## Linear Operators: General theory |

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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 ...Page 689

the

union of the ranges of all the operators T ( u ) -1 . If æ * is a functional vanishing

on the ranges of all the operators T ( u ) - I then X * = x * T ( u ) ** A ( u ) = x * E

and ...

the

**semi**-**group**. Since E ' ( T ( u ) -1 ) = T ( u ) -1 , the range of E ' contains theunion of the ranges of all the operators T ( u ) -1 . If æ * is a functional vanishing

on the ranges of all the operators T ( u ) - I then X * = x * T ( u ) ** A ( u ) = x * E

and ...

Page 697

Let ( S , E , u ) be a positive measure space and let { T ( 1 , ... , tk ) , ty , ... tk > 0 }

be a strongly measurable

ck , which is independent of the

Let ( S , E , u ) be a positive measure space and let { T ( 1 , ... , tk ) , ty , ... tk > 0 }

be a strongly measurable

**semi**-**group**of ... ak Then there is an absolute constantck , which is independent of the

**semigroup**and independent of f , such that f .### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain 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 mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero