The space \(bv(p)\), its \(\beta\)-dual and matrix transformations
Let \(p= (p_k)\) be a bounded sequence of positive numbers. The sequence spaces of Maddox, \(\ell(p)= \{x= (x_k)\mid\sum_k |x_k|^{p_k}< \infty\}\), \(c_0(p)= \{x= (x_k)\mid\lim_k| x_k|^{p_k}= 0\}\) and \(c(p)= c_0(p)\oplus \text{span}\{e\}\) with \(e:= (1,1,\dots)\) have been studied by various researchers [i.e., \textit{K.-G. Grosse-Erdmann}, Can. J. Math. 44, 298--307 (1992; Zbl 0777.46008); J. Math. Anal. Appl. 180, 223--238 (1993; Zbl 0791.47029)]. In this paper the authors define a new space, \[ bv(p)= \Biggl\{x= (x_k)\,\left|\;\sum_k|x_k- x_{k-1}|^{p_k}< \infty\right.\Biggr\}, \] and describe the \(\beta\)-dual space of it. Furthermore, matrix transformations \(A: bv(p)\to Y\) with \(Y\in \{\ell(s),\ell^\infty(s), c_0(s), c(s)\}\) are characterized. For example, the following result is proved (Theorem 3.1 (1)): For \(p_k\leq 1\) and \(s_k\geq 1\), a matrix \(A= (a_{nk})\) maps \(bv(p)\) into \(\ell(s)\) if and only if \(\sup_k \sum^\infty_{n=0}|\sum^\infty_{j=k} a_{nj} B^{-1/p_k}|^{s_n}< \infty\) for some \(B> 1\).
scientific article; zbMATH DE number 2123815