## Linear operators: Spectral operators |

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Page 2118

A generalized

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

...

A generalized

**scalar operator**T e B(X) is said to be regular if it has a regularspectral distribution. Although it is not known whether or not every generalized

**scalar operator**is regular (unless the spectrum is sufficiently "thin"), given any two...

Page 2119

It is proved (see Foias [9] or Colojoara and Foias [4]) that if U is a spectral

distribution on C°°(Q) to B(X) and if <p eC°°(Q), then U(<p) is a generalized

U. In ...

It is proved (see Foias [9] or Colojoara and Foias [4]) that if U is a spectral

distribution on C°°(Q) to B(X) and if <p eC°°(Q), then U(<p) is a generalized

**scalar operator**with spectrum contained in the image, under <p, of the support ofU. In ...

Page 2120

An

91 -+B(X) such that S = Un (where ^(Aj^A). Every

SH-sccdar, with 91 = the algebra of bounded Borel functions. Similarly, if X ...

An

**operator**S 6 B(X) is called Sll-**scalar**if there exists an 91-spectral function U:91 -+B(X) such that S = Un (where ^(Aj^A). Every

**scalar**type spectral**operator**isSH-sccdar, with 91 = the algebra of bounded Borel functions. Similarly, if X ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

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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