Proceedings of the American Mathematical Society

Author(s): M. González; F. León-Saavedra
Journal: Proc. Amer. Math. Soc. 137 (2009), 2049-2055.
MSC (2000): Primary 47B37; Secondary 47B38, 47B99
Posted: January 29, 2009
Retrieve article in: PDF

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.

1.: Shamim I. Ansari.
Existence of hypercyclic operators on topological vector spaces.
J. Funct. Anal., 148(2):384-390, 1997. MR 1469346 (98h:47028a)
2.: Frédéric Bayart and Étienne Matheron.
Hyponormal operators, weighted shifts and weak forms of supercyclicity.
Proc. Edinb. Math. Soc. (2), 49(1):1-15, 2006. MR 2202138 (2006k:47017)
3.: Luis Bernal-González.
On hypercyclic operators on Banach spaces.
Proc. Amer. Math. Soc., 127(4):1003-1010, 1999. MR 1476119 (99f:47010)
4.: Arlen Brown, Paul R. Halmos, and Allen L. Shields.
Cesàro operators.
Acta Sci. Math. (Szeged), 26:125-137, 1965. MR 0187085 (32:4539)
5.: Manuel González.
The fine spectrum of the Cesàro operator in $l\sb p\;(1<p<\infty)$ .
Arch. Math. (Basel), 44(4):355-358, 1985. MR 788950 (86h:47036)
6.: G. H. Hardy, J. E. Littlewood, and G. Pólya.
Inequalities.
Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1988.
Reprint of the 1952 edition. MR 944909 (89d:26016)
7.: Domingo A. Herrero.
Limits of hypercyclic and supercyclic operators.
J. Funct. Anal., 99(1):179-190, 1991. MR 1120920 (92g:47026)
8.: H. M. Hilden and L. J. Wallen.
Some cyclic and non-cyclic vectors of certain operators.
Indiana Univ. Math. J., 23:557-565, 1973/74. MR 0326452 (48:4796)
9.: Fernando León-Saavedra and Vladimır Müller.
Rotations of hypercyclic and supercyclic operators.
Integral Equations Operator Theory, 50(3):385-391, 2004. MR 2104261 (2005g:47009)
10.: Fernando León-Saavedra and Antonio Piqueras-Lerena.
On weak positive supercyclicity.
Israel Journal of Math., 167:303-313, 2008. MR 2448027
11.: Fernando León-Saavedra, Antonio Piqueras-Lerena, and J.B. Seoane.
Orbits of Cesàro type operators.
Math. Nachr. (to appear).
12.: Héctor N. Salas.
Hypercyclic weighted shifts.
Trans. Amer. Math. Soc., 347(3):993-1004, 1995. MR 1249890 (95e:47042)
13.: Héctor N. Salas.
Supercyclicity and weighted shifts.
Studia Math., 135(1):55-74, 1999. MR 1686371 (2000b:47020)
14.: Rebecca Sanders.
Weakly supercyclic operators.
J. Math. Anal. Appl., 292(1):148-159, 2004. MR 2050222 (2005a:47012)
15.: Allen L. Shields.
Weighted shift operators and analytic function theory.
In Topics in operator theory, pages 49-128. Math. Surveys, No. 13. Amer. Math. Soc., Providence, RI, 1974. MR 0361899 (50:14341)
16.: Stanislav Shkarin.
Non-sequential weak supercyclicity and hypercyclicity.
J. Funct. Anal., 242(1):37-77, 2007. MR 2274015 (2007i:47010)

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47B37, 47B38, 47B99

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: 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
Posted: 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
Copyright of article: Copyright 2009, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS