On Topological Numerical Transitivity, C*-Transitivity and generalizations

Abstract

Motivated by the concept of numerical hypercyclicity, in this paper, we introduce three new notions in linear dynamics: topological numerical transitivity, generalized numerical transitivity, 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. 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. 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 nonsupercyclic operators on the standard Hilbert C*-module. In addition, we study diagonal operators on finite-dimensional and separable Hilbert spaces, and we obtain complete characterizations of both numerical hypercyclicity and topological numerical transitivity in terms of coefficient-simplex criteria. Further, we provide some sufficient conditions for diagonal operators on the standard Hilbert C*-module to be C*-transitive. All these results are illustrated with concrete examples. At the end of the paper, we compare topological numerical transitivity and C*-transitivity with numerous standard notions in linear dynamics, and we prove that topological numerical transitivity and C*-transitivity genuinely differ from all these standard concepts in dynamics.