Abstract
It was recently proved by Bernal-González et al. (Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 112(2):341–345, 2018) that for any Toeplitz–Silverman matrix A, there exists a dense linear subspace of the space of all sequences, all of whose nonzero elements are divergent yet whose images under A are 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 matrix A cannot in general be omitted. The case in which the field of the entries of the matrix is ultrametric is also considered.
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L. Bernal-González was supported by the Plan Andaluz de Investigación de la Junta de Andalucía FQM-127 Grant P08-FQM-03543 and by MICINN Grant PGC2018-098474-B-C21. J. B. Seoane-Sepúlveda was supported by Grant PGC2018-097286-B-I00. W. Trutschnig gratefully acknowledges the support of the WISS 2025 project ‘IDA-lab Salzburg’ (20204-WISS/225/197-2019 and 20102-F1901166-KZP).
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Bernal-González, L., Fernández-Sánchez, J., Seoane-Sepúlveda, J.B. et al. Highly tempering infinite matrices II: From divergence to convergence via Toeplitz–Silverman matrices. RACSAM 114, 202 (2020). https://doi.org/10.1007/s13398-020-00934-z
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DOI: https://doi.org/10.1007/s13398-020-00934-z