Quasi-reflexive Fréchet spaces and contractively power bounded operators

Abstract

Abstract. In this paper we introduce a new class of operators acting on a locally convex space. We show that for some Fréchet spaces all these operators are mean ergodic. This leads to the conclusion that the classes of reflexive and non-reflexive Fréchet spaces are, in a sense, close to each other. Mathematics Subject Classification (2010). Primary 46A04, 47A35.Keywords. Fréchet space, Quasi-reflexive space, Mean ergodic operator, Power bounded operator. Introduction. In [3] the following definition is introduced: a Banach spaceThis definition arose from the very well known paper of R. C. James who constructed in [9] an example of a non-reflexive Banach space X of codimension 1 in its bidual. Properties of such spaces have been studied in [4,8,10,17,18]. One of the outstanding profits of these spaces is that they have served as examples to settle several conjectures. Recently Fonf, Lin, and Wojtaszczyk have constructed in [7] a quasi-reflexive Banach space of order 1 with all contractions on the space itself and on its dual mean ergodic. Thus answering negatively a question of Sucheston posed in [19].The aim of this paper is to consider (after introducing a reasonable notation) an analogous problem in the Fréchet space setting. Here the notion of quasi-reflexivity is also available since, whenever we start with an arbitrary Fréchet space X, by [15, 25.3 and 25.10] all the spaces π(X), X and X /π(X) are Fréchet. Examples of such spaces without infinite-dimensional Banach subspaces may be found in [14]. Let us recall them for the convenience of the