Topologically mixing hypercyclic operators
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- by George Costakis and Martín Sambarino PDF
- Proc. Amer. Math. Soc. 132 (2004), 385-389 Request permission
Abstract:
Let $X$ be a separable Fréchet space. We prove that a linear operator $T:X\to X$ satisfying a special case of the Hypercyclicity Criterion is topologically mixing, i.e. for any given open sets $U,V$ there exists a positive integer $N$ such that $T^n(U)\cap V\neq \emptyset$ for any $n\ge N.$ We also characterize those weighted backward shift operators that are topologically mixing.References
- Shamim I. Ansari, Existence of hypercyclic operators on topological vector spaces, J. Funct. Anal. 148 (1997), no. 2, 384–390. MR 1469346, DOI 10.1006/jfan.1996.3093
- Luis Bernal-González, On hypercyclic operators on Banach spaces, Proc. Amer. Math. Soc. 127 (1999), no. 4, 1003–1010. MR 1476119, DOI 10.1090/S0002-9939-99-04657-2
- Juan Bès and Alfredo Peris, Hereditarily hypercyclic operators, J. Funct. Anal. 167 (1999), no. 1, 94–112. MR 1710637, DOI 10.1006/jfan.1999.3437
- José Bonet and Alfredo Peris, Hypercyclic operators on non-normable Fréchet spaces, J. Funct. Anal. 159 (1998), no. 2, 587–595. MR 1658096, DOI 10.1006/jfan.1998.3315
- Karl-Goswin Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 3, 345–381. MR 1685272, DOI 10.1090/S0273-0979-99-00788-0
- Robert M. Gethner and Joel H. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc. 100 (1987), no. 2, 281–288. MR 884467, DOI 10.1090/S0002-9939-1987-0884467-4
- Kitai, Carol, “Invariant Closed Sets for Linear Operators”, Ph.D. thesis, Univ. of Toronto, 1982.
- Shamim I. Ansari, Existence of hypercyclic operators on topological vector spaces, J. Funct. Anal. 148 (1997), no. 2, 384–390. MR 1469346, DOI 10.1006/jfan.1996.3093
- Héctor N. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc. 347 (1995), no. 3, 993–1004. MR 1249890, DOI 10.1090/S0002-9947-1995-1249890-6
Additional Information
- George Costakis
- Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
- Address at time of publication: Vitinis 25 N. Philadelphia, Athens, Greece
- Email: geokos@math.umd.edu
- Martín Sambarino
- Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
- Address at time of publication: IMERL, Fac. Ingenieria, University de la República, CC30 Montevideo, Uruguay
- Email: samba@fing.edu.uy
- Received by editor(s): May 13, 2002
- Received by editor(s) in revised form: September 18, 2002
- Published electronically: June 10, 2003
- Communicated by: Joseph A. Ball
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 385-389
- MSC (2000): Primary 47A16, 47B37; Secondary 37B05
- DOI: https://doi.org/10.1090/S0002-9939-03-07016-3
- MathSciNet review: 2022360