Hypercyclic weighted shifts
Author:
Héctor N. Salas
Journal:
Trans. Amer. Math. Soc. 347 (1995), 9931004
MSC:
Primary 47B37; Secondary 47A99
MathSciNet review:
1249890
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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 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 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.
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 R. M. Gethner and J. H. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proc Amer. Math. Soc. 100 (1987), 281288. MR 884467 (88g:47060)
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 G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), 229269. MR 1111569 (92d:47029)
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 I. Halperin, C. Kitai, and P. Rosenthal, On orbits of linear operators, J. London Math. Soc. 31 (1985), 561565. MR 812786 (87e:47025)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199512498906
PII:
S 00029947(1995)12498906
Keywords:
Cyclic and hypercyclic vectors,
bilateral weighted shifts,
unilateral backward weighted shifts
Article copyright:
© Copyright 1995
American Mathematical Society
