## Linear Operators: Spectral operators |

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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 ) = { 212 - 1€ o } is a compact subset of o ( R ) ,

open ...

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 ) = { 212 - 1€ o } is a compact subset of o ( R ) ,

open ...

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### Contents

SPECTRAL OPERATORS | 1924 |

An Operational Calculus for Bounded Spectral | 1941 |

Part | 1950 |

Copyright | |

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adjoint operator analytic apply arbitrary assume B-space Banach space Boolean algebra Borel sets boundary conditions bounded bounded operator Chapter clear closed commuting compact complex consider constant contained continuous converges Corollary corresponding countably additive defined Definition denote dense determined differential operator discrete 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 Math Moreover multiplicity norm positive preceding present problem projections PROOF properties proved range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows similar spectral measure spectral operator spectrum subset subspace sufficiently Suppose Theorem theory topology unbounded uniform uniformly unique valued vector weakly zero