Numerically hypercyclic polynomials

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 ∈ 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 _∞.