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

Abstract

Passive systems \(\tau = \{T,{\mathfrak{M}},{\mathfrak{N}},{\mathfrak{H}}\}\) with \({\mathfrak{M}}\) and \({\mathfrak{N}}\) as an input and output space and \({\mathfrak{H}}\) 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 \(\tau\) with \({\mathfrak{M}} = {\mathfrak{N}}\) is said to be quasi-selfadjoint if ran \((T - T^*) \subset {\mathfrak{N}}\). The subclass \({\bf S}^{qs}({\mathfrak{N}})\) of the Schur class \({\bf S}({\mathfrak{N}})\) is the class formed by all transfer functions of quasi-selfadjoint passive systems. The subclass \({\bf S}^{qs}({\mathfrak{N}})\) is characterized and minimal passive quasi-selfadjoint realizations are studied. The connection between the transfer function belonging to the subclass \({\bf S}^{qs}({\mathfrak{N}})\) and the Q-function of T is given.