  The conditions for sequences $\{f_{k}\}_{k=1}^{\infty}$ and
$\{g_{k}\}_{k=1}^{\infty}$ being Bessel sequences, frames or Riesz bases, can
be expressed in terms of the so-called cross-Gram matrix. In this paper we
investigate the cross-Gram operator, $G$, associated to the sequence $\{\langle
f_{k}, g_{j}\rangle\}_{j, k=1}^{\infty}$ and sufficient and necessary
conditions for boundedness, invertibility, compactness and positivity of this
operator are determined depending on the associated sequences. We show that
invertibility of $G$ is not possible when the associated sequences are frames
but not Riesz Bases or at most one of them is Riesz basis. In the special case
we prove that $G$ is a positive operator when $\{g_{k}\}_{k=1}^{\infty}$ is the
canonical dual of $\{f_{k}\}_{k=1}^{\infty}$.
