On certain forms of transitivities for linear operators

In this article, we give several characterizations for various transitivity properties for linear operators. We define a general form of ‘Hypercyclicity Criterion’ using a Furstenberg family F to characterize F -transitive operators. In particular, we find an equivalent characterization for mixing operators. We study proximal and asymptotic relations for linear operators and prove that the difference between mixing operators and Kitai's Criterion can be presented through these relations. Finally, we find equivalent characterization of strongly transitive and strongly product transitive operators.