The Numerically Hypercyclic Semigroup

Let X be a Banach space and G be a semigroup, Let T= (Tg)g∈G be a semigroup of operators on a Banach space X. The T is a numerically hypercyclic semigroup of operators,or is provided with the numerical hypercyclicity, If there exists (x,x*)∈Ⅱ(X) such that the{x*(Tgx): g∈G} is dense in C. In this case, we call the vector (x,x*) to be a numerically hypercyclic vector associated with it, and the symbol nHC(T) is employed to represent the set of all numerically hypercyclic vectors for it.This paper focus on the following several topics:Every finite-dime nsional Banach space with dimension at least 2 supports a numerical-ly hypercyclic semigroup of operators. On a reflexive Banach space X, if T is a numerically hypercyclic semigroup of operators, then T*={Tg*:Tg∈T} is also a numerically hypercyclic semigroup of operators. If semigroup of operators T is hypercyclic or a weakly hy-percyclic, then it is numerically hypercyclic, Meanwhile we can find the set of all numerically hypercyclic vectors for semigroup of opera-tors, nHC(T), according to hypercyclic vectors for it. For a strongly continuous semigroup (Co-semigroup) of operators, we can obtain the conditions for an affine semigroup of operators to be numerical hy-percyclic. And we can find the numerical hypercyclic vectors for T according to the numerical hypercyclic vectors for each member of T.