## Some inclusion theorems for absolute summability.(English)Zbl 0879.40004

Let $\sum_{n=0}^\infty x_n \tag{1}$ be an infinite numerical series with partial sums $$s_n$$, $$n=0,1,\dots$$, and let $$A=(a_{nr})$$, $$n=0,1,\dots$$, $$r=0,1,\dots$$, be a lower semi-matrix with nonzero diagonal entries. Let $$T_n=\sum_{r=0}^n a_{nr}s_r,\;n=0,1,\dots$$. The series (1) is said summable $$|A|_k$$ $$(k\geq 1)$$, if $\sum_{n=1}^\infty n^{k-1}|T_n-T_{n-1}|^k <\infty.\tag{2}$ In the case of absolute Riesz summability, i.e., $$a_{nr}= p_r \cdot \Big (\sum_{j=0}^n p_j\Big)^{-1},\;0\leq r\leq n$$, $$a_{nr}=0,\;r> n$$, where $$p_n$$, $$n=0,1,\dots$$, is a sequence of positive real numbers such that $$\lim_{n\to \infty}\sum_{j=0}^n p_j =\infty$$, we write $$|R,p_n|_k$$ for summability $$|A|_k$$.
The aim of this paper is to give necessary and sufficient conditions for the series (1) to be summable $$|A|_k$$, whenever it is summable $$|R,p_n|_1$$. In the proof of this main result (i.e. Theorem on p. 600) a little functional analytic apparatus is used. More precisely, e.g. in the necessity-part of the proof just mentioned the Banach-Steinhaus theorem on certain $$BK$$-spaces of sequences is suitably applied. As corollaries, the authors deduce from their Theorem some known inclusion theorems for absolute Riesz (particularly Cesàro) summability.
Reviewer’s remark. The absolute summability $$|A|_1$$ (i.e. condition (2) with $$k=1$$) for an infinite square matrix $$A$$ of real or complex numbers appeared as early as in the paper by R. P. Cesco [Univ. Nac. La Plata., Publ. Fac. Ci. fis-mat., Revista 2, 147-156 (1941; Zbl 0061.12001)].

### MSC:

 40D25 Inclusion and equivalence theorems in summability theory 40F05 Absolute and strong summability 40C05 Matrix methods for summability 40G05 Cesàro, Euler, Nörlund and Hausdorff methods 40H05 Functional analytic methods in summability

Zbl 0061.12001
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### References:

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