The Backward Shift and Two Infinite-Dimension Phenomena in Banach Spaces

Abstract

We consider the backward shift operator on a sequence Banach space in the context of two infinite-dimensional phenomena: the existence of topologically transitive operators, and the existence of entire analytic functions of the unbounded type. It is well known that the weighted backward shift (for an appropriated weight) is topologically transitive on 1≤p<∞ and on c0. We construct some generalizations of the weighted backward shift for non-separable Banach spaces, which remains topologically transitive. Also, we show that the backward shift, in some sense, generates analytic functions of the unbounded type. We introduce the notion of a generator of analytic functions of the unbounded type on a Banach space and investigate its properties. In addition, we show that, using this operator, one can obtain a quasi-extension operator of analytic functions in a germ of zero for entire analytic functions. The results are supported by examples.