In this paper, we give necessary and sufficient conditions on $\left(p_n\right)$ for $|R,p_n{|}_k$, $k\ge 1$, to be translative. So we extend the known results of Al-Madi [1] and Cesco $\left[4\right]$ to the case $k>1$.

The main result of this paper is the following: A separable Banach space X is reflexive if and only if the infimum of the Gelfand numbers of any bounded linear operator defined on X can be computed by means of just one sequence on nested, closed, finite codimensional subspaces with null intersection.

In this paper, following the methods of Connor [connor], we extend the idea of statistical convergence of a double sequence (studied by Muresaleen and Edely [moe]) to $\mu$-statistical convergence and convergence in $\mu$-density using a two valued measure $\mu$. We also apply the same methods to extend the ideas of divergence and Cauchy criteria for double sequences. We then introduce a property of the measure $\mu$ called the (APO${}_2$) condition, inspired by the (APO) condition of Connor [jc]. We mainly investigate...