On the Weyl-Titchmarsh and Liv\v{s}ic functions

Abstract

We establish a mutual relationship between main analytic objects for the dissipative extension theory of a symmetric operator $\dot A$ with deficiency indices $(1,1)$. In particular, we introduce the Weyl-Titchmarsh function $\cM$ of a maximal dissipative extension $\hat A$ of the symmetric operator $\dot A$. Given a reference self-adjoint extension $A$ of $\dot A$, we introduce a von Neumann parameter $\kappa$, $|\kappa|<1$, characterizing the domain of the dissipative extension $\hat A$ against $\Dom (A)$ and show that the pair $(\kappa, \cM)$ is a complete unitary invariant of the triple $(\dot A, A, \hat A)$, unless $\kappa=0$. As a by-product of our considerations we obtain a relevant functional model for a dissipative operator and get an analog of the formula of Krein for its resolvent.