Operators with hypercyclic Cesaro means

• Autorzy: Fernando León-Saavedra
• Języki publikacji: EN

Abstrakty

EN

An operator T on a Banach space ℬ is said to be hypercyclic if there exists a vector x such that the orbit ${Tⁿx}_{n≥1}$ is dense in ℬ. Hypercyclicity is a strong kind of cyclicity which requires that the linear span of the orbit is dense in ℬ. If the arithmetic means of the orbit of x are dense in ℬ then the operator T is said to be Cesàro-hypercyclic. Apparently Cesàro-hypercyclicity is a strong version of hypercyclicity. We prove that an operator is Cesàro-hypercyclic if and only if there exists a vector x ∈ ℬ such that the orbit ${n^{-1}Tⁿx}_{n≥1}$ is dense in ℬ. This allows us to characterize the unilateral and bilateral weighted shifts whose arithmetic means are hypercyclic. As a consequence we show that there are hypercyclic operators which are not Cesàro-hypercyclic, and more surprisingly, there are non-hypercyclic operators for which the Cesàro means of some orbit are dense. However, we show that both classes, the class of hypercyclic operators and the class of Cesàro-hypercyclic operators, have the same norm-closure spectral characterization.

Kategorie tematyczne

• 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
• 47B38: Operators on function spaces (general)
• 47B99: None of the above, but in this section

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2002
• Tom: 152
• Numer: 3
• Strony: 201-215

Daty

wydano

2002

Twórcy

autor

Fernando León-Saavedra

• Departamento de Matemáticas, Escuela Superior de Ingeniería, Universidad de Cádiz, C/Sacramento 82, 11003 Cádiz, Spain

Identyfikatory

DOI

10.4064/sm152-3-1