Cross-Gram Matrix Associated to Two Sequences in Hilbert Spaces

The conditions for sequences { f k } k = 1 ∞ and { g k } k = 1 ∞ 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 { ⟨ f k , g j ⟩ } j , k = 1 ∞ 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 ∞ is the canonical dual of { f k } k = 1 ∞ .