## Linear Operators: Spectral Theory : Self Adjoint Operators in Hilbert Space, Volume 2 |

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Page 2084

If k is a natural number then there exists a constant Mk such that if 0 SES 1 and o

is a Borel set with diameter at most ε , then | N * E ( 0 ) S M € * + 1 - m ( where N is

the radical part of T . ) 56 ( McCarthy ) Let T be a spectral operator in a

If k is a natural number then there exists a constant Mk such that if 0 SES 1 and o

is a Borel set with diameter at most ε , then | N * E ( 0 ) S M € * + 1 - m ( where N is

the radical part of T . ) 56 ( McCarthy ) Let T be a spectral operator in a

**complex**...Page 2171

Exercises Some of the exercises will use the following notation . The symbol T is

a bounded linear operator on a

[ x ] will be used for the closed linear manifold determined by all the vectors RTÉ

...

Exercises Some of the exercises will use the following notation . The symbol T is

a bounded linear operator on a

**complex**B - space X . For each x in X the symbol[ x ] will be used for the closed linear manifold determined by all the vectors RTÉ

...

Page 2188

Let E be a spectral measure in the

countably additive on a o - field of subsets of a set 1 and let g be a bounded Borel

measurable function defined on the

u ) ...

Let E be a spectral measure in the

**complex**B - space X which is defined andcountably additive on a o - field of subsets of a set 1 and let g be a bounded Borel

measurable function defined on the

**complex**plane . Then s g ( f ( ) ) E ( da ) = g (u ) ...

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### Contents

SPECTRAL OPERATORS XV Spectral Operators | 1924 |

Introduction | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

32 other sections not shown

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### Common terms and phrases

analytic apply arbitrary assumed B-space Banach space Boolean algebra Borel sets boundary conditions bounded bounded Borel bounded operator Chapter clear clearly closure commuting compact complex consider constant contained converges Corollary corresponding countably additive defined Definition denote dense determined differential operator domain elements equation equivalent established example exists extension fact finite follows formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator manifold Math Moreover multiplicity norm normal positive preceding present problem projections PROOF properties prove range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows spectral measure spectral operator spectrum statement strongly subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector weakly zero