Abstract
In this paper, we show that every complex Banach space X with dimension at least 2 supports a numerically hypercyclic d-homogeneous polynomial P for every \({d\in \mathbb{N}}\). Moreover, if X is infinite-dimensional, then one can find hypercyclic non-homogeneous polynomials of arbitrary degree which are at the same time numerically hypercyclic. We prove that weighted shift polynomials cannot be numerically hypercyclic neither on c 0 nor on ℓ p for 1 ≤ p < ∞. In contrast, we characterize numerically hypercyclic weighted shift polynomials on ℓ∞.
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S. G. Kim was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (2010-0009854) and by Kyungpook National University Research Fund, 2012. A. Peris was supported in part by MICINN and FEDER, Project MTM2010-14909, and by Generalitat Valenciana, Project PROMETEO/2008/101. H. G. Song is partially supported by the BK21 program (KNU) of the government of the republic of Korea.
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Kim, S.G., Peris, A. & Song, H.G. Numerically hypercyclic polynomials. Arch. Math. 99, 443–452 (2012). https://doi.org/10.1007/s00013-012-0445-4
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DOI: https://doi.org/10.1007/s00013-012-0445-4