×

zbMATH — the first resource for mathematics

Hypercyclic weighted shifts. (English) Zbl 0822.47030
Summary: An operator \(T\) acting on a Hilbert space is hypercyclic if, for some vector \(x\) in the space, the orbit \(\{T^ n x: n\geq 0\}\) is dense. In this paper we characterize hypercyclic weighted shifts in terms of their weight sequences and identify the direct sums of hypercyclic weighted shifts which are also hypercyclic.
As a consequence, we show within the class of weighted shifts that multihypercyclic shifts and direct sums of fixed hypercyclic shifts are both hypercyclic. For general hypercyclic operators the corresponding questions were posed by D. A. Herrero, and they still remain open. Using a different technique we prove that \(I+ T\) is hypercyclic whenever \(T\) is a unilateral backward weighted shift, thus answering in more generality a question recently posed by K. C. Chan and J. H. Shapiro.

MSC:
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47B99 Special classes of linear operators
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Constantin Apostol, Lawrence A. Fialkow, Domingo A. Herrero, and Dan Voiculescu, Approximation of Hilbert space operators. Vol. II, Research Notes in Mathematics, vol. 102, Pitman (Advanced Publishing Program), Boston, MA, 1984. · Zbl 0572.47001
[2] Paul S. Bourdon and Joel H. Shapiro, Cyclic composition operators on \?², Operator theory: operator algebras and applications, Part 2 (Durham, NH, 1988) Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp. 43 – 53.
[3] Kit C. Chan and Joel H. Shapiro, The cyclic behavior of translation operators on Hilbert spaces of entire functions, Indiana Univ. Math. J. 40 (1991), no. 4, 1421 – 1449. · Zbl 0771.47015 · doi:10.1512/iumj.1991.40.40064 · doi.org
[4] S. M. Duyos-Ruis, Universal functions of the structure of the space of entire functions, Soviet Math. Dokl. 30 (1984), no. 3, 713-716. · Zbl 0599.30059
[5] 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. · Zbl 0618.30031
[6] Gilles Godefroy and Joel H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), no. 2, 229 – 269. · Zbl 0732.47016 · doi:10.1016/0022-1236(91)90078-J · doi.org
[7] Israel Halperin, Carol Kitai, and Peter Rosenthal, On orbits of linear operators, J. London Math. Soc. (2) 31 (1985), no. 3, 561 – 565. · Zbl 0578.47001 · doi:10.1112/jlms/s2-31.3.561 · doi.org
[8] Domingo A. Herrero, Possible structures for the set of cyclic vectors, Indiana Univ. Math. J. 28 (1979), no. 6, 913 – 926. · Zbl 0447.47001 · doi:10.1512/iumj.1979.28.28064 · doi.org
[9] -, Approximation of Hilbert space operators, Vol. I, 2nd ed., Pitman Research Notes in Math. Ser., vol. 224, Longman Sci. Tech., Harlow and Wiley, New York, 1989.
[10] Domingo A. Herrero, Limits of hypercyclic and supercyclic operators, J. Funct. Anal. 99 (1991), no. 1, 179 – 190. · Zbl 0758.47016 · doi:10.1016/0022-1236(91)90058-D · doi.org
[11] Domingo A. Herrero, Hypercyclic operators and chaos, J. Operator Theory 28 (1992), no. 1, 93 – 103. · Zbl 0806.47020
[12] Domingo A. Herrero and Zong Yao Wang, Compact perturbations of hypercyclic and supercyclic operators, Indiana Univ. Math. J. 39 (1990), no. 3, 819 – 829. · Zbl 0724.47009 · doi:10.1512/iumj.1990.39.39039 · doi.org
[13] C. Kitai, Invariant closed sets for linear operators, Thesis, Univ. of Toronto, 1982.
[14] S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17 – 22. · Zbl 0174.44203
[15] Héctor Salas, A hypercyclic operator whose adjoint is also hypercyclic, Proc. Amer. Math. Soc. 112 (1991), no. 3, 765 – 770. · Zbl 0748.47023
[16] Allen L. Shields, Weighted shift operators and analytic function theory, Topics in operator theory, Amer. Math. Soc., Providence, R.I., 1974, pp. 49 – 128. Math. Surveys, No. 13. · Zbl 0303.47021
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.