## Linear Operators: Spectral theory |

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Page 1175

Then Ki is a bounded

each real 5o , let Hy be the

( 4 , 1 ) ( 5 ) = f ( 5 ) , § > 50 , = 0 otherwise . By Corollary 22 , it follows that there

is ...

Then Ki is a bounded

**mapping**of the space L , ( L ( S ) ) into itself . Proof . Foreach real 5o , let Hy be the

**mapping**in L , ( L , ( S ) ) defined by the formula ( 47 )( 4 , 1 ) ( 5 ) = f ( 5 ) , § > 50 , = 0 otherwise . By Corollary 22 , it follows that there

is ...

Page 1401

j - dimensional subspace S ; of Dr , and let D , be its orthocomplement in Dt .

Define an isometric

x = - Ux , XED ; . Let l ; be the graph of U ; : By Theorem XII . 4 . 12 ( b ) , D ( T . ) T '

, is ...

j - dimensional subspace S ; of Dr , and let D , be its orthocomplement in Dt .

Define an isometric

**mapping**U , of Dt onto D _ as follows : U ; x = Ux , XES ; , U ;x = - Ux , XED ; . Let l ; be the graph of U ; : By Theorem XII . 4 . 12 ( b ) , D ( T . ) T '

, is ...

Page 1669

The next topic on which we wish to touch is that of the behavior of distributions

under changes of variable . 44 DEFINITION . Let I , be a domain in Eại , and let I ,

be a domain in En2 . Let M : 14 → 1 , be a

The next topic on which we wish to touch is that of the behavior of distributions

under changes of variable . 44 DEFINITION . Let I , be a domain in Eại , and let I ,

be a domain in En2 . Let M : 14 → 1 , be a

**mapping**of Iį into 1 , such that ( a ) M ...### What people are saying - Write a review

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### Contents

IX | 859 |

extensive presentation of applications of the spectral theorem | 911 |

Miscellaneous Applications | 937 |

Copyright | |

20 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

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 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