Holomorphic operator valued functions generated by passive selfadjoint systems

TL;DR

This paper characterizes a class of holomorphic operator functions that are both Schur and Nevanlinna functions, providing explicit forms, dilations, and transformations, with applications to passive selfadjoint systems.

Contribution

It introduces and characterizes the class al Ral S(rak M), including explicit inner function forms, dilations, and fixed points, advancing the understanding of holomorphic operator functions.

Findings

Explicit form for inner functions in al Ral S(rak M)

Inner dilation construction for functions in al Ral S(rak M)

Identification of fixed points under transformations

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.