Sequences of differential operators: exponentials, hypercyclicity and equicontinuity (Q2759190)

If \(N\) is a positive integer, then the real and complex Euclidean \(n\)-dimensional spaces are denoted by \(\mathbb{R}^N\) and \(\mathbb{C}^N\), with \(\mathbb{R}^1=\mathbb{R}\), \(\mathbb{C}^1=\mathbb{C}\). If \(X\), \(Y\) are linear topological spaces, \(J\) is a parameter set and \(T_j: X\to Y\), \(j\in J\), are continuous linear mappings, then a vector \(x\) in \(X\) is said to be hypercyclic or universal for \(\{T_{ij}\}\) if the `orbit' \(\{T_j(x), j\in J\}\) is dense in \(Y\), and the class \(\{T_j\}\) is said to be hypercyclic. The set of vectors in \(X\) which are hypercyclic for \(\{T_j\}\) is denoted by \(HC(\{T_j\})\). If \(T_j= T^j\), \(j\in J\), for some continuous linear map \(T\), then \(T\) is said to be hypercyclic and \(HC(\{T_j\})\) is denoted by \(HC(T)\).NEWLINENEWLINENEWLINEIf \(G\) is a Runge domain in \(\mathbb{C}^N\), so that each holomorphic function on \(G\) may be uniformly approximated by polynomials on compact subsets of \(G\), then the results of this paper include statements:NEWLINENEWLINENEWLINE(1) equivalent conditions relating to a subset \(S\) of \(\mathbb{C}^N\) include,NEWLINENEWLINENEWLINE(i) \(S\) is unbounded,NEWLINENEWLINENEWLINE(ii) the class \((\tau_\alpha)_{\alpha\in S}\) of translation operators is hypercyclic on \(H(\mathbb{C}^N)\),NEWLINENEWLINENEWLINE(iii) \(HC((\tau_\alpha)_{\alpha\in S})\) is `residual' in \(H(\mathbb{C}^N)\),NEWLINENEWLINENEWLINE(iv) \((\tau_\alpha)_{\alpha\in S}\) is not equicontinuous on \(H(\mathbb{C}^N)\);NEWLINENEWLINENEWLINE(2) if \(\{c_n\}\) is a sequence of complex numbers, \(G\) is a Runge domain in \(\mathbb{C}^N\), and the holomorphic function \(\vartheta\) is non-constant, and if \(0< c_{\inf}\leq c_{\sup}<\infty\), where \(c_{\text{inf/sup}}=\liminf/\sup\{|c_n|^{1/n}\), \(n\to\infty\}\), then \(HC((c_n\vartheta_n(D)))\) is residual in \(H(G)\), where \(D\) is a differential operator;NEWLINENEWLINENEWLINE(3) the class of operators \(\{\vartheta_j(D),j\in\vartheta\}\) is equicontinuous on \(H(\mathbb{C}^N)\) if and only if \(\{\vartheta_j\}\) has a majorant entire function of exponential type.NEWLINENEWLINENEWLINEStatements in the introduction of the paper indicate that the main results provide sufficient conditions for sequences of differential operators to be hypercyclic.