  This paper proves sharp bounds on the tails of the L\'evy exponent of an
operator semistable law on $\mathbb R^d$. These bounds are then applied to
explicitly compute the Hausdorff and packing dimensions of the range, graph,
and other random sets describing the sample paths of the corresponding operator
semi-selfsimilar L\'evy processes. The proofs are elementary, using only the
properties of the L\'evy exponent, and certain index formulae.
