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

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Page 907

negative real axis respectively . PROOF . If N is a bounded normal operator then ,

by Corollary IX.3.15 , NN * - N * N = I if and only if tā = 1 for every spectral point ...

**positive**if and only if its spectrum lies on the unit circle , the real axis , or the non -negative real axis respectively . PROOF . If N is a bounded normal operator then ,

by Corollary IX.3.15 , NN * - N * N = I if and only if tā = 1 for every spectral point ...

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... z), it follows that z = 0. Q.E.D. Next we shall require some information on

adjoint transformation T is

oo). Pnoor.

... z), it follows that z = 0. Q.E.D. Next we shall require some information on

**positive**self adjoint transformations and their square roots. 2 LEMMA. A selfadjoint transformation T is

**positive**if and only if a(T) is a subset of the interval [0,oo). Pnoor.

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Let { Mis } be a

respect to a

defined by the equations Mijle ) S.am mij ( 2 ) u ( da ) , where e is any bounded

Borel ...

Let { Mis } be a

**positive**matrix measure whose elements Mis are continuous withrespect to a

**positive**o - finite measure kl . If the matrix of densities { mi ; } isdefined by the equations Mijle ) S.am mij ( 2 ) u ( da ) , where e is any bounded

Borel ...

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### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Preliminary Notions | 865 |

Copyright | |

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