Cross-Gram Matrix Associated to Two Sequences in Hilbert Spaces

Abstract

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