## Linear Operators: Spectral theory |

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Page 897

It follows then from ( iii ) that the projections E ( d ) also commute with T ( f ) and

this completes the proof of the theorem . Q . E . D . 3 COROLLARY . The spectral

measure is

...

It follows then from ( iii ) that the projections E ( d ) also commute with T ( f ) and

this completes the proof of the theorem . Q . E . D . 3 COROLLARY . The spectral

measure is

**countably additive**in the strong operator topology . Proof . If { 8n } is a...

Page 932

Let S be an abstract set and 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 ) = I . Then there

exists ...

Let S be an abstract set and 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 ) = I . Then there

exists ...

Page 958

This argument shows that the vector valued additive set function y is weakly

theorem of Pettis ( IV . 10 . 1 ) it is

, 14 ( rn ) ...

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 . By atheorem of Pettis ( IV . 10 . 1 ) it is

**countably additive**in the strong topology , i . e ., 14 ( rn ) ...

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

15 other sections not shown

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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 Russian satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero