A theorem of Mazur-Orlicz type in summability (Q1330817)

The authors consider a possible generalization of the classical Mazur- Orlicz theorem: Let \(X\) and \(A\) be conservative matrices. If there exists a bounded (divergent) sequence \(s\) such that \(As\) is bounded, divergent and \(X\)- summable, then there exists also an (unbounded) sequence \(s'\) such that \(As'\) is unbounded and \(X\)-summable. The Mazur-Orlicz theorem is the case \(X=I\) (the identity matrix). The authors note that the possible generalization is true if \(A\) is a normal conservative matrix, and prove it when \(X = C_ 1\), the standard Cesàro matrix. From this they deduce that the generalization is also true for multiplicative matrices.