  We discuss the classes $\fC$, $\fM$, and $\fS$ of analytic functions that can
be realized as the Liv\v{s}ic characteristic functions of a symmetric densely
defined operator $\dot A$ with deficiency indices $(1,1)$, the Weyl-Titchmarsh
functions associated with the pair $(\dot A, A)$ where $A$ is a self-adjoint
extension of $\dot A$, and the characteristic function of a maximal dissipative
extension $\widehat A$ of $\dot A$, respectively. We show that the class $\fM$
is a convex set, both of the classes $\fS$ and $\fC$ are closed under
multiplication and, moreover, $\fC\subset \fS$ is a double sided ideal in the
sense that $\fS\cdot \fC=\fC\cdot \fS\subset \fS$.
  The goal of this paper is to obtain these analytic results by providing
explicit constructions for the corresponding operator realizations.
  In particular, we introduce the concept of an operator coupling of two
unbounded maximal dissipative operators and establish an analog of the
Liv\v{s}ic-Potapov multiplication theorem [14] for the operators associated
with the function classes $\fC$ and $\fS$. We also establish that the modulus
of the von Neumann parameter characterizing the domain of $\widehat A$ is a
multiplicative functional with respect to the operator coupling.
