A generalized product on some sequences and matrix transformations (Q1369015)

Let \(p= (p_n)\) be a sequence of strictly positive real numbers and \(\upsilon= (\upsilon_n)\) a fixed sequence of non-zero complex numbers such that the sequence \((|\upsilon_n|)\) is nondecreasing and \(\liminf_n|\upsilon_n|^{- p_n}< \infty\). In a previous work with \textit{A. Ratha} [same J. 23, 223-234 (1991; Zbl 0757.40003)], the first author introduced the sequence spaces \(D^\wedge_\infty(p):= \{x= (x_k)\mid| v_kx_k|= O(1)\}\) and \(D^\wedge_0(p):= \{x= (x_k)\mid| v_kx_k|= o(1)\}\) and gives necessary and sufficient conditions for both of these spaces to be closed under the Cauchy and the Dirichlet product. In the paper under review, the authors consider the following generalized product defined by \textit{P. N. Natarajan} [same J. 26, 993-1001 (1995; Zbl 0855.40001)]. For every \(n= 1,2,\dots\) let \(b_n\) be a mapping of \(\{1,2,\dots, n\}\) into \(\{0,1,2,\dots, n\}\) with \(b_n(1)= 1\) and \(b_n(k)\to \infty\) for \(n\to\infty\) and \(k= 1,2,\dots\)\ . For two sequences \(x= (x_k)\) and \(y= (y_k)\), the generalized product \(x*_B y\) is defined as the sequence \(\left(\sum^n_{k=1} x_k y_{b_n(k)}\right)_{n= 1,2,\dots}\). It is shown (Theorem 2.1) that \(D^\wedge_\infty(p)\) is closed under the product \(*_B\) if and only if \[ \Biggl(\sum_{k=1, b_k(k)\neq 0} N^{\pi_k+ \pi_{b_n(k)}}/|\upsilon_k \upsilon_{b_n(k)}|\Biggr)^{p_n}= O(|\upsilon_n|^{- p_n})\quad\text{for every }N>1, \] where \(\pi_k= 1/p_k\), \(k= 1,2,\dots\)\ . An analogous result (Theorem 3.1) is obtained for the space \(D^\wedge_0(p)\). These theorems contain the results of Srivastava and Ratha [op. cit.] on the closedness of the spaces \(D^\wedge_\infty(p)\) and \(D^\wedge_0(p)\) under the Cauchy and the Dirichlet product.