Conservative L-systems and the Liv\v{s}ic function

TL;DR

This paper explores the relationship between Livšic functions, dissipative extensions, and transfer functions of conservative L-systems, establishing conditions under which these functions are reciprocal and characterizing impedance functions within the Donoghue class.

Contribution

It introduces a natural hypothesis linking Livšic functions and transfer functions, and characterizes impedance functions in the Donoghue class and its generalization based on the von Neumann parameter.

Findings

When the von Neumann parameter $=0$, the transfer function and Liv
61ic function are reciprocals.

The impedance function of a conservative L-system belongs to the Donoghue class if and only if $=0$.

Criteria are established for impedance functions to belong to the generalized Donoghue class.

Abstract

We study the connection between the Liv\v{s}ic class of functions $s(z)$ that are the characteristic functions of densely defined symmetric operators $\dot A$ with deficiency indices $(1, 1)$, the characteristic functions $S(z)$ (the M\"obius transform of $s(z)$) of a maximal dissipative extension $T$ of $\dot A$ (determined by the von Neumann parameter $\kappa$ of the extension relative to an appropriate basis in the deficiency subspaces) and the transfer functions $W_\Theta(z)$ of a conservative L-system $\Theta$ with the main operator $T$. It is shown that under a natural hypothesis $S(z)$ and $W_\Theta(z)$ are reciprocal to each other. In particular, when $\kappa=0$, $W_\Theta(z)=\frac{1}{S(z)}=-\frac{1}{s(z)}$. It is established that the impedance function of a conservative L-system with the main operator $T$ coincides with the function from the Donoghue class if and only if the von Neumann parameter vanishes ($\kappa=0$). Moreover, we introduce the generalized Donoghue class and obtain the criteria for an impedance function to belong to this class. All results are illustrated by a number of examples.