## 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 Tn 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 Tn 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 ( { 2 } ) ( 11 — T ) ) x = ( 21 — T ) x which shows that

the

neighborhood V of a which is disjoint from 0 , , and let f ( u ) = ( 2 - 4 ) - 1 if u € V

and f ( u ) = 0 ...

Then ( E ( Q ) U ) x = ( 1 – E ( { 2 } ) ( 11 — T ) ) x = ( 21 — T ) x which shows that

the

**range**of the projection E ( 0 ) contains the**range**of T . Choose aneighborhood V of a which is disjoint from 0 , , and let f ( u ) = ( 2 - 4 ) - 1 if u € V

and f ( u ) = 0 ...

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 7 * 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 7 * to D ( T ) + N . Then the

...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 869 |

Commutative BAlgebras | 877 |

Copyright | |

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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 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 unit vanishes vector zero