## Linear Operators: Spectral theory |

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Page 1175

Then X is a bounded

each real fo , let H. be the

961 ) ( 5 ) = f ( 5 ) , 6 > 50 = 0 otherwise . By Corollary 22 , it follows that there is a

...

Then X is a bounded

**mapping**of the space L ( L , ( S ) ) into itself . Proof . Foreach real fo , let H. be the

**mapping**in La ( L , ( S ) ) defined by the formula ( 47 )961 ) ( 5 ) = f ( 5 ) , 6 > 50 = 0 otherwise . By Corollary 22 , it follows that there is a

...

Page 1669

Let M : 1 -1 , be a

I , 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 . ) ...

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 . ) ...

Page 1736

The

by Lemmas 3.22 and 3.23 , and evidently

from Definition 3.15 that it

The

**mapping**g → & g C is a continuous**mapping**of H ( P ) ( 8-11 ) into HP ) ( C )by Lemmas 3.22 and 3.23 , and evidently

**maps**CO ( 1 ) into C ( C ) . It followsfrom Definition 3.15 that it

**maps**H !!! ( 8-11 ) into HP ( C ) . Thus , te = ( fosz ) C ...### 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