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

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Page 2094

12 ) that if X is a weakly complete B - space , then any prespectral

automatically

and Dowson [ 1 ] have considered prespectral

...

12 ) that if X is a weakly complete B - space , then any prespectral

**operator**isautomatically

**spectral**, and so has a unique resolution of the identity . Berksonand Dowson [ 1 ] have considered prespectral

**operators**in some detail and have...

Page 2118

A generalized scalar

scalar

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 generalizedscalar

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

Page 2591

countably additive, XV.2 (1930), XV.2.3 (1930), XV.2.4 (1931) definition of, XV.2.1

(1929) integral with respect to, XV.2 (1929)

2148), XVI.5. 16 (2162) canonical reduction of, XV.4 (1937), XV.4.5 (1939), ...

countably additive, XV.2 (1930), XV.2.3 (1930), XV.2.4 (1931) definition of, XV.2.1

(1929) integral with respect to, XV.2 (1929)

**Spectral operator**, adjoint of, XVI.4.6 (2148), XVI.5. 16 (2162) canonical reduction of, XV.4 (1937), XV.4.5 (1939), ...

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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