  In this paper, a positive operator is given. It is shown that the product of
this positive operator and the convolution operator is a trace class
Hilbert-Schmidt integral operator and has nonnegative eigenvalues. A formula is
given for the trace of this product operator. It seems that this product
operator is the closest trace class integral operator which has nonnegative
eigenvalues and is related to the Weil distribution. A relation is given
between the trace of the product operator and the Weil distribution.
