  We study the contraction semigroups of elliptic quadratic differential
operators. Elliptic quadratic differential operators are the non-selfadjoint
operators defined in the Weyl quantization by complex-valued elliptic quadratic
symbols. We establish in this paper that under the assumption of ellipticity,
as soon as the real part of their Weyl symbols is a non-zero non-positive
quadratic form, the norm of contraction semigroups generated by these operators
decays exponentially in time.
