Passive Systems with a Normal Main Operator and Quasi-selfadjoint Systems

Passive systems t = {T, M, N, H} with M and N as an input and output space and H as a state space are considered in the case that the main operator on the state space is normal. Basic properties are given and a general unitary similarity result involving some spectral theoretic conditions on the main operator is established. A passive system t with M = N is said to be quasi-selfadjoint if ran (T - T*). N. The subclass S(qs)(N) of the Schur class S(N) is the class formed by all transfer functions of quasi-selfadjoint passive systems. The subclass S(qs)(N) is characterized and minimal passive quasi-selfadjoint realizations are studied. The connection between the transfer function belonging to the subclass S(qs) (N) and the Q-function of T is given.