Cross-Gram Matrix associated to two sequences in Hilbert spaces

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 $\{f_{k}\}_{k=1}^{\infty}$.