Mathematische Zeitschrift

, Volume 249, Issue 1, pp 85–96

Topologically transitive extensions of bounded operators

Article

DOI: 10.1007/s00209-004-0690-8

Cite this article as:
Grivaux, S. Math. Z. (2005) 249: 85. doi:10.1007/s00209-004-0690-8

Abstract.

Let X be any Banach space and T a bounded operator on X. An extensionhttps://static-content.springer.com/image/art%3A10.1007%2Fs00209-004-0690-8/MediaObjects/s00209-004-0690-8flb1.gif of the pair (X,T) consists of a Banach space https://static-content.springer.com/image/art%3A10.1007%2Fs00209-004-0690-8/MediaObjects/s00209-004-0690-8flb2.gif in which X embeds isometrically through an isometry i and a bounded operatorhttps://static-content.springer.com/image/art%3A10.1007%2Fs00209-004-0690-8/MediaObjects/s00209-004-0690-8flb3.gif on https://static-content.springer.com/image/art%3A10.1007%2Fs00209-004-0690-8/MediaObjects/s00209-004-0690-8flb2.gif such that https://static-content.springer.com/image/art%3A10.1007%2Fs00209-004-0690-8/MediaObjects/s00209-004-0690-8flb4.gif When X is separable, it is additionally required that https://static-content.springer.com/image/art%3A10.1007%2Fs00209-004-0690-8/MediaObjects/s00209-004-0690-8flb2.gif be separable. We say that https://static-content.springer.com/image/art%3A10.1007%2Fs00209-004-0690-8/MediaObjects/s00209-004-0690-8flb1.gif is a topologically transitive extension of (X, T) when https://static-content.springer.com/image/art%3A10.1007%2Fs00209-004-0690-8/MediaObjects/s00209-004-0690-8flb3.gif is topologically transitive on https://static-content.springer.com/image/art%3A10.1007%2Fs00209-004-0690-8/MediaObjects/s00209-004-0690-8flb2.gif, i.e. for every pair https://static-content.springer.com/image/art%3A10.1007%2Fs00209-004-0690-8/MediaObjects/s00209-004-0690-8flb5.gif of non-empty open subsets of https://static-content.springer.com/image/art%3A10.1007%2Fs00209-004-0690-8/MediaObjects/s00209-004-0690-8flb2.gif there exists an integer n such that https://static-content.springer.com/image/art%3A10.1007%2Fs00209-004-0690-8/MediaObjects/s00209-004-0690-8flb6.gif is non-empty. We show that any such pair (X,T) admits a topologically transitive extension https://static-content.springer.com/image/art%3A10.1007%2Fs00209-004-0690-8/MediaObjects/s00209-004-0690-8flb1.gif, and that when H is a Hilbert space, (H,T) admits a topologically transitive extension https://static-content.springer.com/image/art%3A10.1007%2Fs00209-004-0690-8/MediaObjects/s00209-004-0690-8flb7.gif where https://static-content.springer.com/image/art%3A10.1007%2Fs00209-004-0690-8/MediaObjects/s00209-004-0690-8flb8.gif is also a Hilbert space. We show that these extensions are indeed chaotic.

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Équipe d’AnalyseUniversité Paris 6, Case 186Paris Cedex 05France