## 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 2256

Let T be an operator in a weakly complete B - space X .

totally disconnected . Then T is a spectral operator if and only if ( a ) the family of

projections E ( 0 ; T ' ) corresponding to compact spectral sets of T is uniformly ...

Let T be an operator in a weakly complete B - space X .

**Suppose**that o ( T ) istotally disconnected . Then T is a spectral operator if and only if ( a ) the family of

projections E ( 0 ; T ' ) corresponding to compact spectral sets of T is uniformly ...

Page 2284

However , rather than seek the maximum generality , it will be convenient to

Both these properties hold for B if X is separable , so this will be assumed for the

rest of ...

However , rather than seek the maximum generality , it will be convenient to

**suppose**that B is itself complete and satisfies the countable chain condition .Both these properties hold for B if X is separable , so this will be assumed for the

rest of ...

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enumeration of its spectrum . Let dn denote the distance from an to o ( T ) – { n } .

**Suppose**that E is its resolution of the identity , and**suppose**that { an } is anenumeration of its spectrum . Let dn denote the distance from an to o ( T ) – { n } .

**Suppose**that for all but a finite number of n , Elan ) has a onedimensional range .### What people are saying - Write a review

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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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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