is called the measure of the ordered

will ...

A spectral

**representation**of a Hilbert space $5 onto X. La(u,) relative to a selfadjoint operator T in S) is said to be an ordered

**representation**of X relative to T.The measure u is called the measure of the ordered

**representation**. The sets e,will ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

48 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero