Hypercyclic tuples of operators and somewhere dense orbits
An \(n\)-tuple of operators is a finite sequence of length \(n\) of commuting continuous linear operators on a locally convex space \(X\). Let \(T=(T_1,T_2,\dots,T_n)\) be an \(n\)-tuple of operators on a locally convex space \(X\) and let \({\mathcal F}={\mathcal F}_T=\{T_1^{k_1}T_2^{k_2}\dots T_n^{k_n}:k_i\geq 0\}\) be the semigroup generated by \(T\). Then the tuple \(T\) is called hypercyclic if there is a vector \(x\in X\) whose orbit \(\text{Orb}(T,x)=\text{Orb}({\mathcal F},x)=\{Sx:S\in {\mathcal F}\}\) under \(T\) is dense in \(X\). In the paper under review, the author gives some examples of hypercyclic tuples of operators. In particular, he shows that there are hypercyclic tuples of diagonal matrices on \({\mathbb R}^n\) and \({\mathbb C}^n\). He uses this result to show that there are no hypercyclic tuples of normal operators on an infinite-dimensional Hilbert space, and even no hypercyclic tuples of subnormal operators having commuting subnormal extensions. The author also proves that on Hilbert spaces over the field of real numbers, there are tuples of operators that have a somewhere dense orbit that is not dense. Furthermore, examples are given of tuples \(T\) that are hypercyclic, while \(T^n\) is not hypercyclic.
scientific article; zbMATH DE number 5303195
scientific article