Matrix maps into the space of statistically convergent bounded sequences. (English) Zbl 0865.40001

Proc. Est. Acad. Sci., Phys. Math. 45, No. 2-3, 187-192 (1996); corrigendum ibid. 46, No. 1-2, 150 (1997).
A sequence space is any linear subspace of the vector space of all complex-valued sequences. For two sequence spaces \(X\) and \(Y\), denote by \((X,Y)\) the set of all doubly-infinite matrices \(A=(a_{nk})\) such that, for each \(x\in X\), \(A_nx\equiv\sum^\infty_{k=1}a_{nk}x_k\) exists and the sequence \((A_nx)\in Y\). For such a matrix \(A\), a sequence \(x\) is \(A\)-statistically convergent to \(\ell\) if, for every \(\varepsilon>0\), \(\lim_{n\to\infty} \sum_{k\in V}a_{nk}=0\), where \(V=\{k:|x_k-\ell|\geq\varepsilon\}\). This paper characterizes the matrix class \((X,Y_1)\), where \(Y_1\) is the space of all uniformly-bounded \(A\)-statistically convergent sequences and \(X\) is a separable BK-space (i.e., a Banach sequence space with continuous coordinate functionals) with a countable fundamental set. The special cases where \(X\) is the space of all convergent sequences and where \(X=\ell^p\), \(p\geq 1\), are considered.


40C05 Matrix methods for summability
40A05 Convergence and divergence of series and sequences