## Linear Operators: Spectral theory |

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Page 950

Instead of restricting our consideration to the case of the additive group of real

numbers, we shall discuss the case of a locally

denote by R. We assume throughout that R is o-

Instead of restricting our consideration to the case of the additive group of real

numbers, we shall discuss the case of a locally

**compact**Abelian group which wedenote by R. We assume throughout that R is o-

**compact**, i.e., the union of ...Page 1150

ence of Haar measure on a locally

remarked in the text, the development presented in this section is valid for a

general non-discrete locally

are a few ...

ence of Haar measure on a locally

**compact**, o-**compact**Abelian group. Asremarked in the text, the development presented in this section is valid for a

general non-discrete locally

**compact**, o-**compact**Abelian group. However, thereare a few ...

Page 1331

|Kı". –. s,s,. K(,. s). asdi. o. z. is

but, for the sake of completeness, a proof will be given here. Note first, that by

Schwarz' inequality,. |s,. Kūs)sode'd. s. s,s,. K(,. )*dsall's,. foods). Hence, the norm

...

|Kı". –. s,s,. K(,. s). asdi. o. z. is

**compact**. This is a special case of Exercise VI.9.52,but, for the sake of completeness, a proof will be given here. Note first, that by

Schwarz' inequality,. |s,. Kūs)sode'd. s. s,s,. K(,. )*dsall's,. foods). Hence, the norm

...

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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