## Linear Operators: Spectral theory |

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Page 995

If y and f are in L . ( R ) and L ( R ) respectively and if f ( m ) = 0 for every m in the

spectral set o ( 9 ) , then ol * )

with himo ) = 1 and h vanishing on an open set

) .

If y and f are in L . ( R ) and L ( R ) respectively and if f ( m ) = 0 for every m in the

spectral set o ( 9 ) , then ol * )

**contains**no isolated points . ... Let h be in L ( R )with himo ) = 1 and h vanishing on an open set

**containing**the remainder of olf * °) .

Page 996

From Lemma 12 ( b ) it is seen that olf * 9 ) Colp ) and from Lemma 12 ( c ) and

the equation of = tf it follows that o ( f * )

o ( f * ) is a closed subset of the boundary of o ( q ) . Since f * 9 = 0 it follows from ...

From Lemma 12 ( b ) it is seen that olf * 9 ) Colp ) and from Lemma 12 ( c ) and

the equation of = tf it follows that o ( f * )

**contains**no interior point of o ( 9 ) . Henceo ( f * ) is a closed subset of the boundary of o ( q ) . Since f * 9 = 0 it follows from ...

Page 1397

The method of proof is the following : it will be shown that if the theorem is false ,

then a proper symmetric extension T , of T can be constructed whose domain

properly

The method of proof is the following : it will be shown that if the theorem is false ,

then a proper symmetric extension T , of T can be constructed whose domain

properly

**contains**both D ( T ) and the null - space of T * . This readily yields a ...### 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