A characterization of $(\mathcal{I}, \mathcal{J})$-regular matrices

Abstract

Let I, J be two ideals on N which contain the family Fin of finite sets. We provide necessary and sufficient conditions on the entries of an infinite real matrix A = (a n,k ) which maps I-convergent bounded sequences into J -convergent bounded sequences and preserves the corresponding ideal limits. The well-known characterization of regular matrices due to Silverman-Toeplitz corresponds to the case I = J = Fin.Lastly, we provide some applications to permutation and diagonal matrices, which extend several known results in the literature.Theorem 1.1. A matrix A is regular if and only if :The aim of this work is to generalize Theorem 1.1 in the context of ideal convergence.Recall that an ideal I ⊆ P(N) is a family of subsets of N closed under taking finite unions and subsets. Unless otherwise stated, it is also assumed that I is admissible, i.e., it contains the family of finite sets Fin and I = P(N). Let I ⋆ = {A ⊆ N : A c ∈ I} be its dual filter. An important example of ideal is the family of asymptotic density zero sets, that is,Accordingly, Z-convergence is usually termed statistical convergence. We let c(I) be the vector space I-convergent sequences and c 0 (I) be its subspace of sequences with I-limit 0. Structural properties of the set of bounded I-convergent sequences c(I) ∩ ℓ ∞ and its subspace c 0 (I) ∩ ℓ ∞ have been recently studied in the literature, sometimes providing answers to longstanding questions, see e.g. [5,14,18,22].