On a spectral mapping theorem for scalar type spectral operators, the weak spectral mapping theorem for and stability of $C_0$-semigroups of such operators

We prove a spectral mapping theorem for scalar type spectral operators and apply it to extend the weak spectral mapping theorem and the generalized Lyapunov stability theorem, known to hold for the $C_0$-semigroups of normal operators on complex Hilbert spaces, to the more general case of $C_0$-semigroups of scalar type spectral operators on complex Banach spaces. We also establish for the latter an analogue of the Gearhart-Pr\"uss-Greiner characterization of the uniform exponential stability for $C_0$-semigroups on complex Hilbert spaces.