## 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 2197

... and this contradiction completes the proof . Q . E . D . 4 LEMMA . Let B be a

complete ( o - complete ) Boolean algebra of projections in the B - space X and

let { Ex } be a monotone generalized

Then ...

... and this contradiction completes the proof . Q . E . D . 4 LEMMA . Let B be a

complete ( o - complete ) Boolean algebra of projections in the B - space X and

let { Ex } be a monotone generalized

**sequence**( a monotone**sequence**) in B .Then ...

Page 2218

If a generalized

projections in a B - space converges weakly to a projection , then it converges

strongly . PROOF . In view of Lemma 23 , the proof may be restricted to the case ...

If a generalized

**sequence**of projections in a o - complete Boolean algebra ofprojections in a B - space converges weakly to a projection , then it converges

strongly . PROOF . In view of Lemma 23 , the proof may be restricted to the case ...

Page 2450

Let rn be the distance from the eigenvalue in to the other points on the spectrum

of T , so that rn > 0 , while , of course , the

is bounded . Let R be the closed , densely defined , unbounded operator in X ...

Let rn be the distance from the eigenvalue in to the other points on the spectrum

of T , so that rn > 0 , while , of course , the

**sequence**{ rn } , like the**sequence**in ,is bounded . Let R be the closed , densely defined , unbounded operator in X ...

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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