We deal with the sum and the product of particular BK spaces and give necessary and sufficient conditions to have wα(λ) + wβ(μ) = wα+β(μ). Some results on matrix transformations mapping the space wα(λ) + wβ(μ) into a given BK space are also given. This study generalizes some results obtained in  and .
In this paper, we characterize classes of matrix transformations from BK spaces into spaces of bounded sequences and their
subclasses of infinite matrices that define compact operators. Furthermore, using these results and the solvability of certain
infinite linear systems we give necessary and sufficient conditions for A to be a compact operator on spaces that are strongly α-bounded or summable.