## 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

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 operators satisfying the conditions of the following definition . 1 DEFINITION .Page 685

We recall that the

We recall that the

**semi**-**group**{ T ( t ) , 0 St } is said to be strongly continuous if its dependence upon t is continuous in the strong operator topology , i.e. , if lim ( T ( t ) x — T ( u ) x ) = 0 for each x in X and each t - u u ...Page 697

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

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