Hypercyclic weighted shifts

Salas, Héctor N.

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

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: 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.

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