Anote on discretely compact operators

Abstract

It is the aim of ibis note to prove an important result in the framework of discrete approximations and discrete convergente. The underlying general perturbation theory applies to sequences of linear and nonlinear operators and solutions of operator equations. The theory was originally developed by Stummel[23-31] with contributions by Grigorieff [6-10], Rannacher [18], Wolf [38,39] and the author [19-22]. Similar approaches are used by Vainikko [32-37] and, in special cases, by Anselone [1, 2], Aubin [3], Browder [4,5], Petryshyn [15- 16]. At the time the monograph [22] appeared, the theory of discrete convergence was completed. According to the emphasis of the book [22] on nonlinear mappings and applications, not every aspect of the theory is contained in [22], e.g. not the perturbation of eigenvalue problems, not the perturbation of Sobolev spaces and no results on weak convergence. There are a few newer publications known to the author which use the theory of discrete convergence for special problem settings (see e.g. Niepage [13,14]). The result of the present contribution is not yet published; it is interesting by itself and has importance with respect to applications. The main result of this paper, Theorem 3, states the equivalence of weak discrete compactness and discrete compactness of not necessarily linear operators in a special setting of subspaces.Before, it is shown that discrete compactness implies weak discrete compactness in a general setting (see Theorem 1) and boundedness properties are shown (see Theorem 2). Compactness properties are important for the existence of solutions of operator equations and their convergence for examples and applications see [1, 2,7,15,22,23,28,31,33,34,36].