Matrix transformations of double sequences (Q2711471)

The well-known Pringsheim convergence of double sequences \((x_{kl})\) is defined as the convergence of nets, where the set of indices \(\mathbb{N} \times \mathbb{N}\) is ordered in the natural way. Under this convergence, the row index \(k\) and the column index \(l\) independently tend to infinity. The author considers an essentially weaker convergence notion, which is characterized by the dependence of the index \(k\) on the index \(l\) in tending to infinity. More precisely, a double sequence \((x_{kl})\) is said to be \({\mathcal C}_e\)-convergent to a number \(a\) if \(\lim_l\lim_k |x_{kl}- a|=0\). Let \({\mathcal C}_e\) and \({\mathcal C}_{be}\) denote the space of all \({\mathcal C}_e\)-convergent double sequences and of all bounded \({\mathcal C}_e\)-convergent double sequences, respectively. The main results of the paper (Theorems 3.1, 3.4-3.6) state necessary and sufficient conditions under which a 4-dimensional matrix \(A= (a_{mnkl})\) maps \(X\) into \(Y\), where \(X,Y\in \{{\mathcal C}_e, {\mathcal C}_{be}\}\). The proofs use the gliding hump method.