Statistical approximation with a sequence of 2-Banach spaces

Let (X) over bar = {(X) over bar (n) is an element of N} be a sequence of 2-Banach spaces, where X denotes an arbitrary 2-Banach space, and let {T-n} be a sequence of linear operators T-n : X -> (X) over bar. First, a relationship is constructed between Xn and X by means of Tn and next the notion of statistical T-convergence is introduced. Hence, a statistical approximation theory is constructed between the elements of (X) over bar (n) and X. Then, the properties of this statistical approximation theory are examined. On the other hand, the necessary and sufficient conditions for statistical T-convergence of the elements of (X) over bar (n) to an element of X are investigated, and the methods of the determination of the statistical convergence velocity are examined. Finally, we define the statistical approximation and statistical stability conditions of linear operators, and give an application of our results. (C) 2011 Elsevier Ltd. All rights reserved.