## Linear Operators: Spectral Theory : Self Adjoint Operators in Hilbert Space, Volume 2 |

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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

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 ' ) . It follows that the set t ( o ) = { z12 - 1€ 0 } is a compact subset of o ( R )

...

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 ' ) . It follows that the set t ( o ) = { z12 - 1€ 0 } is a compact subset of o ( R )

...

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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 formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator manifold Math Moreover multiplicity norm normal 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