A note on generalized Köthe-Toeplitz duals (Q1206279)

In 1980 Ivor J. Maddox determined the necessary and sufficient conditions for \((A_ k)^ \infty_ 0\) to belong to \(Z^ \beta(X)\), where \(Z= c_ 0\), \(c\), \(\ell_ \infty\), \(A_ k\in B(X,Y)\), \(X\) and \(Y\) are any Banach spaces. The purpose of this paper is firstly to determine the most general continuous linear functional \(f\) in \(X^*\), the dual space of \(X\), where \(X\) is a semi-conservative BK-space with \(\Delta^ += \{\delta,\delta^ 0,\delta^ 1,\delta^ 2,\dots\}\) as its Schauder basis, and secondly to determine the necessary and sufficient conditions for \((A_ k)^ \infty_ 0\) to belong to \(\text{bv}^ \beta_ 0(X)\), \(\text{bv}^ \beta(X)\), where \(A_ k\in B(X,Y)\), \(X\) and \(Y\) any Banach spaces.