zbMATH — the first resource for mathematics

A Banach space which admits no chaotic operator. (English) Zbl 1046.47008
An operator \(T\) on a Banach space \(X\) is called hypercyclic if there is a vector \(x\) in \(X\) such that its orbit \({\mathcal O}(x,T):= \{x, Tx,T^2x,\dots\}\) is dense in \(X\). In [“An introduction to chaotic dynamical systems” (Addison-Wesley, Redwood City) (1989; Zbl 0695.58002)], R. L. Devaney defined analogously that a continuous map \(f\) on a metric space \(E\) is chaotic if it is topologically transive and if the set of periodic points is dense.
It was proved by F. Martinez-Giménez and A. Peris [Int. J. Bifurcation Chaos Appl. Sci. Eng. 12, 1703–1715 (2002; Zbl 1070.47024)] that no \(T\in{\mathcal L}(E)\) of the form \(\lambda I+ K\), where \(K\) is a compact operator, is chaotic on a locally convex space \(E\). Since all the hypercyclic operators constructed on a separable Banach space are of this type, it is a natural question whether every infinite-dimensional separable Banach space admits a chaotic operator. In this paper, the authors show that if \(X\) is a separable complex Banach space with hereditarily indecomposable dual, then \(X\) carries no chaotic linear operator.
The existence of such spaces is a consequence of the work of W. T. Gowers and B. Maurey [J. Am. Math. Soc. 6, 851–874 (1993; Zbl 0827.46008)].

47A16 Cyclic vectors, hypercyclic and chaotic operators
46B20 Geometry and structure of normed linear spaces
Full Text: DOI