Holomorphic Operator-valued Functions Generated by Passive Selfadjoint Systems

Abstract

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