  The Gram matrix is defined for Bessel sequences by combining synthesis with
subsequent analysis operators. If different sequences are used and an operator
U is inserted we reach so called U-cross Gram matrices. This can be seen as
reinterpretation of the matrix representation of operators using frames. In
this paper we investigate some necessary or sufficient conditions for Schatten
p-class properties and the invertibility of U-cross Gram matrices. In
particular, we show that under mild conditions the pseudo-inverse of a U-cross
Gram matrix can always be represented as a U-cross Gram matrix with dual frames
of the given ones. We link some properties of U-cross Gram matrices to
approximate duals. Finally, we state several stability results. More precisely,
it is shown that the invertibility of U-cross Gram matrices is preserved under
small perturbations.
