## Linear Operators: General theory |

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Page 262

Since the integral [ st ( s ) u ( ds )

) 0 ( yu , S ) , it is clear that the integral ... there is an isometric isomorphism

between C * ( S ) and rba ( S ) such that corresponding elements ~ * and u

the ...

Since the integral [ st ( s ) u ( ds )

**satisfies**the inequality | 1st ( 8 ) ( ds ) < sup | t ( s) 0 ( yu , S ) , it is clear that the integral ... there is an isometric isomorphism

between C * ( S ) and rba ( S ) such that corresponding elements ~ * and u

**satisfy**the ...

Page 495

Evidently B ( h ) C B. , and it is clear that B ( h )

above so that B ( h ) = B .. This proves that if h e C ( S ) and f e Bo , then sh e Bo .

Now let f be a fixed element of Bo , and let B ( 1 ) = { g € B lg € B. } . We have just

...

Evidently B ( h ) C B. , and it is clear that B ( h )

**satisfies**properties ( i ) and ( ii )above so that B ( h ) = B .. This proves that if h e C ( S ) and f e Bo , then sh e Bo .

Now let f be a fixed element of Bo , and let B ( 1 ) = { g € B lg € B. } . We have just

...

Page 613

... the most general continuous real ( or complex ) function of the nonnegative

real variable t which

operator valued functions defined on the range t 2 0 , which

( i ) T ( t ...

... the most general continuous real ( or complex ) function of the nonnegative

real variable t which

**satisfies**the functional ... the most general continuousoperator valued functions defined on the range t 2 0 , which

**satisfy**the equations( i ) T ( t ...

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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