## Linear Operators: General theory |

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Page 614

Throughout this section , X will denote a complex B - space , and { T ( t ) } a

strongly continuous

satisfying the conditions of the following definition . 1 DEFINITION . A family { T ( t

) } , 0 < t ...

Throughout this section , X will denote a complex B - space , and { T ( t ) } a

strongly continuous

**semi**-**group**of operators in x ; i.e. , a family of operatorssatisfying the conditions of the following definition . 1 DEFINITION . A family { T ( t

) } , 0 < t ...

Page 685

The

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

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

**semi**-**group**is ...Page 697

Nelson Dunford, Jacob T. Schwartz. 11 LEMMA . Let ( S , E , u ) be a positive

measure space and let { T ( 4 , ... , tx ) , 41 , ... , tx > 0 } be a strongly measurable

Nelson Dunford, Jacob T. Schwartz. 11 LEMMA . Let ( S , E , u ) be a positive

measure space and let { T ( 4 , ... , tx ) , 41 , ... , tx > 0 } be a strongly measurable

**semi**-**group**of operators in L ( S , E , u ) with T ( tj , ... , tx ) , 51 , \ T ( 1 , ... , telo sl .### What people are saying - Write a review

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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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