  We study an unbounded operator arising naturally after linearizing the system
modelling the motion of a rigid body in a viscous incompressible fluid. We show
that this operator is $\mathcal{R}$ sectorial in $L^q$ for every $q\in
(1,\infty)$, thus it has the maximal $L^p$-$L^q$ regularity property. Moreover,
we show that the generated semigroup is exponentially stable with respect to
the $L^q$ norm. Finally, we use the results to prove the global existence for
small initial data, in an $L^p$-$L^q$ setting, for the original nonlinear
problem.
