Passive systems with a normal main operator and quasi-selfadjoint systems

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Passive systems τ=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 τ with M=N is said to be quasi-selfadjoint if ran(T-T*)⊂ N. The subclass Sqs of the Schur class S is the class formed by all transfer functions of quasi-selfadjoint passive systems. The subclass Sqs is characterized and minimal passive quasi-selfadjoint realizations are studied. The connection between the transfer function belonging to the subclass Sqs and the Q-function of T is given.