## Linear operators: Spectral operators |

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Page 1930

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

set and the whole plane, in short, if 2 is a field of sets in the complex plane, then a

spectral measure E on 2 is called a resolution of the

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

set and the whole plane, in short, if 2 is a field of sets in the complex plane, then a

spectral measure E on 2 is called a resolution of the

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

12) that if J 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

obtained ...

12) that if J 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

obtained ...

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

resolution ...

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

resolution ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

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### Common terms and phrases

adjoint operator algebra of projections Amer analytic arbitrary asymptotic B-space Banach space Boolean algebra Borel sets boundary conditions bounded Borel function bounded linear operator bounded operator commuting compact complex numbers complex plane constant contains continuous functions converges Corollary countably additive Definition denote dense differential operator disjoint Doklady Akad domain eigenvalues elements equation exists finite number Foias follows from Lemma follows from Theorem formal differential operator formula Hence Hilbert space hypothesis identity inequality inverse Lebesgue Math matrix measurable functions multiplicity Nauk SSSR norm normal operators operators in Hilbert perturbation polynomial Proof properties prove quasi-nilpotent restriction Russian satisfies scalar operator scalar type operator scalar type spectral Section semi-group sequence shows spectral measure spectral operator spectral theory spectrum strong operator topology subset Suppose trace class type spectral operator unbounded uniformly bounded unique vector weakly complete zero