Hypercyclic weighted shifts

Author:
Héctor N. Salas

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

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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]**C. Apostol, L. A. Fialkow, D. A. Herrero, and D. Voiculescu,*Approximation of Hilbert space operators*, Vol. II, Research Notes in Math., vol. 102, Pitman, Boston, London, and Melbourne, 1984. MR**735080 (85m:47002)****[2]**P. S. Bourdon and J. H. Shapiro,*Cyclic composition operators in*, Proc. Sympos. Pure Math., vol. 51, Part 2, Amer. Math. Soc., Providence, RI, 1990. MR**1077418 (91h:47028)****[3]**K. C. Chan and J. H. Shapiro,*The cyclic behavior of translation operators on Hilbert spaces of entire functions*, Indiana Univ. Math. J.**40**(1991), 1421-1449. MR**1142722 (92m:47060)****[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]**R. M. Gethner and J. H. Shapiro,*Universal vectors for operators on spaces of holomorphic functions*, Proc Amer. Math. Soc.**100**(1987), 281-288. MR**884467 (88g:47060)****[6]**G. Godefroy and J. H. Shapiro,*Operators with dense, invariant, cyclic vector manifolds*, J. Funct. Anal.**98**(1991), 229-269. MR**1111569 (92d:47029)****[7]**I. Halperin, C. Kitai, and P. Rosenthal,*On orbits of linear operators*, J. London Math. Soc.**31**(1985), 561-565. MR**812786 (87e:47025)****[8]**D. A. Herrero,*Possible structures for the set of cyclic vectors*, Indiana Univ. Math. J.**28**(1979), 913-926. MR**551155 (81k:47023)****[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]**-,*Limits of hypercyclic and supercyclic operators*, J. Funct. Anal.**99**(1991), 179-190. MR**1120920 (92g:47026)****[11]**-,*Hypercyclic operators and chaos*, J. Operator Theory**28**(1992), 93-103. MR**1259918 (95g:47031)****[12]**D. A. Herrero and Z. Y. Wang,*Compact perturbation of hypercyclic and cyclic operators*, Indiana Univ. Math. J.**3**(1990), 819-829. MR**1078739 (91k:47042)****[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 (39:3292)****[15]**H. Salas,*A hypercyclic operator whose adjoint is also hypercyclic*, Proc. Amer. Math. Soc.**112**(1991), 765-770. MR**1049848 (91j:47016)****[16]**A. L. Shields,*Weighted shifts operators and analytic function theory*, Topics of Operator Theory, Math. Surveys Monographs, vol. 13, Amer. Math. Soc., Providence, RI, 1974, pp. 49-128. MR**0361899 (50:14341)**

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Additional Information

DOI:
https://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