一类算子的谱理论(Ⅲ)—(AC)算子、可分解算子和譜算子

A CLASS OF OPERATOR AND THE SPECTRAL THEORY (Ⅲ)——(AC) OPERATOR, DECOMPOSABLE OPERATOR AND SPECTRAL OPERATOR

本文是文献[9],[10]的继续。在本文中,我们研究了(AC)算子,可分解算子,谱算子以及它们之间的关系。证明了:(1)若T∈B(X)是(AC)算子,对于每个E,F∈F,有则T是可分解算子。(2)T∈B(X)是谱算子当且仅当T是(AC)算子且满足下述条件:(ⅰ)对每个Borel子集δ,δ∈B,有X_T(δ)=X_T((δ∩δ)⊕此处⊕表示直接和;(ⅱ)对每个x∈X,数集是有界的,此处(3)若是(H)空间,是可分解算子,则下述条件是等价的:(ⅰ)(E)(ⅱ)①从推出(此处P_F是从到_T(F)上直交射影,⊕表示直交和)。它是B.L.Wadhwa定理的新形式。

This paper is a continuation of the papers [9],[10]. In this paper we investigated(AC)operators, decomposable operators, spectral operators and their relationships.We proved that: (1) If T∈B(X) is (AC) operator, for every, then T is a decomposable operator. (2) T∈B(X) is spectral operator if and only if T is (AC) operator and T satisfies the following conditions: (i) For every Borel subsets σ,δ∈B has where denoted direct sum; (ii) For every x∈X, the set of numbers is bounded, where (3) If is (H) space, T∈B() is a decomposable operator, then the following conditions are equivelent: (i) where P_F is an orthogonal projection on (F), the denotes orthogonal sum. This is a new form of B.L.Wadhwa[8]theorem.