## 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 analytic

X - valued function f defined and analytic on an open set D ( ) 2 p ( T ) and such

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

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 such

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

**extension**, f ( $ ) =R ...

Page 2092

The single valued

not have the single valued

S . Kakutani ( see Dunford [ 18 ] ) . Kesel ' man ( 1 ) gave necessary conditions for

...

The single valued

**extension**property . The example of an operator which doesnot have the single valued

**extension**property that is given in Section 2 is due toS . Kakutani ( see Dunford [ 18 ] ) . Kesel ' man ( 1 ) gave necessary conditions for

...

Page 2095

Consequently , the analogs of the results stated in the preceding paragraph for

restrictions of spectral and scalar type operators also hold for their quotients .

Although the restrictions of operators with the single valued

have ...

Consequently , the analogs of the results stated in the preceding paragraph for

restrictions of spectral and scalar type operators also hold for their quotients .

Although the restrictions of operators with the single valued

**extension**propertyhave ...

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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