## 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 | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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