Numerically hypercyclic polynomials
Let \(E\) be a complex Banach space, \(E^{*}\) be its dual space and \(S_E, S_{E^{*}}\) be their respective unit spheres. The state set \(\Pi(E)\) is defined by \[ \Pi(E)=\{(x,x^{*}):x\in S_E,\, x^{*}\in S_{E^{*}},\, x^{*}(x)=1\}. \] A map \(f: E\longrightarrow E\) is called \textit{numerically hypercyclic} if there exists \((x,x^{*})\in\Pi(E)\) such that the \textit{numerical orbit} \[ \text{norb}((x,x^{*}),f):=\{x^{*}f^n(x): n=0,1, \dots \} \] is dense in \(\mathbb{C}\). This concept was recently introduced in [\textit{S. G. Kim} et al., Integral Equations Oper. Theory 72, No. 3, 393--402 (2012; Zbl 1298.47016)], where one can find a proof that hypercyclic continuous linear operators are numerically hypercyclic. The paper under review is devoted to the study of numerically hypercyclic polynomials \(P: E\rightarrow E\). Some conditions under which a hypercyclic polynomial is numerically hypercyclic are established and examples of polynomials satisfying these conditions are given. Let us describe these examples. Consider \(E=c_0\) or \(E=\ell_q\), \(1\leq q<\infty\), and, given a complex polynomial \(p: \mathbb{C}\rightarrow \mathbb{C}\) of degree at least 2 such that \(p(0)=0\), define the polynomial \(P: E\rightarrow E\) by \(P(z_1,z_2,\dots)=(p(z_2),p(z_3),\dots)\). Then \(P\) is hypercyclic if and only if 0 belongs to the Julia set of \(p\) [\textit{A. Peris}, J. Math. Anal. Appl. 287, No. 2, 487--493 (2003; Zbl 1042.46019)]. Once these conditions are satisfied, \(P\) is numerically hypercyclic if and only if 0 belongs to the Julia set of \(p\). As to the general case, it remains unknown if every hypercyclic polynomial is numerically hypercyclic. In the case of a separable infinite-dimensional complex Banach space \(E\), the authors show that there exist (not necessarily homogeneous) polynomials of arbitrary degree on \(E\) which are numerically hypercyclic. On the other hand, if \(E\) is a Banach space with dimension at least 2, no matter if it is infinite-dimensional or finite-dimensional, then \(E\) supports a numerically hypercyclic \(d\)-homogeneous polynomial for every \(d\in \mathbb{N}\). The article ends with interesting results related to weighted shift polynomials, which are the polynomial version of weighted shift operators. Let \(E\) be a Banach sequence space, \(d\in \mathbb{N}\) with \(d\geq 2\), \((w_k)\in \mathbb{C}^n\) with \(\sup_{k}|w_k|<\infty\), and \(w_k\neq 0\) for \(k\geq 2\), and let \(P\) be the \(d\)-homogeneous polynomial on \(E\) defined by \[ P((x_i)_i)=(w_2x_2^d, w_3x_3^d, \dots , w_nx_n^d, \dots) \] \noindent for \((x_i)_i\in E\). It is proved that, if \(E=c_0\) or \(E=\ell_p\), \(1\leq p<\infty\), then \(P\) cannot be numerically hypercyclic. In contrast, if \(E=\ell_\infty\), \(P\) is numerically hypercyclic if and only if \((w_k)_k\) satisfies a certain condition; and if \(P\) is numerically hypercyclic, then \(P^\ell\) is numerically hypercyclic for every \(\ell\in \mathbb{N}\).
scientific article; zbMATH DE number 6116964
scientific article