Hypercyclic weighted shifts

Salas, Héctor N.

doi:10.1090/S0002-9947-1995-1249890-6

Hypercyclic weighted shifts
HTML articles powered by AMS MathViewer

by Héctor N. Salas PDF

Trans. Amer. Math. Soc. 347 (1995), 993-1004 Request permission

Abstract:

An operator $T$ acting on a Hilbert space is hypercyclic if, for some vector $x$ in the space, the orbit $\{ {T^n}x:n \geqslant 0\}$ 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 multi-hypercyclic 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.

References

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. MR 735080
Paul S. Bourdon and Joel H. Shapiro, Cyclic composition operators on $H^2$, 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. MR 1077418
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. MR 1142722, DOI 10.1512/iumj.1991.40.40064

Universal functions of the structure of the space of entire functions

30

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
Gilles Godefroy and Joel H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), no. 2, 229–269. MR 1111569, DOI 10.1016/0022-1236(91)90078-J
Israel Halperin, Carol Kitai, and Peter Rosenthal, On orbits of linear operators, J. London Math. Soc. (2) 31 (1985), no. 3, 561–565. MR 812786, DOI 10.1112/jlms/s2-31.3.561
Domingo A. Herrero, Possible structures for the set of cyclic vectors, Indiana Univ. Math. J. 28 (1979), no. 6, 913–926. MR 551155, DOI 10.1512/iumj.1979.28.28064

Approximation of Hilbert space operators

Domingo A. Herrero, Limits of hypercyclic and supercyclic operators, J. Funct. Anal. 99 (1991), no. 1, 179–190. MR 1120920, DOI 10.1016/0022-1236(91)90058-D
Domingo A. Herrero, Hypercyclic operators and chaos, J. Operator Theory 28 (1992), no. 1, 93–103. MR 1259918
Domingo A. Herrero and Zong Yao Wang, Compact perturbations of hypercyclic and supercyclic operators, Indiana Univ. Math. J. 39 (1990), no. 3, 819–829. MR 1078739, DOI 10.1512/iumj.1990.39.39039

Invariant closed sets for linear operators

S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17–22. MR 241956, DOI 10.4064/sm-32-1-17-22
Héctor Salas, A hypercyclic operator whose adjoint is also hypercyclic, Proc. Amer. Math. Soc. 112 (1991), no. 3, 765–770. MR 1049848, DOI 10.1090/S0002-9939-1991-1049848-8
Allen L. Shields, Weighted shift operators and analytic function theory, Topics in operator theory, Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. MR 0361899