## Linear Operators: Part III: Spectral Operators [by] Nelson Dunford and Jacob T. Schwartz, with the Assistance of William G. Bade and Robert G. Bartle, Volume 1 |

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Page 1931

If x is a vector in X , then by an

X - valued function f defined and

that ( ŠI — T ) f ( 6 ) = x , ŠE D ( f ) . It is clear that , for such an extension , f ( $ ) =

R ...

If x is a vector in X , then by an

**analytic**extension of R ( É ; T ) x will be meant anX - valued function f defined and

**analytic**on an open set D ( ) 2 p ( T ) and suchthat ( ŠI — T ) f ( 6 ) = x , ŠE D ( f ) . It is clear that , for such an extension , f ( $ ) =

R ...

Page 1932

In this case x ( 5 ) is a single valued

X ( $ ) = R ( É ; T ) x , Šep ( T ) . It will be shown in the next section that , if T is a

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

valued ...

In this case x ( 5 ) is a single valued

**analytic**function with domain p ( x ) and withX ( $ ) = R ( É ; T ) x , Šep ( T ) . It will be shown in the next section that , if T is a

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

valued ...

Page 2248

Let f be a function

number of exceptional points p , includes a neighborhood of o ( T ) and a

neighborhood of the point at infinity . Suppose that each exceptional point p

satisfies E ...

Let f be a function

**analytic**in a domain U which , when taken together with a finitenumber of exceptional points p , includes a neighborhood of o ( T ) and a

neighborhood of the point at infinity . Suppose that each exceptional point p

satisfies E ...

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### Contents

SPECTRAL OPERATORS XV Spectral Operators | 1924 |

Introduction | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

32 other sections not shown

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### Common terms and phrases

analytic apply arbitrary assumed B-space Banach space Boolean algebra Borel sets boundary conditions bounded bounded Borel bounded operator Chapter clear clearly closure commuting compact complex consider constant contained converges Corollary corresponding countably additive defined Definition denote dense determined differential operator domain elements equation equivalent established example exists extension fact finite follows formal formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator manifold Math Moreover multiplicity norm positive preceding present problem projections PROOF properties prove range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows spectral measure spectral operator spectrum statement strongly subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector weakly zero