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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
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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
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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) $\sum^\infty_{k = 1}|a_{n, k}| \rightarrow 1$ 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).

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