×

zbMATH — the first resource for mathematics

Hypercyclicity on the algebra of Hilbert–Schmidt operators. (English) Zbl 1080.47013
Authors’ abstract: We prove that the hypercyclicity criterion for any operator \(T\) on a Hilbert space is equivalent to the hypercyclicity of the left multiplication operator induced by \(T\) on the algebra of Hilbert–Schmidt operators.

MSC:
47A16 Cyclic vectors, hypercyclic and chaotic operators
47L10 Algebras of operators on Banach spaces and other topological linear spaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] S. Rolewicz, On orbits of elements, Studia Math, 32 (1969), 17–22. · Zbl 0174.44203
[2] C. Kitai, Invariant closed sets for linear operators, Dissertation, Univ. of Toronto, 1982.
[3] R. M. Gethner and J. H. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc, 100 (1987), 281–288. · Zbl 0618.30031 · doi:10.1090/S0002-9939-1987-0884467-4
[4] P. S. Bourdon and J. H. Shapiro, Cyclic phenomena for composition operators, Memoirs of the Amer. Math. Soc. 125, Amer. Math. Soc. Providence, RI, 1997. · Zbl 0996.47032
[5] H. N. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc, 347 (1995), 993–1004. · Zbl 0822.47030 · doi:10.1090/S0002-9947-1995-1249890-6
[6] P. S. Bourdon and J. H. Shapiro, Hypercyclic operators that commute with the Bergman backward shift, Trans. Amer. Math. Soc, 352 (2000) no.11, 5293–5316. · Zbl 0960.47003 · doi:10.1090/S0002-9947-00-02648-9
[7] P. S. Bourdon, Orbits of hyponormal operators, Mich. Math. Journal, 44 (1997), 345–353. · Zbl 0896.47020 · doi:10.1307/mmj/1029005709
[8] G. Godefroy and J. H. Shapiro, Operators with dense invariant cyclic manifolds, J. Func. Anal, 98 (1991), 229–269. · Zbl 0732.47016 · doi:10.1016/0022-1236(91)90078-J
[9] J. Bes, Three problems on hypercyclic operators, PhD thesis, Kent State University 1998.
[10] D. Herrero, Hypercyclic operators and chaos, J. Operator Theory, 28 (1992), 93–103. · Zbl 0806.47020
[11] J. Bes, and A. Peris, Hereditarily hypercyclic operators, J. Func. Anal, no.1, 167 (1999), 94–112. · Zbl 0941.47002 · doi:10.1006/jfan.1999.3437
[12] K. C. Chan, Hypercyclicity of the operator algebra for a separable Hilbert space, J. Operator Theory, 42 (1999), 231–244. · Zbl 0997.47058
[13] K. C. Chan and R. D. Taylor, Hypercyclic subspaces of a Banach space, Integral Equations Operator Theory, 41 (2001), 381–388. · Zbl 0995.46014 · doi:10.1007/BF01202099
[14] F. Martinez-Gimenez and A. Peris, Universality and chaos for tensor products of operators, J. Approx. Theory, 124 (2003), 7–24. · Zbl 1062.47014 · doi:10.1016/S0021-9045(03)00118-7
[15] J. Bonet, F. Martinez-Gimenez and A. Peris, Universal and chaotic multipliers on spaces of opera tors, J. Math. Anal. Appl (to appear).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.