  The paper gives a short account of some basic properties of
\textit{Dirichlet-to-Neumann} operators $\Lambda_{\gamma,\partial\Omega}$
including the corresponding semigroups motivated by the Laplacian transport in
anisotropic media ($\gamma \neq I$) and by elliptic systems with dynamical
boundary conditions. For illustration of these notions and the properties we
use the explicitly constructed \textit{Lax semigroups}. We demonstrate that for
a general smooth bounded convex domain $\Omega \subset \mathbb{R}^d$ the
corresponding {Dirichlet-to-Neumann} semigroup $\left\{U(t):= e^{-t
\Lambda_{\gamma,\partial\Omega}}\right\}_{t\geq0}$ in the Hilbert space
$L^2(\partial \Omega)$ belongs to the \textit{trace-norm} von Neumann-Schatten
ideal for any $t>0$. This means that it is in fact an \textit{immediate Gibbs}
semigroup. Recently Emamirad and Laadnani have constructed a
\textit{Trotter-Kato-Chernoff} product-type approximating family
$\left\{(V_{\gamma, \partial\Omega}(t/n))^n \right\}_{n \geq 1}$
\textit{strongly} converging to the semigroup $U(t)$ for $n\to\infty$. We
conclude the paper by discussion of a conjecture about convergence of the
\textit{Emamirad-Laadnani approximantes} in the the {\textit{trace-norm}}
topology.

Functional Analysis