Existence of disjoint weakly mixing operators that fail to satisfy the disjoint hypercyclicity criterion
Let \(T_1,\dots,T_N\) be bounded, linear operators on a separable infinite-dimensional Banach space \(X\). According to \textit{L. Bernal-González} [Stud. Math. 182, No. 2, 113--131 (2007; Zbl 1134.47006)], \textit{J. Bès} and \textit{A. Peris} [J. Math. Anal. Appl. 336, No. 1, 297--315 (2007; Zbl 1129.47007)], \(T_1,\dots,T_N\) are disjoint or \(d\)-hypercyclic if there is a (\(d\)-hypercyclic) vector \(x \in X\) such that \((x,\dots, x)\) is a hypercyclic vector for the direct sum operator \(\bigoplus_{i=1}^N T_i\) on the space \(\bigoplus_{i=1}^N X_i\) with respect to the product topology. If the set of \(d\)-hypercyclic vectors is dense in \(X\), then \(T_1,\dots,T_N\) are called densely \(d\)-hypercyclic. The aim of this deep, interesting paper is to show that several dynamical properties of a single operator fail to hold true in the disjoint setting. Several questions formulated by \textit{J. Bés} et al. [J. Oper. Theory 72, No. 1, 15--40 (2014; Zbl 1318.47010)] are answered: Every separable, infinite-dimensional Banach space admits \(d\)-weakly mixing operators which fail the \(d\)-hypercyclicity criterion, as well as densely \(d\)-hypercyclic operators which fail to be \(d\)-weakly mixing. Moreover, in every separable, infinite-dimensional Banach space there exist \(d\)-hypercyclic operators \(T_1,T_2\) which fail to be densely \(d\)-hypercyclic. Two families of \(d\)-hypercyclic operators that fail to have any \(d\)-hypercyclic vector in common are exhibited. The behavior of the inverse of invertible, densely \(d\)-hypercyclic operators is also investigated.
scientific article; zbMATH DE number 6380659
scientific article