On the Köthe-Toeplitz duals of some generalized sets of difference sequences (Q2708133)

Let \(p= (p_k)^\infty_{k=1}\) be a sequence of positive real numbers. Denote by \(\omega\) the set of all sequences \(x= (x_k)^\infty_{k=1}\) of complex numbers. PutNEWLINENEWLINENEWLINE\(\ell_\infty(p)= \{x\in \omega:\sup_k|x_k|^{p_k}<+ \infty\}\),NEWLINENEWLINENEWLINE\(c(p)= \{x\in\omega:|x_k- \ell|^{p_k}\to 0\) for some \(\ell\}\),NEWLINENEWLINENEWLINE\(c_0(p)= \{x\in\omega:|x_k|^{p_k}\to 0\}\),NEWLINENEWLINENEWLINE\(\ell(p)= \{x\in\omega: \sum^\infty_{k=1}|x_k|^{p_k}<+ \infty\}\).NEWLINENEWLINENEWLINEFor \(u= (u_k)^\infty_{k= 1}\) with \(u_k\neq 0\) \((k= 1,2,\dots)\) we setNEWLINENEWLINENEWLINE\(\ell_\infty(u, \Delta,p)= \{x\in\omega: (u_k\Delta x_k)\in \ell_\infty(p)\}\),NEWLINENEWLINENEWLINE\(c(u,\Delta, p)= \{x\in \omega:(u_k\Delta x_k)\in c(p)\}\),NEWLINENEWLINENEWLINE\(c_0(u,\Delta, p)= \{x\in\omega: (u_k\Delta x_k)\in c_0(p)\}\),NEWLINENEWLINENEWLINEwhere \(\Delta x_k= x_k- x_{k+1}\) \((k= 1,2,\dots)\).NEWLINENEWLINENEWLINEIf \(p= (p_k)\) is a bounded sequence then these sets are linear spaces under the usual definition \(x+y\) and \(\gamma x\), \(\gamma\) being an arbitrary complex number.NEWLINENEWLINENEWLINEFor \(x\subseteq\omega\), we setNEWLINENEWLINENEWLINE\(X^\alpha= \{a= (a_k)\in\omega: \sum^\infty_{k=1}|a_k x_k|<+\infty\) for each \(x\in X\}\),NEWLINENEWLINENEWLINE\(X^\beta= \{a= (a_k)\in \omega: \sum^\infty_{k=1} a_k x_k\) converges for each \(x\in X\}\).NEWLINENEWLINENEWLINEThe sets \(X^\alpha\), \(X^\beta\) are called \(\alpha\)- and \(\beta\)-duals of \(X\), respectively. In the paper \(\alpha\)- and \(\beta\)-duals of the sets \(\ell_\infty(u, \Delta,p)\), \(c_0(u,\Delta, p)\) and \(c(u,\Delta, p)\) are determined. Similar results are established for the sequence space \(\ell_\infty(\Delta_r, p)\) investigated in the paper by \textit{Mursaleen}, \textit{A. K. Gaur} and \textit{A. H. Saifi} [Bull. Calcutta Math. Soc. 88, No. 3, 207-212 (1996; Zbl 0891.46003)].