## Linear Operators: Spectral theory |

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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 zā = 1 for every spectral ...

**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 zā = 1 for every spectral ...

Page 1247

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

on

adjoint transformation T is

...

... ( x , z ) , it follows that x = 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 o ( T ) is a subset of the interval [ 0...

Page 1338

( ii ) we have Möjl Uem ) = Mislem ) m = 1 m = 1 for each sequence of disjoint

Borel sets with bounded union . 7 LEMMA . Let { uis } be a

measure whose elements Mis are continuous with respect to a

measure u .

( ii ) we have Möjl Uem ) = Mislem ) m = 1 m = 1 for each sequence of disjoint

Borel sets with bounded union . 7 LEMMA . Let { uis } be a

**positive**matrixmeasure whose elements Mis are continuous with respect to a

**positive**o - finitemeasure u .

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 869 |

Commutative BAlgebras | 877 |

Copyright | |

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additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f give given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero