  Let $X=\{X(t)\}_{t\geq0}$ be an operator semistable L\'evy process on
$\mathbb{R}^d$ with exponent $E$, where $E$ is an invertible linear operator on
$\mathbb{R}^d$. In this paper we determine exact Hausdorff measure functions
for the range of $X$ over the time interval $[0,1]$ under certain assumptions
on the principal spectral component of $E$. As a byproduct we also present
Tauberian results for semistable subordinators and sharp bounds for the
asymptotic behavior of the expected sojourn times of $X$.
