✦ LIBER ✦

Existence of Hypercyclic Operators on Topological Vector Spaces

✍ Scribed by Shamim I. Ansari

Publisher: Elsevier Science
Year: 1997
Tongue: English
Weight: 261 KB
Volume: 148
Category: Article
ISSN: 0022-1236
DOI: 10.1006/jfan.1996.3093

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✦ Synopsis

The present paper introduces a very simple, but very useful notion of the so called quasi-extension of l 1 -operators and proves that a large class of topological vector spaces admit continuous hypercyclic operators. In particular, it answers in the affirmative a question of S. Rolewicz, posed in 1969, whether or not every infinite dimensional separable Banach space admits a continuous hypercyclic operator.

1997 Academic Press J. H. Shapiro [7] have dealt with hypercyclicity in topological vector spaces. In 1969 S. Rolewicz [15] asked the question if every infinite dimensional separable Banach space admits a hypercyclic operator. This question appeared also in [9], [12], and [16]. G. Herzog [12] proved that every infinite dimensional Banach space admits a supercyclic operator, but the question of S. Rolewicz regarding hypercyclic operators had remained open. K. Chan and J. H. Shapiro [5] proved that for certain types of unilateral weighted backward shifts T on Hilbert space the operator I+T is hypercyclic. G. Godefroy and J. H. Shapiro [8] asked the question article no.

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