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On the supercyclicity and hypercyclicity of the operator algebra. (English) Zbl 1154.47004
Let \(H\) be a separable infinite-dimensional Hilbert space. The \(*\)-strong operator topology on the space \(B(H)\) of all the bounded operators on \(H\) is defined by the family of continuous seminorms \(q_h(T):=\| Th\| +\| T^* h\| \) as \(h\) varies in \(H\). This topology is coarser than the operator norm on \(B(H)\) and finer than the strong operator topology. In this paper, the authors obtain sufficient conditions to ensure that a bounded linear map \(L\) on \(B(H)\) is supercyclic or hypercyclic for the \(*\)-strong operator topology. These results are then applied to left multiplication maps \(L_T(S):=TS\), \(S \in B(H)\). Analogous results for the strong operator topology were discovered by K. C. Chan [J. Oper. Theory 42, No. 2, 231–244 (1999; Zbl 0997.47058)], whose ideas are exploited in the paper under review.

MSC:
47A16 Cyclic vectors, hypercyclic and chaotic operators
47L10 Algebras of operators on Banach spaces and other topological linear spaces
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[1] Birkhoff, G. D.: Demonstration d’un theoreme elementaire surles functions entiéres. C. R. Acad. Sci. Paris, 189, 473–475 (1929) · JFM 55.0192.07
[2] Maclane, G. R.: Sequences of derivatives and normal families. J. Anal. Math., 2, 72–87 (1952) · Zbl 0049.05603 · doi:10.1007/BF02786968
[3] Rolewics, S.: On orbits of elements. Studia Math., 32, 17–22 (1969)
[4] Kitai, C.: Invariant closed sets for linear operators, Dissertation, Univ. of Toronto, 1982
[5] Gethner, R. M., Shapiro, J. H.: Universal vectors for operators on spaces of holomorphic functions. Proc. Amer. Math. Soc., 100, 281–288 (1987) · Zbl 0618.30031 · doi:10.1090/S0002-9939-1987-0884467-4
[6] Bourdon, P. S., Shapiro, J. H.: Cyclic phenomena for composition operators, Memoirs of the Amer. Math. Soc., 125, Amer. Math. Soc. Providence, RI, 1997 · Zbl 0996.47032
[7] Salas, H. N.: Hypercyclic weighted shifts. Trans. Amer. Math. Soc., 347, 993–1004 (1995) · Zbl 0822.47030 · doi:10.2307/2154883
[8] Bourdon, P. S., Shapiro, J. H.: Hypercyclic operators that commute with the Bergman backward shift. Trans. Amer. Math. Soc., 352, 5293–5316 (2000) · Zbl 0960.47003 · doi:10.1090/S0002-9947-00-02648-9
[9] Bourdon, P. S.: Orbits of hyponormal operators. Mich. Math. Journal, 44, 345–353 (1997) · Zbl 0896.47020 · doi:10.1307/mmj/1029005709
[10] Gross-Erdmann, K. G.: Universal families and hypercyclic operators. Bull. Amer. Math. Soc., 36, 345–381 (1999) · Zbl 0933.47003 · doi:10.1090/S0273-0979-99-00788-0
[11] Peris, A.: Hypercyclicity criteria and Mittag-Leffler theorem. Bull. Soc. Roy. Sci. Liege, 70, 365–371 (2001) · Zbl 1046.47009
[12] Feldman, N. S.: Perturbations of hypercyclic vectors. J. Math. Anal. Appl., 273, 67–74 (2002) · Zbl 1039.47008 · doi:10.1016/S0022-247X(02)00207-X
[13] Bermudez, T., Bonilla, A., Peris, A.: On hypercyclicity and supercyclicity criteria. Bull. Austral. Math. Soc., 70, 45–54 (2004) · Zbl 1071.47007 · doi:10.1017/S0004972700035802
[14] Bernal-Gonzalez, L., Grosse-Erdmann, K. G.: The Hypercyclicity Criterion for sequences of operators. Studia Math, 157, 17–32 (2003) · Zbl 1032.47006 · doi:10.4064/sm157-1-2
[15] Leon-Saavedra, F.: Notes about the hypercyclicity criterion. Math. Slovaca, 53, 313–319 (2003) · Zbl 1072.47008
[16] Yousefi, B., Rezaei, H.: Some necessary and sufficient conditions for Hypercyclicity Criterion. Proc. Indian Acad. Sci. (Math. Sci.), 115(2), 209–216 (2005) · Zbl 1084.47003 · doi:10.1007/BF02829627
