## Linear Operators: Spectral theory |

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Results 1-3 of 64

Page 893

Returning now to the general integral s f(s)|E(ds) where E is merely a bounded

defined in terms of the uniform operator topology. It is clear that if v is a bounded

...

Returning now to the general integral s f(s)|E(ds) where E is merely a bounded

**additive**operator valued set function, we observe that the integral has beendefined in terms of the uniform operator topology. It is clear that if v is a bounded

...

Page 932

Let S be an abstract set and 2 a field (resp. o-field) of subsets of S. Let F be an

operators on a Hilbert space $5 satisfying F(q) = 0 and F(S) = I. Then there exists

a ...

Let S be an abstract set and 2 a field (resp. o-field) of subsets of S. Let F be an

**additive**(resp. weakly countably**additive**) function on 2 to the set of positiveoperators on a Hilbert space $5 satisfying F(q) = 0 and F(S) = I. Then there exists

a ...

Page 958

Hence if e1 and ea are disjoint then op(ei U ex) = E(ei U ex)y(ei U ea) = [E(e1)+E

(eg)]"p(ei U ea) = E(e)y(ei U ea)+E(es)w(ei U ea) = p(e1)+"p(ee), so that the

vector valued set function p is

function ...

Hence if e1 and ea are disjoint then op(ei U ex) = E(ei U ex)y(ei U ea) = [E(e1)+E

(eg)]"p(ei U ea) = E(e)y(ei U ea)+E(es)w(ei U ea) = p(e1)+"p(ee), so that the

vector valued set function p is

**additive**on 30. Therefore, if ei nea = b, the setfunction ...

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

34 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

additive Akad algebra Amer analytic applied assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complete 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 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