## 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 1687

tinuous in the

theorem . First suppose that U is disjoint from the boundary of D . Then h ; F is a

distribution whose carrier is a compact set contained in U ( cf . Lemma 3 . 13 ( iv )

) .

tinuous in the

**closure**of D . This will evidently imply the truth of the presenttheorem . First suppose that U is disjoint from the boundary of D . Then h ; F is a

distribution whose carrier is a compact set contained in U ( cf . Lemma 3 . 13 ( iv )

) .

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### Contents

IX | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Compact Groups | 945 |

Copyright | |

46 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 |

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