## 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 2118

A generalized scalar operator T e B ( X ) is said to be

spectral distribution . Although it is not known whether or not every generalized

scalar operator is

two ...

A generalized scalar operator T e B ( X ) is said to be

**regular**if it has a**regular**spectral distribution . Although it is not known whether or not every generalized

scalar operator is

**regular**( unless the spectrum is sufficiently " thin " ) , given anytwo ...

Page 2158

If the set of points

subinterval of I ' , whose end points are

Borel subset of the plane is measurable T . PROOF . Let y be a closed subinterval

of To ...

If the set of points

**regular**relative to T is dense on To , then every closedsubinterval of I ' , whose end points are

**regular**relative to T is in S ( T ) and everyBorel subset of the plane is measurable T . PROOF . Let y be a closed subinterval

of To ...

Page 2160

If X is reflexive and if the adjoint T * satisfies the boundedness condition ( B ) ,

then the

interval of constancy relative to T consists entirely of

view of ...

If X is reflexive and if the adjoint T * satisfies the boundedness condition ( B ) ,

then the

**regular**points relative to T are dense in To and , in particular , everyinterval of constancy relative to T consists entirely of

**regular**points . PROOF . Inview of ...

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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 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