[17] Bonet, J., Martinez-Gimenez, F., Peris, A.: Universal and chaotic multipliers on spaces of operators. J. Math. Anal. Appl., 297, 599–611 (2004) · Zbl 1062.47011 · doi:10.1016/j.jmaa.2004.03.073
[18] Yousefi, B., Rezaei, H.: Hypercyclicity on the algebra of Hilbert-Schmidt operators. Journal of Results in Mathematics, 46, 174–180 (2004) · Zbl 1080.47013
[19] Hilden, H. M., Wallen, L. J.: Some cyclic and non-cyclic vectors of certain operators. Indiana Univ. Math. J., 24, 557–565 (1974) · Zbl 0274.47004 · doi:10.1512/iumj.1974.23.23046
[20] Salas, H. N.: Supercyclicity and weighted shifts. Studia Mathematica, 135(1), (1999) · Zbl 0940.47005
[21] Montes-Rodriguez, A.: Banach spaces of hypercyclic vectors. Michigan Math. J., 43, 419–436 (1996) · Zbl 0907.47023 · doi:10.1307/mmj/1029005536
[22] Chan, K. C.: Hypercyclicity of the operator algebra for a separable Hilbert space. J. Operator Theory, 42, 231–244 (1999) · Zbl 0997.47058
[23] Bes, J.: Three problems on hypercyclic operators, PhD thesis, Kent State University, 1998
[24] Bes, J., Peris, A.: Hereditarily hypercyclic operators J. Func. Anal., 167(1), 94–112 (1999) · Zbl 0941.47002 · doi:10.1006/jfan.1999.3437
[25] Bonet, J.: Hypercyclic and chaotic convolution operators, J. London Math. Soc., 62(2), 253–262 (2000) · Zbl 0956.46029 · doi:10.1112/S0024610700001174
[26] Costakis, G., Sambarino, M.: Topologically mixing hypercyclic operators. Proc. Amer. Math. Soc., 132, 385–389 (2003) · Zbl 1054.47006 · doi:10.1090/S0002-9939-03-07016-3
[27] Godefroy, G., Shapiro, J. H.: Operators with dense invariant cyclic manifolds. J. Func. Anal., 98, 229–269 (1991) · Zbl 0732.47016 · doi:10.1016/0022-1236(91)90078-J
[28] Grosse-Erdmann, K. G.: Holomorphic Monster und universelle Funktionen. Mitt. Math. Sem. Giessen, 176, 1–84 (1987)
[29] Herrero, D.: Hypercyclic operators and chaos. J. Operator theory, 28, 93–103 (1992) · Zbl 0806.47020
[30] Montes-Rodriguez, A., Romero-Moreno, M. C.: Supercyclicity in the operator algebra. Studia Math., 150(3), 201–213 (2002) · Zbl 1006.47009 · doi:10.4064/sm150-3-1
[31] Montes-Rodriguez, A., Salas, H. N.: Supercyclic subspaces: spectral theory and weighted shifts. Advances in Mathematics, 163(1), 74–134 (2001) · Zbl 1008.47010 · doi:10.1006/aima.2001.2001
[32] Grosse-Erdmann, K. G., Peris, A.: Frequently dense orbits. Comptes Rendus Mathematique, 341(20), 123–128 (2005) · Zbl 1068.47012 · doi:10.1016/j.crma.2005.05.025
[33] Shkarin, S.: Non-sequential weak supercyclicity and hypercyclicity. Journal of Functional Analysis, 242(1), 37–77 (2007) · Zbl 1114.47007 · doi:10.1016/j.jfa.2006.04.021
[34] Petersson, H.: A hypercyclicity criterion with applications. Journal of Mathematical Analysis and Applications, 327(2), 1431–1443 (2007) · Zbl 1108.47009 · doi:10.1016/j.jmaa.2006.05.019
[35] Grivaux, S.: Hypercyclic operators, mixing operators and the bounded steps problem. J. Operator Theory, 54, 147–168 (2005) · Zbl 1104.47010
[36] Salas, H.: Pathological hypercyclic operators. Arch. Math., 86, 241–250 (2006) · Zbl 1097.47007 · doi:10.1007/s00013-005-1511-y
[37] Petersson, H.: Hypercyclic sequence of PDE-preserving operators. J. Approx. Theory, 138(20), 168–183 (2006) · Zbl 1087.47008 · doi:10.1016/j.jat.2005.11.004
[38] Bayart, F., Grivaux, S.: Frequently hypercyclic operators. Transactions of the American Mathematical Society, 358(11), 5083–5117 (2006) · Zbl 1115.47005 · doi:10.1090/S0002-9947-06-04019-0
[39] Yousefi, B., Rezaei, H.: Hypercyclic property of weighted composition operators. Proc. Amer. Math. Soc., 135(10), 3263–3271 (2007) · Zbl 1129.47010 · doi:10.1090/S0002-9939-07-08833-8
[40] Petersson, H.: Topologies for which every nonzero vector is hypercyclic, preprint · Zbl 1413.47021
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