A Class of Regular Matrices

Abstract

Let m be the space of real, bounded sequences $x = \{ {x_k}\}$ with the sup norm, and let $A = ({a_{n,k}})$ be a regular (i.e., Toeplitz) matrix. We consider the following two possible conditions for A: (1) $\Sigma _{k = 1}^\infty |{a_{n,k}}| \to 1$ as $n \to \infty$, (2) $\Sigma _{k = 1}^\infty |{a_{n,k}} - {a_{n,k + 1}}| \to 0$ as $n \to \infty$. G. Das [J. London Math. Soc. (2) 7 (1974), 501-507] proved that if a regular matrix A satisfies both (1) and (2) then (3) ${\overline {\lim } _{n \to \infty }}{(Ax)_n} \leqslant q(x)$ for all $x \in m$, where $q(x) = {\inf _{{n_i},p}}{\overline {\lim } _{k \to \infty }}{p^{ - 1}}\Sigma _{i = 1}^p{x_{{n_i} + k}}$. Das used “Banach limits” and Hahn-Banach techniques, and stated that he thought it would be “difficult to establish the result... by direct method". In the present paper an elementary proof of the result is given, and it is shown also that the converse holds, i.e., for a regular A, (3) implies (1) and (2). Hence (3) completely characterizes the class of regular matrices satisfying (1) and (2).