Absence of L-infinity-contractivity of semi groups associated with complexelliptic divergence operators

Consider a second order elliptic operator with complex coefficients on an o pen set of R-N. We obtain necessary and sufficient conditions on the coeffi cients for the contractivity in L-infinity of the semigroup defined under d ifferent boundary conditions. In particular, we show that this property is closely related to the fact that the coefficients are actually real valued.