Bishop’s Property (β), Svep and Dunford Property (c) ∗

If S, T ∈ B(H) have Bishop's property (β), does S + T have Bishop's property (β)? In this paper, a special case of this question is studied. Also given are a necessary and sufficient condition for a 2 × 2 operator matrix to have Bishop's property (β). Finally, the Helton class of an operator which has Bishop's property (β) is studied. 1. Introduction. Let B(H) be the algebra of all bounded linear operators acting on infinite dimensional separable complex Hilbert space H. An operator T ∈ B(H) is said to have the single-valued extension property (or SVEP) if for every open subset G of C and any analytic function f : G → H such that (T − z)f (z) ≡ 0 on G, we have f (z) ≡ 0 on G. For T ∈ B(H) and x ∈ H, the set ρ T (x) is defined to consist of elements z 0 ∈ C such that there exists an analytic function f (z) defined in a neighborhood of z 0 , with values in H, which verifies (T − z)f (z) = x, and it is called the local resolvent set of T at x. We denote the complement of ρ T (x) by σ T (x), called the local spectrum of T at x, and define the local spectral subspace of T , H T (F) = {x ∈ H : σ T (x) ⊂ F } for each subset F of C. Bishop [1] introduced Bishop's property (β). The study of operators satisfying Bishops property (β) is of significant interest and is currently being done by a number of mathematicians around the world [12, 13]. An operator T ∈ B(H) is said to have Bishop's property (β) if for every open subset G of C and every sequence f n : G → H of H-valued analytic functions such that (T − z)f n (z) converges uniformly to 0 in norm on compact subsets of G, f n (z) converges uniformly to 0 in norm on compact subsets of G. An operator T ∈ B(H) is said to have Dunford's property (C) if H T (F) is closed for each closed subset F of C. It is well known that