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

Cross-Gram Matrix associated to two sequences in Hilbert spaces