## Linear Operators: Spectral theory |

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Page 1642

Let I , be the bounded open

such that U , K , = 1 .. Using Lemma 1 , let { w } be a sequence of functions in C (

1. ) ...

Let I , be the bounded open

**subset**of I where Im ( x ) + 0. Since 1 , is an open**subset**of E " , there exists an increasing sequence { K , } of compact**subsets**of 1 ,such that U , K , = 1 .. Using Lemma 1 , let { w } be a sequence of functions in C (

1. ) ...

Page 1650

which y vanishes . ... Let F be a distribution in the open

closed set Cp in I which is the complement in I of the largest open set in I in which

...

**subsets**of I and let F be in D ( I ) . ... Let K be a compact**subset**of Ual , outside ofwhich y vanishes . ... Let F be a distribution in the open

**subset**I of En . Then theclosed set Cp in I which is the complement in I of the largest open set in I in which

...

Page 1669

Let M : 1 -1 , be a mapping of I , into I , such that ( a ) M - ' C is a compact

I , whenever C is a compact

Then ( i ) for each q in Co ( 12 ) , po M will denote the function y in CR ( 1 . ) ...

Let M : 1 -1 , be a mapping of I , into I , such that ( a ) M - ' C is a compact

**subset**ofI , whenever C is a compact

**subset**of Iz ; ( b ) ( M ( :) ) , € CR ( 1 ) , j = 1 , ... , No.Then ( i ) for each q in Co ( 12 ) , po M will denote the function y in CR ( 1 . ) ...

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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