Topologically transitive extensions of bounded operators
- First Online:
- 92 Downloads
Let X be any Banach space and T a bounded operator on X. An extensionOpen image in new window of the pair (X,T) consists of a Banach space Open image in new window in which X embeds isometrically through an isometry i and a bounded operatorOpen image in new window on Open image in new window such that Open image in new window When X is separable, it is additionally required that Open image in new window be separable. We say that Open image in new window is a topologically transitive extension of (X, T) when Open image in new window is topologically transitive on Open image in new window, i.e. for every pair Open image in new window of non-empty open subsets of Open image in new window there exists an integer n such that Open image in new window is non-empty. We show that any such pair (X,T) admits a topologically transitive extension Open image in new window, and that when H is a Hilbert space, (H,T) admits a topologically transitive extension Open image in new window where Open image in new window is also a Hilbert space. We show that these extensions are indeed chaotic.
Unable to display preview. Download preview PDF.
- 9.Devaney, R.L.: An Introduction to Chaotic Dynamical Systems. 2nd ed. Reading, MA: Addison-Wesley, 1989Google Scholar
- 10.Feldman, N.: Linear Chaos. Preprint, http://home.wlu.edu/~feldmann/research.html.
- 13.Godefroy, G., Kalton, N.J.: Lipschitz-free Banach spaces. Studia Math. 159, 121–141 (2003)Google Scholar
- 21.Kitai, C.: Invariant closed sets for linear operators. Ph. D. thesis. Univ. of Toronto, 1982Google Scholar
- 24.Shapiro, J.H.: Notes on the dynamics of linear operators. Unpublished notes, http://www.math.msu.edu/~shapiro/Pubvit/LecNotes.html.