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

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 $\text{cM}$ of a maximal dissipative extension $\widehat{A}$ of the symmetric operator $\dot{A}$. Given a reference self-adjoint extension $A$ of $\dot{A}$, we introduce a von Neumann parameter $\kappa$, $\mathrm{\mid }\kappa \mathrm{\mid }<1$, characterizing the domain of the dissipative extension $\widehat{A}$ against $\text{Dom}(A)$ and show that the pair $(\kappa ,\text{cM})$ 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.