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

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Page 932

Let S be an abstract set and E a field ( resp . o - field ) of subsets of S. Let F be an

additive ( resp . weakly

operators on a Hilbert space H satisfying F ( $ ) = 0 and F ( S ) = 1 . Then there

exists ...

Let S be an abstract set and E a field ( resp . o - field ) of subsets of S. Let F be an

additive ( resp . weakly

**countably additive**) function on to the set of positiveoperators on a Hilbert space H satisfying F ( $ ) = 0 and F ( S ) = 1 . Then there

exists ...

Page 958

... is additive on Bo . To see that Mo is

disjoint sets ... This argument shows that the vector valued additive set function y

is weakly

... is additive on Bo . To see that Mo is

**countably additive**on B , let en , n 2 1 , bedisjoint sets ... This argument shows that the vector valued additive set function y

is weakly

**countably additive**on the o - field consisting of all Borel subsets of e .Page 959

Since Mo is

for some m , Moleem ) Moleembn ) > k - ε . This shows that the set function u is

uniquely defined on B. Next we show that u is

purpose ...

Since Mo is

**countably additive**on Bo Mo ( ebn ) = limm Mo ( eembn ) 2k , and sofor some m , Moleem ) Moleembn ) > k - ε . This shows that the set function u is

uniquely defined on B. Next we show that u is

**countably additive**. For thispurpose ...

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### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Preliminary Notions | 865 |

Copyright | |

61 other sections not shown

### Common terms and phrases

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 Nauk 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