## Linear Operators: Spectral theory |

### From inside the book

Results 1-3 of 78

Page 961

7 LEMMA . If f is in L ( R ) , L ( R ) and if e is a Borel set in M whose closure does

not contain poo , then S. ( tt ) ( m ) u ( dm ) = 0 [ E ( e ) } ] ( t , y ( e ) ) where y ( e ) is

the vector in Ly ( R ) defined in Lemma 6 . 4

7 LEMMA . If f is in L ( R ) , L ( R ) and if e is a Borel set in M whose closure does

not contain poo , then S. ( tt ) ( m ) u ( dm ) = 0 [ E ( e ) } ] ( t , y ( e ) ) where y ( e ) is

the vector in Ly ( R ) defined in Lemma 6 . 4

**PROOF**. If we write y for yle ) , then ...Page 1459

present corollary follows from Corollary 7 and Definition 25 ( b ) . Q.E.D. 31

COROLLARY . Suppose in addition to the hypotheses of Theorem 8 that the

coefficients Pre ...

**PROOF**. It is obvious from Definition 20 that t is bounded below . Thus thepresent corollary follows from Corollary 7 and Definition 25 ( b ) . Q.E.D. 31

COROLLARY . Suppose in addition to the hypotheses of Theorem 8 that the

coefficients Pre ...

Page 1750

We shall see , however , that this fact is needed in the course of the

Theorem 1 , and shall prove it by a direct method where it is needed . Remark 2.

The theorem is false if no boundedness restriction is imposed on the coefficient ...

We shall see , however , that this fact is needed in the course of the

**proof**ofTheorem 1 , and shall prove it by a direct method where it is needed . Remark 2.

The theorem is false if no boundedness restriction is imposed on the coefficient ...

### What people are saying - Write a review

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

### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Copyright | |

57 other sections not shown

### 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 Nauk 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