, Volume 99, Issue 5, pp 443-452
Date: 16 Nov 2012

Numerically hypercyclic polynomials

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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.

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.