A hypercyclicity criterion with applications
The following hypercyclicity criterion for continuous linear operators \(T: X \rightarrow X\) defined on a Hausdorff locally convex space \((X,\tau)\) is proved: Assume that \(X\) admits a finer topology \(\mu\) such that \((X,\mu)\) is a Fréchet space and \(T\) is \(\mu\)-continuous. Suppose that there is a countable \(\tau\)-dense subset \(Y\) of \(X\) and there is a sequence \((S_n)_n\) of maps \(S_n:Y \rightarrow X\) such that, for all \(y \in Y\), we have \(T_n(y) \rightarrow 0\), \(S_n(y) \rightarrow 0\) in \((X,\mu)\) and \(T^n S_n(y)=y\). Then the operator \(T\) is hypercyclic for \((X,\tau)\), i.e., there is \(x \in X\) such that its orbit \(\{x, T(x), T^2(x),\dots\}\) is dense in \((X,\tau)\). The proof is constructive and does not require that \((X,\mu)\) is separable. The criterion is used to show weak-* hypercyclicity of weighted backward shift operators on the dual of Banach sequence spaces. As the main application, the author exhibits hypercyclic left multipliers \(L_T: S \rightarrow TS\) on spaces of operators \(L(X,Y)\), endowed with the topology of uniform convergence on bounded sets, for several pairs \((X,Y)\) of spaces of holomorphic functions of several variables. It is also investigated when \(L_T\) admits a complemented infinite-dimensional subspace of hypercyclic vectors.
scientific article; zbMATH DE number 5083361
scientific article