On Convergence with respect to an Ideal and a Family of Matrices

P. Das et al. recently introduced and studied the notions of strong -summability with respect to an Orlicz function and -statistical convergence, where is a nonnegative regular matrix and is an ideal on the set of natural numbers. In this paper, we will generalise these notions by replacing with a family of matrices and with a family of Orlicz functions or moduli and study the thus obtained convergence methods. We will also give an application in Banach space theory, presenting a generalisation of Simons' sup-limsup-theorem to the newly introduced convergence methods (for the case that the filter generated by the ideal has a countable base), continuing some of the author's previous work. 1. Introduction Let us begin by recalling that an ideal on a nonempty set is a nonempty set of subsets of such that and is closed under the formation of subsets and finite unions. The ideal is called admissible if for each . For example, if is infinite, then the set of all finite subsets of forms an ideal on . If is an ideal, then is a filter on . Now if is a sequence in a topological space and is an ideal on the set of natural numbers, then is said to be -convergent to if for every neighbourhood of the set belongs to (equivalently, ). In a Hausdorff space the -limit is unique if it exists. It will be denoted by . If is the ideal of all finite subsets of , then -convergence is equivalent to the usual convergence. Thus if is admissible, the usual convergence implies -convergence. For a normed space the set of all -convergent sequences in is a subspace of and the map is linear. We refer the reader to [1–4] for more information on -convergence. Recall now that for a given infinite matrix with real or complex entries a sequence of (real or complex) numbers is said to be -summable to the number provided that each of the series is convergent and . The matrix is called regular if every sequence that is convergent in the ordinary sense is also -summable to the same limit. A well-known theorem of Toeplitz states that is regular if and only if the following holds:(i) ,(ii) ,(iii) . Let us suppose for the moment that is regular and also nonnegative (i.e., for all ). We will denote by the set for every . Then is said to be -statistically convergent to if for every we have , where the symbol denotes the characteristic function of the set . If one takes to be the Cesàro matrix (i.e., for and for ) one gets the usual notion of statistical convergence as it was introduced by Fast in [5]. Note that the set of all subsets for which holds is an ideal on and -statistical convergence is