Abstract: In the paper we consider the space \(\mathcal{L}_{0,1}\) of infinite matrices \((a_{ij})_{i,j=1}^\infty\), where \(a_{ij}\in\mathbb{R}\), equipped with a cut-norm. The cut-norm of a matrix appears naturally in various areas of discrete mathematics, and is also used in functional analysis. It is equivalent to the operator norm of a matrix considered as a mapping from the space \(c_0\) of sequences converging to zero (or from \(l^\infty\)) to \(l^1\). It is proved that the set of matrix units \(\{E_{ij}\}_{i,j=1}^\infty\) forms a system of random unconditional convergence (RUC) in the space \(\mathcal{L}_{0,1}\), but is not a system of unconditional convergence. This system is linearly full in \(\mathcal{L}_{0,1}\) and, being enumerated in a certain order, may turn out to be a conditional basis. We obtain both necessary and sufficient conditions for a bijection \(\ell\colon \mathbb{N}\to \mathbb{N}^2\), under which the sequence \((E_{\ell(k)})\) forms a basis in \(\mathcal{L}_{0,1}\). In particular, any successive filling of the corner squares leads to a basis, while the triangular filling not. It is shown that random permutations of the sequence \((E_{\ell(k)})\) almost surely do not form a basis. The problem of a complete characterization of all bijections \(\ell\colon \mathbb{N}\to \mathbb{N}^2\) that lead to the basis \((E_{\ell(k)})\) is set up, and it is constructively shown that the proposed necessary and sufficient conditions do not provide a complete answer to this problem.
Funding: The work of the first author was completed as a part of the implementation of the development program of the Volga Region Scientific and Educational Mathematical Center (agreement no. 075-02-2025-1791). The second author was supported by the State Research Program "Convergence-2030", Project no. 1.08.3. The third author was supported by the State Research Program "Convergence-2025", Project no. 1.3.05, and by the State Research Program "Convergence-2030", Project no. 1.08.1.
For citation: Astashkin, S. V., Bakhtin, V. I. and Lykov, K. V. Conditional Bases in the Matrix Space with the Cut-Norm, Vladikavkaz Math. J., 2026, vol. 28, no. 2, pp. 5-19. DOI 10.46698/b6298-9094-1158-n
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