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 R sectorial in L q for every q ∈ (1, ∞), 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.