Hypercyclic and chaotic generalized convolution operators generated by Gel'fond-Leont'ev operators
by
Vitaly Kim
Institute of Mathematics of the Russian Academy of Science, Ufa

In this communication we introduce a new class of hypercyclic operators on a space of all entire functions. More precisely, we show that every non-trivial generalized convolution operator, generated by Gel'fond-Leont'ev operators [1], is hypercyclic. This class of operators includes, as a partial case, classical convolution operators, proved to be hypercyclic in [2], and Dunkl convolution operators, proved to be hypercyclic in [3]. We also show that these operators are chaotic.

1. A. O. Gel'fond, A. F. Leont'ev. "On a generalization of Fourier series" (Russian) // Mat. Sbornik. 1951. V. 29. P. 477--500.

2. G. Godefroy, J. H. Shapiro. "Operators with dense, invariant, cyclic vector manifolds" // J. Funct. Anal. 1991. V. 98. P. 229-269.

3. J. J. Betancor, M. Sifi, K. Trimeche. "Hypercyclic and chaotic convolution operators associated with the Dunkl operators on C" // Acta Math. Hungar. 2005. V. 106. P. 101-116.

Date received: February 15, 2008

Copyright © 2008 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cawh-05.