## Linear Operators: Spectral theory |

### From inside the book

Results 1-3 of 58

Page 893

Thus if E is a bounded

operators in the B - space X , and if x e X , then the integral | | ( 8 ) E ( ds ) x is

defined for every bounded E - measurable function f on S . It follows immediately

...

Thus if E is a bounded

**additive**set function on £ whose values Ed ) are boundedoperators in the B - space X , and if x e X , then the integral | | ( 8 ) E ( ds ) x is

defined for every bounded E - measurable function f on S . It follows immediately

...

Page 932

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

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

Hence if e , and eg are disjoint then yley u ez ) = E ( e u ez ) y ( e , u ez ) = [ E ( 41

) + E ( ez ) ] y ( e , u ez ) = E ( e ) y ( e , u ex ) + E ( e ) y ( e , u ez ) = ylez ) + ylez ) ,

so that the vector valued set function y is

Hence if e , and eg are disjoint then yley u ez ) = E ( e u ez ) y ( e , u ez ) = [ E ( 41

) + E ( ez ) ] y ( e , u ez ) = E ( e ) y ( e , u ex ) + E ( e ) y ( e , u ez ) = ylez ) + ylez ) ,

so that the vector valued set function y is

**additive**on B . Therefore , if ein ez = $ ...### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

15 other sections not shown

### Other editions - View all

### Common terms and phrases

additive adjoint operator 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