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 study the problem of estimation of individual measurements of objects in a biased spring balance weighing design under assumption that the errors are uncorrelated and they have different variances. The lower bound for the variance of each of the estimated measurements for this design and the necessary and sufficient conditions for this lower bound to be attained are given. The incidence matrices of the balanced incomplete block designs are used for construction of the A-optimal biased spring balance weighing design.