Hypercyclic weighted shifts

Author:
Héctor N. Salas

Journal:
Trans. Amer. Math. Soc. **347** (1995), 993-1004

MSC:
Primary 47B37; Secondary 47A99

MathSciNet review:
1249890

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An operator acting on a Hilbert space is *hypercyclic* if, for some vector in the space, the orbit 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 is hypercyclic whenever is a unilateral backward weighted shift, thus answering in more generality a question recently posed by K. C. Chan and J. H. Shapiro.

**[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. MR**735080****[2]**Paul S. Bourdon and Joel H. Shapiro,*Cyclic composition operators on 𝐻²*, 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****[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. MR**1142722**, 10.1512/iumj.1991.40.40064**[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.**[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. MR**884467**, 10.1090/S0002-9939-1987-0884467-4**[6]**Gilles Godefroy and Joel H. Shapiro,*Operators with dense, invariant, cyclic vector manifolds*, J. Funct. Anal.**98**(1991), no. 2, 229–269. MR**1111569**, 10.1016/0022-1236(91)90078-J**[7]**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**, 10.1112/jlms/s2-31.3.561**[8]**Domingo A. Herrero,*Possible structures for the set of cyclic vectors*, Indiana Univ. Math. J.**28**(1979), no. 6, 913–926. MR**551155**, 10.1512/iumj.1979.28.28064**[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. MR**1120920**, 10.1016/0022-1236(91)90058-D**[11]**Domingo A. Herrero,*Hypercyclic operators and chaos*, J. Operator Theory**28**(1992), no. 1, 93–103. MR**1259918****[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. MR**1078739**, 10.1512/iumj.1990.39.39039**[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. MR**0241956****[15]**Héctor Salas,*A hypercyclic operator whose adjoint is also hypercyclic*, Proc. Amer. Math. Soc.**112**(1991), no. 3, 765–770. MR**1049848**, 10.1090/S0002-9939-1991-1049848-8**[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. MR**0361899**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
47B37,
47A99

Retrieve articles in all journals with MSC: 47B37, 47A99

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1995-1249890-6

Keywords:
Cyclic and hypercyclic vectors,
bilateral weighted shifts,
unilateral backward weighted shifts

Article copyright:
© Copyright 1995
American Mathematical Society