On the space of almost convergent sequences (Q2314113)

Let \(\ell_{\infty}\) and \(c\) denote the Banach spaces of all bounded and convergent sequences, respectively. A sequence \(x=\left( x_{k}\right) \in\ell_{\infty}\) is said to be almost convergent if there exists a number \(t\) such that \(\lim_{n\rightarrow\infty}n^{-1}\sum_{k=m+1}^{m+n}x_{k}=t\) uniformly in \(m\). From the text: ``In the first part of the present paper, we give a criterion for the almost convergence of a nonnegative sequence to zero, based on ideas from [\textit{E. M. Semenov} et al., Izv. Math. 78, No. 3, 596--620 (2014; Zbl 1309.40003), Section 5]. In the second part of the paper, we consider the function \[ \alpha\left( x\right) =\limsup_{i\rightarrow\infty}\max_{i\leq j\leq2^{i} }\left\vert x_{i}-x_{j}\right\vert \] on \(\ell^{\infty}\) and study its relationship with the distance to the space \(c\) for almost convergent sequences; we also give a criterion for the convergence of an almost convergent sequence based on this function.''