## 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
S. M. Duyos-Ruis, - 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
—, - 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
C. Kitai, - 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**

*Universal functions of the structure of the space of entire functions*, Soviet Math. Dokl.

**30**(1984), no. 3, 713-716.

*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.

*Invariant closed sets for linear operators*, Thesis, Univ. of Toronto, 1982.

## Additional Information

- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**347**(1995), 993-1004 - MSC: Primary 47B37; Secondary 47A99
- DOI: https://doi.org/10.1090/S0002-9947-1995-1249890-6
- MathSciNet review: 1249890