## Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

## If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

# A Class of Regular Matrices

Godfrey L. Isaacs
Proceedings of the American Mathematical Society
Vol. 60, No. 1 (Oct., 1976), pp. 211-214
DOI: 10.2307/2041144
Stable URL: http://www.jstor.org/stable/2041144
Page Count: 4
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) $\sum^\infty_{k = 1}|a_{n, k}| \rightarrow 1$ as $n \rightarrow \infty$, (2) $\sum^\infty_{k = 1}|a_{n, k} - a_{n, k + 1}| \rightarrow 0$ as $n \rightarrow \infty$. 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) $\overline{\lim}_{n \rightarrow \infty}(Ax)_n \leqslant q(x)$ for all $x \in m$, where $q(x) = \inf_{n_i, p}\overline{\lim}_{k \rightarrow \infty}p^{-1}\sum^p_{i = 1}x_{n_i + k}$. Das used "Banach limits" and Hahn-Banach techniques, and stated that he thought it would be "difficult to establish the result $\ldots$ 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).