Supermixing and hypermixing of strongly continuous semigroups and their direct sum.

Supermixing and hypermixing strongly continuous semigroups are introduced in this paper. It is proved that supermixing preserves under quasiconjugacy. Moreover, it is established that if a strongly continuous semigroup is supermixing(hypermixing), then any discretization of it, is supermixing(hypermixing). Also, it is proved that supermixing(hypermixing) of (T_t)_{t ≥ 0} implies that any of T_t's is a supermixing(hypermixing). Some sufficient conditions for supermixing and hypermixing are stated that based on dense sets and kernels of operators of semigroup. It is proved that the supermixing(hypermixing) of two strongly continuous semigroups imply that their direct sum is supermixing(hypermixing). Also, supermixing(hypermixing) of the direct sum of finite strongly continuous semigroups indicates that any of these semigroups is supermixing(hypermixing). Outcomes of this study have shown that extension of the concepts of supermixing and hypermixing to strongly continuous semigroups leads to get access to a proper subset of the set of hypercyclic semigroups and reach valuable results about them. [ABSTRACT FROM AUTHOR]