A characterization of (I,J)-regular matrices

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.