Holomorphic operator valued functions generated by passive selfadjoint systems

Abstract

In this paper we study a class $\mathcal{R}\mathcal{S}(\mathfrak{M})$ of operator functions that are holomorphic in the domain $\mathbb{C}\setminus \{(-\infty ,-1]\cup [1,+\infty )\}$ and whose values are contractive operators in a Hilbert space $(\mathfrak{M})$. The functions in $\mathcal{R}\mathcal{S}(\mathfrak{M})$ 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 $\mathcal{R}\mathcal{S}(\mathfrak{M})$ and obtain an explicit form for the inner functions from the class $\mathcal{R}\mathcal{S}(\mathfrak{M})$ as well as an inner dilation for any function from $\mathcal{R}\mathcal{S}(\mathfrak{M})$. We also consider various transformations of the class $\mathcal{R}\mathcal{S}(\mathfrak{M})$, construct realizations of their images, and find corresponding fixed points.

Version 2 with 35 pages where Section 6.6 has been added