Matrix Transforms Into the Set of a-absolutely Convergent Sequences with Speed and the Regularity of Matrices on the Sub-spaces of C

Abstract

Let α > 1. The α-absolute convergence with speed, where the speed is defined by a monotonically increasing positive sequence µ, has been studied in the present paper. Let l µ α be the set of all α-absolutely µconvergent sequences and X a sequence space defined by another speed λ. Necessary and sufficient conditions for a matrix A (with real or complex entries) to map X into l µ α have been presented. It is proved as an example that the Zweier matrix Z1/2 satisfies these necessary and sufficient conditions for certain speeds λ and µ. The notion of regularity on the subspace X of the set c of converging sequences is defined, and also, necessary and sufficient conditions for a matrix A to be regular on certain X ⊂ c are presented. It has also been shown that there exists an irregular matrix, which is regular on the subspace X of c.