## Linear Operators: Spectral theory |

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Page 909

Spectral Representation Let u be a finite positive measure defined on the

sets B of the complex plane and vanishing on the complement of a bounded set

S . One of the simplest examples of a bounded normal operator is the operator T

...

Spectral Representation Let u be a finite positive measure defined on the

**Borel**sets B of the complex plane and vanishing on the complement of a bounded set

S . One of the simplest examples of a bounded normal operator is the operator T

...

Page 913

Then { on } is a decreasing sequence of

such that if e is a

E ( 0 ; ) 4 ; . To see that x is maximal let e be a

Then { on } is a decreasing sequence of

**Borel**sets such that - v ; ( 0 % ) = 0 , andsuch that if e is a

**Borel**subset of on and - v ; ( e ) = 0 , then vn ( e ) = 0 . Put x = 2 -E ( 0 ; ) 4 ; . To see that x is maximal let e be a

**Borel**set for which ( E ( e ) x , x ) ...Page 1900

12 . 1 ( 41 )

891 )

Lebesgue measure ) , construction of , ( 139 ) , III . 13 . 8 ( 223 )

measure ...

12 . 1 ( 41 )

**Borel**field of sets , definition , III . 5 . 10 ( 137 )**Borel**function , X . 1 (891 )

**Borel**measurable function , X . I ( 891 )**Borel**measure ( or**Borel**-Lebesgue measure ) , construction of , ( 139 ) , III . 13 . 8 ( 223 )

**Borel**- Stieltjesmeasure ...

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### Contents

BAlgebras | 861 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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