## Linear Operators: Spectral theory |

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Page 1036

Then the infinite product 8 ( T ) = } ( 1 - 1 ) er

analytic for a + 0 . For each fixed a + 0 , 8 ( T ) is a continuous complex valued

function on the B - space of all Hilbert - Schmidt operators . PROOF . First note

that if ...

Then the infinite product 8 ( T ) = } ( 1 - 1 ) er

**converges**and defines a functionanalytic for a + 0 . For each fixed a + 0 , 8 ( T ) is a continuous complex valued

function on the B - space of all Hilbert - Schmidt operators . PROOF . First note

that if ...

Page 1333

Since gn + 0 and TR ( 2 ; T ) is a bounded operator , it follows that R ( 2 ; T ) gn

and TR ( 1 ; T ) gn

( 4 , 9 : 19 ;

of ...

Since gn + 0 and TR ( 2 ; T ) is a bounded operator , it follows that R ( 2 ; T ) gn

and TR ( 1 ; T ) gn

**converge**to zero in L ( I ) . Thus by Lemma 2 . 16 the series on( 4 , 9 : 19 ;

**converges**to f in the topology of Cn - ( J ) for each compact interval Jof ...

Page 1436

Let { en } be a bounded sequence of elements of D ( T ) such that { Tgn }

each j , 1 si s k . Then h ; = h ; - _ * * ( hz ) ; is in D , and Tħ ; = Thị . Thus { ħ ; }

Let { en } be a bounded sequence of elements of D ( T ) such that { Tgn }

**converges**. Find a subsequence { gn ; } = { h ; } such that x ( h ; )**converges**foreach j , 1 si s k . Then h ; = h ; - _ * * ( hz ) ; is in D , and Tħ ; = Thị . Thus { ħ ; }

**converges**...### What people are saying - Write a review

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

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