**MSC:**- 35Q30 Stokes and Navier-Stokes equations, See also {76D05, 76D07,
- 47A10 Spectrum, resolvent
- 35J45 General theory of elliptic systems of PDE
- 35J55 Boundary value problems for elliptic systems

operator between two parallel planes of ${\mathbb R}^3$.

We impose Dirichlet homogeneous boundary-conditions, and we

consider vector fields, with fixed period . We show that the semi

group, generated by the Stokes operator, is analytic with respect to the

$L^p$ norm under these boundary conditions.

This property is closely related to an estimate of the

operator's resolvent. For $p=2$, we prove that this resolvent

exists, then that it satisfies the desired estimate. To do this, we use

a decomposition principle of [1], splitting the solenoidal vector fields

of $\mathbb R^3$ into poloidal and toroidal fields and the mean flow.

For larger values of $p$, we show that the estimate, obtained for

$p=2$, extends to larger values of $p$, in view of a result of [2],

valid in a half-space for vectors with prescribed divergence.