## Linear Operators: General theory |

### From inside the book

Results 1-3 of 34

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**...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 * = * * T ( u ) = 2 * A ( u ) = x * E ...

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 * = * * T ( u ) = 2 * A ( u ) = x * E ...

Page 697

Nelson Dunford. 11 LEMMA . Let ( S , E , u ) be a positive measure space and let

{ T ( 4 , . . . , tk ) , t , . . . , tx > 0 } be a strongly measurable

operators in L ( S , E , u ) with T ( , . . . , tx ) li = 1 , \ T ( , . . . , txllo S 1 . Let 1 sp < 0 ,

f e Lp ...

Nelson Dunford. 11 LEMMA . Let ( S , E , u ) be a positive measure space and let

{ T ( 4 , . . . , tk ) , t , . . . , tx > 0 } be a strongly measurable

**semi**-**group**ofoperators in L ( S , E , u ) with T ( , . . . , tx ) li = 1 , \ T ( , . . . , txllo S 1 . Let 1 sp < 0 ,

f e Lp ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero