## Linear Operators: General theory |

### From inside the book

Results 1-3 of 86

Page 42

This

) is a homomorphism . To show that it is an isomorphism , let Xo = yo . We will

Xo ...

This

**proves**the statement made above . We have seen that the mapping r → H ( r) is a homomorphism . To show that it is an isomorphism , let Xo = yo . We will

**prove**that there exists an ho e H with ho ( ro ) = hy ( yo ) . If Yo = yo , then eitherXo ...

Page 653

a strongly continuous semi - group of bounded operators defined on the interval [

0 , 00 ) . The operator A is the infinitesimal generator of T ( t ) . 2 Suppose lim + 0

...

**Prove**that T ( t ) is strongly continuous on ( 0 , 0 ) . In Exercise 2-9 below T ( t ) isa strongly continuous semi - group of bounded operators defined on the interval [

0 , 00 ) . The operator A is the infinitesimal generator of T ( t ) . 2 Suppose lim + 0

...

Page 670

which , when combined with ( 2 ) ,

ei + 1gi g , = ei + [ 1 - P ( g + 1 + Pg . + 2 + ... ) ] > 0 , which

ii ) we shall first

which , when combined with ( 2 ) ,

**proves**( 3 ) . From ( 1 ) and ( 3 ) it is seen thatei + 1gi g , = ei + [ 1 - P ( g + 1 + Pg . + 2 + ... ) ] > 0 , which

**proves**( i ) . To**prove**(ii ) we shall first

**prove**, by induction downwards , that ( 4 ) ( 8i + Pgi + 1 + .### What people are saying - Write a review

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

### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

50 other sections not shown

### Other editions - View all

### Common terms and phrases

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero