Matrix transformations of Cesàro summable series (Q1820939)

This paper generalizes and extends a result of \textit{L. Alpar} [ibid. 39, 233-243 (1982; Zbl 0493.40003)] that gave necessary and sufficient conditions on an infinite matrix A to transform every (C,\(\alpha)\) summable series into a (C,\(\beta)\) summable series where \(\alpha\geq 0,\beta \geq 0\). We obtain here necessary and sufficient conditions on A in order that it transform every (C,\(\alpha)\) summable series into a B summable series where \(\alpha\geq -1\) and B is any normal matrix. The proof uses the representation of the dual of the space of (C,\(\alpha)\) summable series in the case -1\(\leq \alpha \leq 0\).