Almost convergence and a theorem of Lorentz (Q1338178)

Denote the translates of a sequence \(x = (x_ k)\) by \((T^ rx)_ k : = x_{k+r}\) \((k,r = 0,1, \dots)\) and \(m,c\) the spaces of real-valued bounded, convergent, sequences respectively. For a real matrix \(A = (a_{n,k})\), let \(F(A) : = \{x \in m : \exists \xi\), \(\lim A(T^ rx) = \xi\) uniformly in \(r \geq 0\}\). When \(A\) is the Cesàro matrix \(C_ 1\), we get \(F(C_ 1) = ac\), the space of real almost convergent sequences. It is well-known [\textit{G. G. Lorentz}, Acta Math. Uppsala 80, 167-190 (1948; Zbl 0031.29501)] that if \(A\) is \((c,c)\)-regular then \(c \subseteq F(A) \subseteq ac\). In the present paper, it is proved that if \(\sum^ \infty_{k=0} | a_{nk} | < \infty\) for each \(n \geq 0\), then in order that \(F(A) \subseteq ac\) it is necessary that \(\limsup_ n \sum^ \infty_{k=0} | a_{nk} | > 0\) and sufficient that \(\limsup_ n | \sum^ \infty_{k = 0} a_{nk} | > 0\).