Holomorphic operator-valued functions generated by passive selfadjoint systems

Abstract

Let M be a Hilbert space. In this paper we study a class RS(m) of operator functions that are holomorphic in the domain C∖{(−∞,−1] ∪ [1,+∞)} and whose values are bounded linear operators in m . The functions in RS(m) are Schur functions in the open unit disk D and, in addition, Nevanlinna functions in C+∪C− . Such functions can be realized as transfer functions of minimal passive selfadjoint discrete-time systems.We give various characterizations for the class RS(m) and obtain an explicit form for the inner functions from the class RS(m) as well as an inner dilation for any function from RS(m) . We also consider various transformations of the class RS(m) , construct realizations of their images, and find corresponding fixed points.