Hereditarily Hypercyclic Operators

✍ Scribed by Juan Bès; Alfredo Peris

• Book ID: 102591531
• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 151 KB
• Volume: 167
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1999.3437

✦ Synopsis

We show that a continuous linear operator T on a Fre chet space satisfies the so-called Hypercyclicity Criterion if and only if it is hereditarily hypercyclic, and if and only if the direct sum TÄ T is hypercyclic. In particular, hypercyclic operators with either a dense generalized kernel or a dense set of periodic points (i.e., chaotic in the sense of R. L. Devaney (1989, ``An Introduction to Chaotic Dynamical Systems,'' Addison Wesley, Reading, MA)) must satisfy the Criterion. Finally, we provide a characterization of those weighted shifts T that are hereditarily hypercyclic with respect to a given sequence (n k ) of positive integers, as well as conditions under which T and [T n k ] k 1 share the same set of hypercyclic vectors.