The potency of weighted means: Addendum to a paper of Kuttner and Parameswaran
Let \((A)= m\cap c_A\). A matrix \(A\) is called conservative, if \(c\subset c_A\). By definition, \(A\) is regular, if it is conservative and \(\lim Ax=\lim x\) for each \(x\in c\). A matrix \(A\) is called potent, if \(\chi\cap c_A\subset c_B\) implies \(m\cap c_A\subset c_B\) for each matrix \(B\). By KP we denote the class of all conservative potent matrices. KG denote the class of all conservative matrices such that each matrix \(B\) satisfying \(\chi\cap c_A\subset c_B\) is conservative. Every \(R_p\) is conservative, and \(R_p\) is regular if and only if \(P_n\to \infty\). If \((p_n)= e= (1,1,1,\dots)\), then \(R_p\) equals the Cesàro matrix \(C_1\). Making the general assumptions \[ p_n> 0\;(n\in \mathbb{N}^0) \quad\text{and}\quad P_n\to\infty\;(n\to\infty), \tag{1} \] the authors prove the following results: Theorem 1. If \(A\in \text{KG}\) and the linear span \((\chi\cap c_A)\) of \(\chi\cap c_A\) is dense in \((m\cap c_A, \|\;\|_\infty)\), then \(A\) is potent. Theorem 2. If \(p= (p_n)\) satisfies the condition (1), then the following conditions are equivalent: (a) \(R_p\in \text{KP}\), that is \(R_p\) is potent. (b) \((p_n/P_n)\in c_0\). (c) \(R_p\in \text{KG}\). This paper extends the results of \textit{B. Kuttner} and \textit{M. R. Parameswaran} [J. Anal. 4, 161-172 (1996; Zbl 0880.40003)] by filling a nontrivial gap.
scientific article; zbMATH DE number 1417956