Let X, Y be two subspaces of summability domains of matrices with real or complex entries determined by speeds of the convergence, i.e.; by monotonically increasing positive sequences λ and µ. In this paper, we give necessary and sufficient conditions for a matrix M (with real or complex entries) to transform X into Y, where X is the subspace of summability domain of a λ-reversible or a normal or a λ-perfect matrix A defined by the speed λ and Y is the subspace of a lower triangular matrix B defined by the speed µ.