Generalized spectrum and commuting compact perturbations

Vladimir Rakočević
1993 Proceedings of the Edinburgh Mathematical Society  
Let X be an infinite-dimensional complex Banach space and denote the set of bounded (compact) linear operators on X by B(X) (K(X)). Let N(A) and R(A) denote, respectively, the null space and the range space of Let a g (A) = C\{XeC:R(A -X) is closed and k{A -X) = 0} denote the generalized (regular) spectrum of A. In this paper we study the subset a gb (A) of <s g (A) defined by o gb (A) = C\{XeC:R(A -X) is closed and k(A -A)<oo}. Among other things, we prove that if / is a function analytic in a
more » ... neighborhood of a(A), then <J gb (f(A))= f(a gb (A)). 1991 Mathematics subject classification scheme: 47 A53, 47 A55.
doi:10.1017/s0013091500018332 fatcat:bojv7mv2gvc5lm6itknpieaooa