## 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 ...

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 ...

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**...Page 697

To Then there is an absolute constant Cx , which is independent of the

used for Ck with the additional hypothesis that the operators in the

are positive ...

To Then there is an absolute constant Cx , which is independent of the

**semigroup**and independent of t , such that ule * ( B ) ... The symbol CPR will beused for Ck with the additional hypothesis that the operators in the

**semi**-**group**are positive ...

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### Contents

Metric Spaces | 19 |

Convergence and Uniform Convergence of Generalized | 26 |

Exercises | 33 |

Copyright | |

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