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

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.