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