## Linear Operators: General theory |

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Page 613

exp ( ta ) is the most general continuous real ( or complex ) ... An operator valued

function T ( t ) satisfying these equations is called a

**Semi**-**groups**of Operators It is well known that the exponential function f ( t ) =exp ( ta ) is the most general continuous real ( or complex ) ... An operator valued

function T ( t ) satisfying these equations is called a

**semi**-**group**of operators .Page 655

1 * ( 1 - e - ) x ( s + $ ) dF T ( t ) ( x , s ) xex , 29 J -- 1-2e- cos & te - 2t ' defines a

strongly continuous

where 03 ( s , t ) = 1 + 2 mm - ze - not cos ns , is a strongly continuous

on [ 0 ...

1 * ( 1 - e - ) x ( s + $ ) dF T ( t ) ( x , s ) xex , 29 J -- 1-2e- cos & te - 2t ' defines a

strongly continuous

**semi**-**group**on [ 0 , 0 ) ... 03 ( s- & , t ) x ( 5 ) d , X e X , 27where 03 ( s , t ) = 1 + 2 mm - ze - not cos ns , is a strongly continuous

**semigroup**on [ 0 ...

Page 685

The

function T ( ) x is measurable , with respect to Lebesgue measure , on the infinite

interval 0 St. It was observed in Lemma 1.3 that a strongly measurable

The

**semi**-**group**is said to be strongly measurable if , for each x in X , thefunction T ( ) x is measurable , with respect to Lebesgue measure , on the infinite

interval 0 St. It was observed in Lemma 1.3 that a strongly measurable

**semi**-**group**is ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

81 other sections not shown

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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric neighborhood norm operator positive measure problem Proc proof properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero