We study the stability of a-Browder-type theorems for orthogonal direct sums of operators. We give counterexamples which show that in general the properties (SBaw), (SBab), (SBw) and (SBb) are not preserved under direct sums of operators. However, we prove that if S and T are bounded linear operators acting on Banach spaces and having the property (SBab), then S ⊕ T has the property (SBab) if and only if σSBF− + (S ⊕ T ) = σSBF− + (S) ∪ σSBF− + (T ), where σSBF− + (T ) is the upper semi-B-Weyl spectrum of T . We obtain analogous preservation results for the properties (SBaw), (SBb) and (SBw) with extra assumptions.
In this paper, we give a new approach to the study of Weyl-type theorems. Precisely, we introduce the concepts of spectral valued and spectral partitioning functions. Using two natural order relations on the set of spectral valued functions, we reduce the question of relationship between Weyl-type theorems to the study of the set difference between the parts of the spectrum that are involved. This study solves completely the question of relationship between two spectral valued functions, comparable for one or the other order relation. Then several known results about Weyl-type theorems become corollaries of the results obtained.