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

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Page 1930

If Eis a Boolean algebra of subsets of the complex plane which contains the void

set and the whole plane , in short , if & is a field of sets in the complex plane , then

a spectral measure E on & is called a resolution of the

If Eis a Boolean algebra of subsets of the complex plane which contains the void

set and the whole plane , in short , if & is a field of sets in the complex plane , then

a spectral measure E on & is called a resolution of the

**identity**( or a spectral ...Page 2094

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

automatically spectral , and so has a unique resolution of the

and Dowson [ 1 ] have considered prespectral operators in some detail and have

...

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

automatically spectral , and so has a unique resolution of the

**identity**. Berksonand Dowson [ 1 ] have considered prespectral operators in some detail and have

...

Page 2242

The projection valued measure E is said to be the resolution of the

13 LEMMA . An unbounded spectral operator of scalar type in the sense of

Definition 12 is a spectral operator in the sense of Definition 1 . Moreover , the ...

The projection valued measure E is said to be the resolution of the

**identity**for T .13 LEMMA . An unbounded spectral operator of scalar type in the sense of

Definition 12 is a spectral operator in the sense of Definition 1 . Moreover , the ...

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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