## Linear Operators: Spectral theory |

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Page 893

In summary we state the following theorem . 1 THEOREM . Let E be a

self adjoint spectral measure in Hilbert space defined on a field of subsets of a

set S . Then the map | → T ( 1 ) defined by the equation T ( ) = | - | ( 8 ) E ( ds ) , te

B ...

In summary we state the following theorem . 1 THEOREM . Let E be a

**bounded**self adjoint spectral measure in Hilbert space defined on a field of subsets of a

set S . Then the map | → T ( 1 ) defined by the equation T ( ) = | - | ( 8 ) E ( ds ) , te

B ...

Page 900

and thus there is a

set having E measure zero . If f is E - measurable then fo is a

measurable function , i . e . , an element of the B * - algebra B ( S , E ) . The

algebra EB ...

and thus there is a

**bounded**function to on S with f ( s ) = to ( s ) except for s in aset having E measure zero . If f is E - measurable then fo is a

**bounded**E -measurable function , i . e . , an element of the B * - algebra B ( S , E ) . The

algebra EB ...

Page 1452

Suppose that such a u exists . Then , by Theorem XII . 2 . 6 , ( Tx , x ) = E ( dx ) x2

= ulx | ? , 2 e ( T ) , so that if \ x is

Conversely , suppose that for each n , en = ( - 0 , - n ) n o ( T ) is non - void . By

Theorem ...

Suppose that such a u exists . Then , by Theorem XII . 2 . 6 , ( Tx , x ) = E ( dx ) x2

= ulx | ? , 2 e ( T ) , so that if \ x is

**bounded**, ( Tx , x ) is**bounded**below .Conversely , suppose that for each n , en = ( - 0 , - n ) n o ( T ) is non - void . By

Theorem ...

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

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