Highly tempering infinite matrices II: From divergence to convergence via Toeplitz-Silverman matrices

It was recently proved by Bernal-Gonzalez et al. (Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 112(2):341-345, 2018) that for any Toeplitz-Silverman matrixA, there exists a dense linear subspace of the space of all sequences, all of whose nonzero elements are divergent yet whose images underAare convergent. In this paper, we improve and generalize this result by showing that, under suitable assumptions on the matrix, there are a dense set, a large algebra and a large Banach lattice consisting (except for zero) of such sequences. We show further that one of our hypotheses on the matrixAcannot in general be omitted. The case in which the field of the entries of the matrix is ultrametric is also considered.