## Linear Operators: Spectral theory |

### From inside the book

Results 1-3 of 82

Page 961

dx = ( F ( 0 – 2 ) y ( x ) dx = ( f * y ) ( 0 ) = 8 ( f * w ) . Since the operation T ( f ) of

convolution by f commutes with E ( e ) and since , as we have seen in the

...

**PROOF**. If we write y for y ( e ) , then ( f , x ) = fat ( xy ( x ) dx = Sat ( 2 – 0 ) ¥ ( x )dx = ( F ( 0 – 2 ) y ( x ) dx = ( f * y ) ( 0 ) = 8 ( f * w ) . Since the operation T ( f ) of

convolution by f commutes with E ( e ) and since , as we have seen in the

**proof**of...

Page 1012

from which it follows that | | T - Tmll Sɛ for m > m ( s ) and completes the

HS is a B - space under the Hilbert - Schmidt norm . Finally , let T be in HS and let

B be any bounded linear operator in H . Then GEA αε Α | | BT | | * = E \ BTxal ?

from which it follows that | | T - Tmll Sɛ for m > m ( s ) and completes the

**proof**thatHS is a B - space under the Hilbert - Schmidt norm . Finally , let T be in HS and let

B be any bounded linear operator in H . Then GEA αε Α | | BT | | * = E \ BTxal ?

Page 1708

Hence , since g ( x ) = 1 for x in E1 / 4 , it follows from Lemmas 3 . 9 and 3 . 23 that

the restriction fel E14 belongs to A ( m + P ) ( E14 ) . Thus ( cf . 3 . 48 ) | | 26 / 4

belongs to A ( m + P ) ( E8 / 4 ) , and the

Hence , since g ( x ) = 1 for x in E1 / 4 , it follows from Lemmas 3 . 9 and 3 . 23 that

the restriction fel E14 belongs to A ( m + P ) ( E14 ) . Thus ( cf . 3 . 48 ) | | 26 / 4

belongs to A ( m + P ) ( E8 / 4 ) , and the

**proof**of Lemma 3 is complete . Q . E . D ...### What people are saying - Write a review

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

### Contents

IX | 859 |

extensive presentation of applications of the spectral theorem | 911 |

Miscellaneous Applications | 937 |

Copyright | |

20 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 adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f give given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero