Some local properties of spectrum of linear dynamical systems in Hilbert space

We prove some local properties of the spectrum of a linear dynamical system in Hilbert space. The semigroup generator, the control operator and the observation operator may be unbounded. We consider (i) the PBH test, (ii) the correspondence between the poles of the resolvent of the semigroup generator and the poles of the transfer function, and (iii) pole-zero cancellation between two transfer functions of the cascade connection of two dynamical systems. For our investigation we take well-posed linear systems and a subclass of them called weakly regular systems as the most general setting.