  In a separable Hilbert space, we study supercontractivity and
ultracontractivity properties for a transition semigroups associated with a
stochastic partial differential equations. This is done in terms of exponential
integrability of Lipschitz functions and some logarithmic Sobolev-type
inequalities with respect to invariant measures. The abstract characterization
results concerning the improving of summability can be applied to transition
semigroups associated to a stochastic reaction-diffusion equations.
