Some properties of the matrix class (l suffix alpha,l suffix alpha), 0

Abstract

Throughout the present paper, entries of sequences, infinite series and infinite matrices are real or complex numbers. The class \((\ell_\alpha, \ell_\alpha)\), \(0 < \alpha \leq 1\), of all infinite matrices transforming sequences in \(\ell_\alpha\) to sequences in \(\ell_\alpha\) is characterized. The structure of \((\ell_\alpha, \ell_\alpha)\), \(0 < \alpha \leq 1\), is then discussed. Following Fridy [\emph{Properties of absolute summability matrices}, Proc. Amer. Math. Soc. 24 (1970), 583--585], a Steinhaus type result involving the class \((\ell_\alpha, \ell_\alpha)\) is also proved.