## Linear Operators: Spectral theory |

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Page 978

nondiscrete locally compact Abelian group and integration will always be

performed with respect to a Haar measure on the group . It was observed in

Corollary 5 .

**Closure**Theorems As in the preceding section the letter R will stand for anondiscrete locally compact Abelian group and integration will always be

performed with respect to a Haar measure on the group . It was observed in

Corollary 5 .

Page 1226

This fact leads us to make the following definition . 7 DEFINITION . The minimal

closed symmetric extension of a symmetric operator T with dense domain is

called its

restriction of ...

This fact leads us to make the following definition . 7 DEFINITION . The minimal

closed symmetric extension of a symmetric operator T with dense domain is

called its

**closure**, and written T . 8 LEMMA . ( a ) The**closure**T of T is therestriction of ...

Page 1686

+ 5 THEOREM . Let n 21 , and let D be a bounded open set in Euclidean space E

" . Suppose that the boundary of D is a smooth surface and that no point in the

boundary of D is interior to the

+ 5 THEOREM . Let n 21 , and let D be a bounded open set in Euclidean space E

" . Suppose that the boundary of D is a smooth surface and that no point in the

boundary of D is interior to the

**closure**of D . Let k 21 and m 2 0 be integers .### What people are saying - Write a review

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### Contents

IX | 859 |

extensive presentation of applications of the spectral theorem | 911 |

Miscellaneous Applications | 937 |

Copyright | |

20 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 give 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