## Linear Operators: Self Adjoint Operators in Hilbert Space. Spectral theory. Part II |

### From inside the book

Results 1-3 of 78

Page 1215

... Ss_t ( ) W. ( S , • vv ( ds ) = ( Ulla → ( Uf ) a Thus Sst ( s ) W , ( s , a ) u ( ds )

exists in the mean square sense and equals ( Uf ) , ( 2 ) , proving ( c ) . Q.E.D.

Using the notation of the

S " .

... Ss_t ( ) W. ( S , • vv ( ds ) = ( Ulla → ( Uf ) a Thus Sst ( s ) W , ( s , a ) u ( ds )

exists in the mean square sense and equals ( Uf ) , ( 2 ) , proving ( c ) . Q.E.D.

Using the notation of the

**preceding**proof we let F = ( Uf ) , so that , by Lemma 9 ,S " .

Page 1437

Suppose that a bounded sequence { n } of elements of D ( T ( T ) ) exists such that

{ ( T - 20 ) / n } converges but the sequence { fr } has no convergent subsequence

. Then , since To ( t ) C T ( T ) , it follows immediately from the

Suppose that a bounded sequence { n } of elements of D ( T ( T ) ) exists such that

{ ( T - 20 ) / n } converges but the sequence { fr } has no convergent subsequence

. Then , since To ( t ) C T ( T ) , it follows immediately from the

**preceding**lemma ...Page 1474

By the remark ( a ) made

dt = Si { ( to ) ( t , ) o ( t , ay ) –o ( t , 17 ) ( 70 ) ( t , 2 ) } dt + p ( z ) o ( z , a ) o ' ( 2 , 2 ,

) < 0 . = This contradiction establishes our assertion . Q.E.D. 44 COROLLARY .

By the remark ( a ) made

**preceding**Lemma 41 , 0 < ( 9–22 ) f * o ( t , 1 ) o ( t , 27 )dt = Si { ( to ) ( t , ) o ( t , ay ) –o ( t , 17 ) ( 70 ) ( t , 2 ) } dt + p ( z ) o ( z , a ) o ' ( 2 , 2 ,

) < 0 . = This contradiction establishes our assertion . Q.E.D. 44 COROLLARY .

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Preliminary Notions | 865 |

Copyright | |

61 other sections not shown

### Other editions - View all

### 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