On Topological Numerical Transitivity, C-Transitivity, and Generalized Numerical Transitivity*

Motivated by the concept of numerical hypercyclicity, in this paper we introduce three new notions in linear dynamics: topological numerical transitivity, generalized numerical transitivity, and C*-transitivity. The first notion is defined for operators on general Banach spaces, whereas the latter two are formulated in the setting of operators on Banach algebras. The concept of C*-transitivity is also applicable to operators on the space of Hilbert–Schmidt operators. We prove that these new notions are mutually different, and we also show that they differ from the standard notions in dynamics like hypercyclicity, supercyclicity, and numerical hypercyclicity. In particular, while topological numerical transitivity implies numerical hypercyclicity, as we argue in the paper, we provide examples of numerically hypercyclic and strongly numerically hypercyclic operators that are not topologically numerically transitive. We also provide examples of numerically hypercyclic operators that are neither C*-transitive nor generalized numerically transitive. Also, we construct nonsupercyclic C*-transitive and generalized numerically transitive operators on the C*-algebra of compact operators on a separable Hilbert space, as well as C*-transitive operators on the standard Hilbert C*-module. In addition, we study diagonal operators on finite-dimensional Hilbert spaces and on ℓp(N), 1<p<∞, and we obtain complete characterizations of both numerical hypercyclicity and topological numerical transitivity in terms of coefficient-simplex criteria. As a consequence of this result, we prove that, unlike numerical and strong numerical hypercyclicity, a diagonal operator on a finite-dimensional complex Hilbert space of dimension at least two can never be topologically numerically transitive. Furthermore, we provide some sufficient conditions for diagonal operators on the standard Hilbert C*-module to be C*-transitive, and illustrate this result with examples. At the end of the paper, we compare topological numerical transitivity and C*-transitivity with numerous standard notions in linear dynamics, like N-weak supercyclicity, weak-n supercyclicity, ϵ−cyclicity, Li–Yorke chaos, distributional chaos of type 3, subspace hypercyclicity, and subspace supercyclicity, and we prove that they genuinely differ from all these standard concepts in dynamics. Finally, we also introduce the notion of C*-numerical transitivity, and we argue that many of the results on C*-transitivity hold analogously for C*-numerical transitivity.