Linear Structure of Hypercyclic Vectors

✍ Scribed by Fernando León-Saavedra; Alfonso Montes-Rodrı́guez

• Book ID: 102589437
• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 389 KB
• Volume: 148
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1996.3084

✦ Synopsis

A vector x in a Banach space B is called hypercyclic for a bounded linear operator T : B Ä B if the orbit [T n x: n 1] is dense in B. Our main result states that if T is a compact perturbation of an operator of norm 1 and satisfies an appropiate extra hypothesis, then there is an infinite-dimensional closed subspace consisting, except for zero, entirely of hypercyclic vectors for T. In particular the result applies to compact perturbations of the identity. We also include applications to some weighted backward shifts and compact perturbations of the identity by weighted backward shifts. This last result in combination with a recent one that states that every Banach space admits an operator with a hypercyclic vector proves that in all Banach space there is an operator T with an infinite-dimensional closed subspace consisting, except for zero, of hypercyclic vectors. The main result also applies to the differentiation operator and the translation operator T : f (z) Ä f (z+1) on certain Hilbert spaces consisting of entire functions.