Spectral Measures, Boundedly σ-Complete Boolean Algebras and Applications to Operator Theory

1987 ◽  
Vol 304 (2) ◽  
pp. 819 ◽  
Author(s):  
Werner J. Ricker
Keyword(s):  
Operator Theory ◽  
Boolean Algebras ◽  
Spectral Measures

scholarly journals Spectral measures on spaces not containing l∞

1981 ◽  
Vol 24 (1) ◽  
pp. 41-45 ◽  
Author(s):  
T. A. Gillespie
Keyword(s):  
Banach Space ◽  
Boolean Algebra ◽  
Banach Spaces ◽  
Spectral Measure ◽  
Dual Space ◽  
Boolean Algebras ◽  
Special Role ◽  
Spectral Measures ◽  
Sequential Completeness ◽  
Algebra Of Sets

The property of weak sequential completeness plays a special role in the theory of Boolean algebras of projections and spectral measures on Banach spaces. For instance, if X is a weakly sequentially complete Banach space, then(i) every strongly closed bounded Boolean algebra of projections on X is complete (3, XVII.3.8, p. 2201); from which it follows easily that(ii) every spectral measure on X of arbitary class (Σ, Γ), where Σ is a σ-algebra of sets and Γ is a total subset of the dual space of X, is strongly countably additive; and hence that(iii) every prespectral operator on X is spectral.(See also (1, Theorem 6.11, p. 165) for (iii).)

Cline’s formula for g-Drazin inverses in a ring

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2249-2255
Author(s):  
Huanyin Chen ◽  
Marjan Abdolyousefi
Keyword(s):  
Operator Theory ◽  
Spectral Properties ◽  
Associative Ring ◽  
Linear Operators ◽  
Drazin Inverse ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Cline’S Formula

It is well known that for an associative ring R, if ab has g-Drazin inverse then ba has g-Drazin inverse. In this case, (ba)d = b((ab)d)2a. This formula is so-called Cline?s formula for g-Drazin inverse, which plays an elementary role in matrix and operator theory. In this paper, we generalize Cline?s formula to the wider case. In particular, as applications, we obtain new common spectral properties of bounded linear operators.

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