## 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 ( P ) , then olf * )

make an indirect proof by supposing that mo is an isolated point of olf * ) . Since

olf ...

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 ( P ) , then olf * )

**contains**no isolated points . PROOF . We shallmake an indirect proof by supposing that mo is an isolated point of olf * ) . Since

olf ...

Page 996

From Lemma 12 ( b ) it is seen that olf * o ) Colo ) and from Lemma 12 ( c ) and

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

Hence o ( f * o ) is a closed subset of the boundary of o ( p ) . Since f * q = 0 it

follows ...

From Lemma 12 ( b ) it is seen that olf * o ) Colo ) and from Lemma 12 ( c ) and

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

**contains**no interior point of o ( q ) .Hence o ( f * o ) is a closed subset of the boundary of o ( p ) . Since f * q = 0 it

follows ...

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

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Copyright | |

57 other sections not shown

### Common terms and phrases

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