## Linear Operators: Spectral theory |

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Page 950

... union of countably many compact sets. Every such group has a non-negative

countably additive

compact sets, positive or infinite on open sets, invariant under translation, i.e., A(a

+ E) ...

... union of countably many compact sets. Every such group has a non-negative

countably additive

**measure**which is defined on the Borel sets 2, finite oncompact sets, positive or infinite on open sets, invariant under translation, i.e., A(a

+ E) ...

Page 1152

The existence of an invariant

countability was first shown by Haar [1], and the question ... Other results

concerning

Ulam [1].

The existence of an invariant

**measure**on a group satisfying the second axiom ofcountability was first shown by Haar [1], and the question ... Other results

concerning

**measures**invariant under transformations are found in Oxtoby andUlam [1].

Page 1153

Since the

integration as developed in Chapter III may be used as a basis for the theory

developed in Sections 3–4. In particular we should notice that the product group

R × R is ...

Since the

**measure**space (R, 2, 2) is a g-finite**measure**space the theory ofintegration as developed in Chapter III may be used as a basis for the theory

developed in Sections 3–4. In particular we should notice that the product group

R × R is ...

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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