## Linear Operators: Spectral operators |

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Page 1932

In this case x ( $ ) is a single valued

x ( ) = R ( $ ; T ' ) x , ξερ ( T ' ) . It will be shown in the next section that , if T is a

spectral operator , the function R ( E ; T ' ) . x has , for every x in X , the single

valued ...

In this case x ( $ ) is a single valued

**analytic**function with domain p ( x ) and withx ( ) = R ( $ ; T ' ) x , ξερ ( T ' ) . It will be shown in the next section that , if T is a

spectral operator , the function R ( E ; T ' ) . x has , for every x in X , the single

valued ...

Page 1934

... h ( En ) → E ( E ) h ( E . ) = h ( $ o ) , which contradicts the fact that h ( fo ) +0 .

Q.E.D. Thus , whenever T is spectral , and x e X , RIF ; T ' ) x has a maximal

COROLLARY .

... h ( En ) → E ( E ) h ( E . ) = h ( $ o ) , which contradicts the fact that h ( fo ) +0 .

Q.E.D. Thus , whenever T is spectral , and x e X , RIF ; T ' ) x has a maximal

**analytic**extension defined in p ( x ) , which we shall denote as x ( $ ) . 3COROLLARY .

Page 2087

Then ( i ) The operator As + B with domain ( P ) T ( 20 ) ( RN ) is the infinitesimal

generator of a strongly continuous semi - group S ( t ) , t 20 , of bounded linear

operators in HP and S ( t ) has a strongly

...

Then ( i ) The operator As + B with domain ( P ) T ( 20 ) ( RN ) is the infinitesimal

generator of a strongly continuous semi - group S ( t ) , t 20 , of bounded linear

operators in HP and S ( t ) has a strongly

**analytic**extension to a semi - group SIS...

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### Contents

SPECTRAL OPERATORS 1937 1941 1945 XV Spectral Operators | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

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