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 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.

中文翻译：

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高度调和的无限矩阵 II：通过 Toeplitz-Silverman 矩阵从发散到收敛

Bernal-Gonzalez 等人最近证明了这一点。（Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 112(2):341–345, 2018）对于任何 Toeplitz-Silverman 矩阵 A，都存在所有序列，其所有非零元素都是发散的，但其 A 下的图像是收敛的。在本文中，我们通过证明在矩阵的适当假设下，存在一个稠密集、一个大代数和一个由这些序列组成的大巴拿赫格（除了零）来改进和概括这个结果。我们进一步表明，我们对矩阵 A 的假设之一通常不能省略。还考虑了矩阵条目的字段是超度量的情况。