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.
A surjective bounded homomorphism fails to preserve n-weak amenability, in general. We however show that it preserves the property if the involved homomorphism enjoys a right inverse. We examine this fact for certain homomorphisms on several Banach algebras.