  We study non-elliptic quadratic differential operators. Quadratic
differential operators are non-selfadjoint operators defined in the Weyl
quantization by complex-valued quadratic symbols. When the real part of their
Weyl symbols is a non-positive quadratic form, we point out the existence of a
particular linear subspace in the phase space, intrinsically associated to the
Weyl symbols, called a singular space, such that when the singular space has a
symplectic structure, the associated heat semigroup is smoothing in every
direction of its symplectic orthogonal complement. When the Weyl symbol of such
an operator is elliptic on the singular space, this space is always symplectic
and we prove that the spectrum of the operator is discrete and can be described
as in the globally elliptic case. We also describe the large time behavior of
contraction semigroups generated by these operators.
