In this work we introduce and study the M-Hypercyclicity of C 0-semigroup T = (T t) t≥0 on an infinite-dimensional separable complex Banach space X. We give sufficient conditions of being M-hypercyclic for this semigroup. Moreover, some proprieties and analogous result for the notion of M-Transitive are also obtained.

Let B(X) the Banach algebra of all bounded operators on a Banach space X and let T ∈ B(X). We denote by R alc (X) = {T ∈ B(X) : C(T ) = {0}} and R ac (X) = {T ∈ B(X) : K(T ) = {0}} where C(T ) and K(T ) are respectively the algebraic core and the analytic core. In this paper we show that R alc (X) and R ac (X) are a regularities in Kordula-Müller's sense. Mathematics Subject Classification : 47A10

In this paper, we show a spectral inclusion of a different spectra of a C 0 -quasi-semigroup and its generator and precisely for ordinary, point, approximate point, residual, essential and regular spectra.

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