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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Cyclic behavior of the Cesàro operator on $ L_2(0,\infty)$


Authors: M. González and F. León-Saavedra
Journal: Proc. Amer. Math. Soc. 137 (2009), 2049-2055
MSC (2000): Primary 47B37; Secondary 47B38, 47B99
Published electronically: January 29, 2009
MathSciNet review: 2480286
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Abstract: In this paper we study the cyclic properties of the infinite continuous Cesàro operator defined on $ L^2(0,\infty)$ by $ (C_\infty f)(x)=\frac{1}{x}\int_0^x f(s) ds $. Despite this operator being cyclic, we show that it is not supercyclic; even more, it is not weakly supercyclic. These results complement some recent ones on the cyclic behavior of Cesàro operators.


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Additional Information

M. González
Affiliation: Department of Mathematics, University of Cantabria, Facultad de Ciencias, Avda. de los Castros s/n, E-39071-Santander, Spain
Email: gonzalem@unican.es

F. León-Saavedra
Affiliation: Department of Mathematics, University of Cádiz, Avda. de la Universidad s/n, E-11405-Jerez de la Frontera, Spain
Email: fernando.leon@uca.es

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09833-5
PII: S 0002-9939(09)09833-5
Keywords: Ces\`aro operators, positive supercyclicity, bilateral shifts, commutants
Received by editor(s): July 21, 2008
Published electronically: January 29, 2009
Additional Notes: The first author was partially supported by Plan Nacional I+D, Grant MTM-2007-67994
The second author was partially supported by Plan Nacional I+D, Junta de Andalucía FQM-257, and a Grant of Ministerio de Educación y Ciencia.
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2009 American Mathematical Society