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Some necessary and sufficient conditions for hypercyclicity criterion. (English) Zbl 1084.47003
The authors give necessary and sufficient conditions for a linear continuous operator \(T\) on an infinite-dimensional separable Hilbert space \(H\) to satisfy the hypercyclicity criterion, which states that \(T: H\to H\) is hypercyclic if there exist dense subsets \(Y\), \(Z\) of \(H\), a sequence \((n_k)\), and functions \(S_{n_k}: Z\to X\) such that \(T^{n_k}y\to 0\) \((y\in Y)\), \(S_{n_k}z\to 0\) \((z\in Z)\), and \(T^{n_k}S_{n_k}z\to z\) \((z\in Z)\). Stimulated by the open question whether each hypercyclic operator in \(B(H)\) satisfies this criterion, the authors prove that \(T\in B(H)\) satisfies the hypercyclicity criterion if and only if for each pair \(U\), \(V\) of nonempty open subsets of \(H\) and each open neighborhood \(W\) of zero, \(T^n U\cap W\neq\emptyset\) and \(T^n W\cap V\neq\emptyset\) for some integer \(n\). Based on this result, further characterizations are given.

47A16 Cyclic vectors, hypercyclic and chaotic operators
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[1] Aron R and Bes J, Hypercyclic differentiation operators, Function Spaces, Contemporary Mathematics,Am. Math. Soc. (Providence, RI) (1999) vol. 232, pp. 39–46 · Zbl 0938.47004
[2] Bes J and Peris A, Hereditarily hypercyclic operators,J. Func. Anal. 167(1) (1999) 94–112. · Zbl 0941.47002 · doi:10.1006/jfan.1999.3437
[3] Birkhoff G, Demonstration dun theoreme sur les fonctions entieres,C. R. Acad, Sci. Paris 189 (1929) 473–475 · JFM 55.0192.07
[4] Bonet J and Peris A, Hypercyclic operators on non-normable Frechet spaces,J. Funct. Anal. 159 (1998) 587–595 · Zbl 0926.47011 · doi:10.1006/jfan.1998.3315
[5] Bourdon P S, Orbits of hyponormal operators,Mich. Math. J. 44 (1997) 345–353 · Zbl 0896.47020 · doi:10.1307/mmj/1029005709
[6] Bourdon P S and Shapiro J H, Cyclic phenomena for composition operators, Memoirs of the Amer. Math. Soc.,Am. Math. Soc. (Providence, RI) (1997) vol. 125 · Zbl 0996.47032
[7] Bourdon P S and Shapiro J H, Hypercyclic operators that commute with the Bergman backward shift,Trans. Am. Math. Soc. 352(11) (2000) 5293–5316 · Zbl 0960.47003 · doi:10.1090/S0002-9947-00-02648-9
[8] Chan K C and Shapiro J H, The cyclic behaviour of translation operators on Hilbert spaces of entire functions,Indiana Univ. Math. J. 40 (1991) 1421–1449 · Zbl 0771.47015 · doi:10.1512/iumj.1991.40.40064
[9] Chan K C, Hypercyclicity of the operator algebra for a separable Hilbert space,J. Operator Theory 42 (1999) 231–244 · Zbl 0997.47058
[10] Conway J B, The theory of subnormal operators, Mathematical Surveys and Monographs, American Mathematical Society, 1991
[11] Curto R, Spectral theory of elementary operators, in: Elementary operators and Applications (ed.) Martin Mathiea (World Scientific) (1992)
[12] Gethner R M and Shapiro J H, Universal vectors for operators on spaces of holomorphic functions,Proc. Am. Math. Soc. 100 (1987) 281–288 · Zbl 0618.30031 · doi:10.1090/S0002-9939-1987-0884467-4
[13] Godefroy G and Shapiro J H, Operators with dense invariant cyclic manifolds,J. Func. Anal. 98 (1991) 229–269 · Zbl 0732.47016 · doi:10.1016/0022-1236(91)90078-J
[14] Grosse-Erdmann K-G, Universal families and hypercylic operators,Bull. Am. Math. Soc. 36 (1999) 345–381 · Zbl 0933.47003 · doi:10.1090/S0273-0979-99-00788-0
[15] Herrero D A, Limits of hypercyclic and supercyclic operators,J. Func. Anal. 99 (1991) 179–190 · Zbl 0758.47016 · doi:10.1016/0022-1236(91)90058-D
[16] Herzog G and Schomoeger C, On operatorsT such thatf(T) is hypercyclic,Studia Math. 108 (1994) 209–216
[17] Kitai C, Invariant closed sets for linear operators (Dissertation, Univ. of Toronto) (1982)
[18] Leon-Saavedra F and Montes-Rodriguez A, Linear structure of hypercyclic vectors,J. Funct. Anal. 148 (1997) 524–545 · Zbl 0999.47009 · doi:10.1006/jfan.1996.3084
[19] MacLane G R, Sequences of derivatives and normal families,J. D. Analyse Math. 2 (1952) 72–87 · Zbl 0049.05603 · doi:10.1007/BF02786968
[20] Rolewicz S, On orbits of elements,Studia Math. 32 (1969) 17–22 · Zbl 0174.44203
[21] Salas H N, Hypercyclic weighted shifts,Trans. Am. Math. Soc. 347 (1995) 993–1004 · Zbl 0822.47030 · doi:10.2307/2154883
[22] Salas H, A hypercyclic operator whose adjoint is also hypercyclic,Proc. Am. Math. Soc. 112 (1991) 765–770 · Zbl 0748.47023 · doi:10.1090/S0002-9939-1991-1049848-8
[23] K Zhu, Operator theory in function spaces (New York: Marcel Dekker, Inc.) (1990) · Zbl 0706.47019
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