On the Weyl-Titchmarsh and Livšic 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 $(κ, \cM)$ 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.