## Linear Operators: Spectral theory |

### From inside the book

Results 1-3 of 78

Page 893

In summary we state the following theorem . 1 THEOREM . Let E be a

self adjoint spectral measure in Hilbert space defined on a field of subsets of a

set S . Then the map | → T ( 1 ) defined by the equation T ( ) = | - | ( 8 ) E ( ds ) , te

B ...

In summary we state the following theorem . 1 THEOREM . Let E be a

**bounded**self adjoint spectral measure in Hilbert space defined on a field of subsets of a

set S . Then the map | → T ( 1 ) defined by the equation T ( ) = | - | ( 8 ) E ( ds ) , te

B ...

Page 900

and thus there is a

set having E measure zero . If f is E - measurable then fo is a

measurable function , i . e . , an element of the B * - algebra B ( S , E ) . The

algebra EB ...

and thus there is a

**bounded**function to on S with f ( s ) = to ( s ) except for s in aset having E measure zero . If f is E - measurable then fo is a

**bounded**E -measurable function , i . e . , an element of the B * - algebra B ( S , E ) . The

algebra EB ...

Page 1452

Suppose that such a u exists . Then , by Theorem XII . 2 . 6 , ( Tx , x ) = E ( dx ) x2

= ulx | ? , 2 e ( T ) , so that if \ x is

Conversely , suppose that for each n , en = ( - 0 , - n ) n o ( T ) is non - void . By

Theorem ...

Suppose that such a u exists . Then , by Theorem XII . 2 . 6 , ( Tx , x ) = E ( dx ) x2

= ulx | ? , 2 e ( T ) , so that if \ x is

**bounded**, ( Tx , x ) is**bounded**below .Conversely , suppose that for each n , en = ( - 0 , - n ) n o ( T ) is non - void . By

Theorem ...

### 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 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