The conditions for sequences ffkg1 k¼1 and fgkg1 k¼1 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 fhfk; gjig1 j;k¼1 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 fgkg1 k¼1 is the canonical dual of ffkg1 k¼1.
