Infinite matrices of the type \(M(E)\) (Q1184837)

Let us cite the author's abstract: ``Let \(A=(a_{nk})\) be an infinite matrix and \(x=(x_ k)\) an infinite sequence of complex numbers. Let \(E\) be a sequence space, we write \(E_ A=\{x:\;Ax\in E\}\). Let \(E\subseteq E_ A\), \(A\) is said to be of type \(M(E)\) if whenever \(tA=\theta\) for \(t\in E^ \beta\) (\(\beta\)-dual of \(E\)) then \(t=\theta\), where \(\theta\) denotes the zero scalar sequences. The purpose of this note is to characterize the matrices of the type \(M(E)\).'' The \(\beta\)-dual \(E^ \beta\) of \(E\) is defined as follows. \(E^ \beta=\{t:\;tx\) exists for all \(x\in E\}\), \(\| t\|_{E^ \beta}=\sup_{\| x\|_ E=1}| tx|\), where \(tx=\sum_ k t_ k x_ k\).