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

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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 .

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### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Preliminary Notions | 865 |

Copyright | |

61 other sections not shown

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