## Linear Operators, Part 2 |

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Page 893

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

adjoint spectral measure in Hilbert space defined on a field of subsets of a set S.

Then the map f = T ( 1 ) defined by the equation T ( ) = Sxt ( s ) E ( ds ) , fe B ( S ...

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

**bounded**selfadjoint spectral measure in Hilbert space defined on a field of subsets of a set S.

Then the map f = T ( 1 ) defined by the equation T ( ) = Sxt ( s ) E ( ds ) , fe B ( S ...

Page 900

and thus there is a

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

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

EB ( S ...

and thus there is a

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

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

EB ( S ...

Page 1452

2 u / * l * 2 e ( T ) , so that if \ ~ \ is

Conversely , suppose that for each n , en = ( -0 , -n ) no ( T ) is non - void . By

Theorem XII.2.9 ( b ) , Elen ) # 0 for any n . Using this fact , choose an xn such that

E ( en ) xn ...

2 u / * l * 2 e ( T ) , so that if \ ~ \ is

**bounded**, ( Tx , x ) is**bounded**below .Conversely , suppose that for each n , en = ( -0 , -n ) no ( T ) is non - void . By

Theorem XII.2.9 ( b ) , Elen ) # 0 for any n . Using this fact , choose an xn such that

E ( en ) xn ...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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