Abel-Schur Multipliers on Banach Spaces of Infinite Matrices

Abstract

We consider a more general class than the class of Schur multipliers namely the Abel-Schur multipliers, which in turn coincide with the bounded linear operators on $ell_{2}$ preserving the diagonals. We extend to the matrix framework Theorem 2.4 (a) of a paper of Anderson, Clunie, and Pommerenke published in 1974, and as an application of this theorem we obtain a new proof of the necessity of an old theorem of Hardy and Littlewood in 1941.