×

Matrix transformation between some classes of sequences. (English) Zbl 0907.47021

Throughout this paper let \(p= (p_k)\) be any sequence of strictly positive numbers, the Köthe-Toeplitz dual of the sequence space \[ \ell_\infty(p)= \Biggl\{(x_k)\in w:\sup_k | x_k|^{p_k}< \infty\Biggr\} \] is \[ M_\infty(p)= \bigcap_{N> 1} \Biggl\{(x_k)\in w: \sum_k| x_k| N^{1/p_k}< \infty\Biggr\}. \] Let \(X\) and \(Y\) be sets of sequences, \((X,Y)\) denotes the class of matrices \(A= (a_{nk})\) of complex numbers such that for each \(x= (x_k)\in X\), \(Ax= (A_nx)\in Y\), where \(A_nx= \sum_k a_{nk}x_k\).
The following results are established.
Theorem 1. Let \(p_k> 0\) for every \(k\). Then \(A\in(M_\infty(p),\overline c\cap\ell_\infty)\) if and only if there exists an integer \(B>1\) such that \(D= \sup_{n,k} | a_{nk}| B^{-1/p_k}<\infty\) holds and \(a_k= \text{stat}-\lim_{n\to\infty} a_{nk}\) exists \(\forall k\in \mathbb{N}\).
Theorem 2. \(A\in(\varphi, \overline c\cap\ell_\infty)\) if and only if there are numbers \(t\) and \(r\) \((0<r<1)\) such that \(| a_{nk}|\leq \text{tr}^k\) for all \(n,k= 1,2,\dots\) and \(a_k= \text{stat}- \lim_{n\to\infty} a_{nk}\) exists for each \(k\in\mathbb{N}\).
Reviewer: M.Kutkut (Amman)

MSC:

47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
46A45 Sequence spaces (including Köthe sequence spaces)
PDFBibTeX XMLCite
Full Text: DOI