Matrix transformations between some new sequence spaces. (English) Zbl 0542.40003

Let \(\sigma\) be a mapping of the set of positive integers into itself, let \(V_{\sigma}\) be the space of bounded sequences all of whose \(\sigma\)-means are equal, and let \(\sigma\)-lim x be the common value of all \(\sigma\)-means on x. In this paper the author generalizes the idea of strong almost convergence of I. J. Maddox [Math. Proc. Camb. Philos. Soc. 83, 61-64 (1978; Zbl 0392.40001)]: a bounded sequence \(x=(x_ k)\) is said to be strongly \(\sigma\)-convergent to a number L if and only if \((| x_ k-L|)\in V_{\sigma}\) with \(\sigma\)-limit zero. He characterizes those real infinite matrices which map all convergent sequences (all sequences of \(\sigma\)-bounded variation) into sequences strongly \(\sigma\)-convergent to zero (strongly \(\sigma\)- convergent). The concept of sequences of \(\sigma\)-bounded variation was introduced by the author in an earlier paper [Q. J. Math., Oxf. II. Ser. 34, 77-86 (1983)].
Reviewer: J.Boos


40C05 Matrix methods for summability
40F05 Absolute and strong summability
40C99 General summability methods
40D25 Inclusion and equivalence theorems in summability theory


Zbl 0392.40001