## Linear Operators: Spectral theory |

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Page 1303

Clearly B(f) = 0 for those f which vanish in a

boundary value for t at a. To prove the converse, let B be a boundary value at a.

Choose a function h in C*(I) which is identically equal to one in a

of a ...

Clearly B(f) = 0 for those f which vanish in a

**neighborhood**of a. Thus B is aboundary value for t at a. To prove the converse, let B be a boundary value at a.

Choose a function h in C*(I) which is identically equal to one in a

**neighborhood**of a ...

Page 1403

It is then sufficient to show that each A e A has a

A) e L2(a,c) for ut-almost all A e Ao, since A may then be written as a countable

union of such

It is then sufficient to show that each A e A has a

**neighborhood**Ao such that W,(“,A) e L2(a,c) for ut-almost all A e Ao, since A may then be written as a countable

union of such

**neighborhoods**Ao. We shall show below that for each A e A there ...Page 1678

Let is be a second function in Co(I) such that f(r) = 1 for a in a

. Then pop–9p vanishes in a

op ...

Let is be a second function in Co(I) such that f(r) = 1 for a in a

**neighborhood**of K1. Then pop–9p vanishes in a

**neighborhood**of K n C(F), and vanishes in a**neighborhood**of C(F)—K since p vanishes in the complement of K. Hence opp—op ...

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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