  In this note, we study the integrodifferential operator $(I-\Delta)^{\log}$
corresponding to the logarithmic symbol $\log(1+|\xi|^2)$, which is a singular
integral operator given by $$
  (I-\Delta)^{\log}
u(x)=d_{N}\int_{\mathbb{R}^N}\frac{u(x)-u(x+y)}{|y|^{N}}\omega(|y|)\, dy, $$
where $d_N=\pi^{-\frac{N}{2}}$,
$\omega(r)=2^{1-\frac{N}{2}}r^{\frac{N}{2}}K_{\frac{N}{2}}(r)$ and $K_{\nu}$ is
the modified Bessel function of second kind with index $\nu$. This operator is
the L\'evy generator of the variance gamma process and arises as derivative
$\partial_s\Big|_{s=0}(I-\Delta)^s$ of fractional relativistic Schr\"{o}dinger
operators at $s=0$. In order to study associated Dirichlet problems in bounded
domains, we first introduce the functional analytic framework and some
properties related to $(I-\Delta)^{\log}$, which allow to characterize the
induced eigenvalue problem and Faber-Krahn type inequality. We also derive a
decay estimate in $\mathbb{R}^N$ of the Poisson problem and investigate small
order asymptotics $s\to 0^+$ of Dirichlet eigenvalues and eigenfunctions of
$(I-\Delta)^s$ in a bounded open Lipschitz set.
