## 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 2150

Since the spectrum is totally disconnected , every spectral point is

spectral set of arbitrarily small diameter and thus in an S ( T ) set of arbitrarily

small diameter . Since it is clear that every subset of the resolvent set is an S ( T )

set ...

Since the spectrum is totally disconnected , every spectral point is

**contained**in aspectral set of arbitrarily small diameter and thus in an S ( T ) set of arbitrarily

small diameter . Since it is clear that every subset of the resolvent set is an S ( T )

set ...

Page 2234

The first statement follows from Definition 8 and the three paragraphs of

explanation which precede it , and from Lemma 6 . Statement ( i ) follows from

Corollary 7 . If e is a bounded Borel set with closure

supposed ...

The first statement follows from Definition 8 and the three paragraphs of

explanation which precede it , and from Lemma 6 . Statement ( i ) follows from

Corollary 7 . If e is a bounded Borel set with closure

**contained**in U , it may besupposed ...

Page 2256

Let R = T - 1 . Then , by Theorem VII . 9 . 5 , o ( R ) = { 212 - 1 € ( T ) } U { 0 } .

Since o ( T ) is totally disconnected , each point , in o ( T ) is

arbitrarily small compact subset o of o ( T ) which is open in the relative topology

of o ( T ' ) ...

Let R = T - 1 . Then , by Theorem VII . 9 . 5 , o ( R ) = { 212 - 1 € ( T ) } U { 0 } .

Since o ( T ) is totally disconnected , each point , in o ( T ) is

**contained**in anarbitrarily small compact subset o of o ( T ) which is open in the relative topology

of o ( T ' ) ...

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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