Based on the spectral inclusion and mapping theorems for scalar type spectral operators (discussed in my prior talk of May 4, 2025), we extend a precise weak spectral mapping theorem along with the spectral bound equal growth bound condition and a generalized Lyapunov stability theorem from the known case of $C_0$-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. For such semigroups, we obtain exponential estimates with the best stability constants.

We also extend to a Banach space setting a celebrated characterization of uniform exponential stability for $C_0$-semigroups on complex Hilbert spaces and thereby acquire a characterization of uniform exponential stability for scalar type spectral and eventually norm-continuous $C_0$-semigroups.

The finer spectrum structure is given itemized consideration.