## 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

### MSC:

 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