Wigner weight functions and Weyl symbols of non-negative definite linear operators

In this paper we present several necessary and, for radially symmetric functions, necessary and sufficient conditions for a function of two variables to be a Wigner weight function (Weyl symbol of a non-negative definite linear operator of L²(R)). These necessary conditions are in terms of spread and p-norms of the weight functions. We also conjecture an entropy inequality for such weight functions. It is shown by examples that none of the necessary conditions is consistently weaker or stronger than any of the others. Hence each condition represents a particular feature of Wigner weight functions.