## Linear Operators: Spectral theory |

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Page 1092

By Lemma 5 and Corollary 4 , and the elementary fact that any compact operator

may be approximated in norm by a sequence of operators T , with

finitedimensional

T has finite ...

By Lemma 5 and Corollary 4 , and the elementary fact that any compact operator

may be approximated in norm by a sequence of operators T , with

finitedimensional

**range**, it is enough to prove the lemma in the special case thatT has finite ...

Page 1395

Then ( E ( Q ) U ) x = ( 1 - E ( { 4 } ) ( 11 – T ) ) x = ( 11 — T ) x which shows that

the

neighborhood V of 2 which is disjoint from g , and let f ( u ) = ( 2 - u ) -1 if u € V

and f ( u ) = 0 if ...

Then ( E ( Q ) U ) x = ( 1 - E ( { 4 } ) ( 11 – T ) ) x = ( 11 — T ) x which shows that

the

**range**of the projection E ( 0 ) contains the**range**of T. Choose aneighborhood V of 2 which is disjoint from g , and let f ( u ) = ( 2 - u ) -1 if u € V

and f ( u ) = 0 if ...

Page 1397

This readily yields a contradiction as follows : the assumption that 0 € 0 ( T )

implies that the

easily seen to be symmetric obtained by restricting T * to D ( T ) + N . Then the

This readily yields a contradiction as follows : the assumption that 0 € 0 ( T )

implies that the

**range**R ( T ) of T is closed . Let T , be the extension – which iseasily seen to be symmetric obtained by restricting T * to D ( T ) + N . Then the

**range**R ...### 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