Holomorphic operator valued functions generated by passive selfadjoint systems

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 $\mathcal R\mathcal S(\mathfrak M)$areSchurfunctionsintheopenunitdisk$\mathbb D$and,inaddition,Nevanlinnafunctionsin$\mathbb C_+\cup\mathbb C_-$. 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.