SOME THEOREMS ON MATRIX TRANSFORMS BETWEEN SPEED-MADDOX SPACES

Abstract

First, in this paper, the notions of convergence and boundedness with speed, and the notion of speed-Maddox spaces are recalled. Let X,Y be two sets of sequences with real or complex entries, and (X,Y) the set of matrices (with real or complex entries) to map X into Y. Let λ and μ be speeds of the convergence, i.e.; monotonically increasing positive sequences. Necessary and sufficient conditions for a matrix A ∈ (X,Y), if X is the certain speed-Maddox space defined by λ, and Y is another speed-Maddox space defined by μ are proved. As an application of main results, one example where A is the Zweier matrix Z_1/2 is presented.