Cross-Gram Matrix associated to two sequences in Hilbert spaces

TL;DR

This paper studies the properties of the cross-Gram operator associated with two sequences in Hilbert spaces, providing conditions for its boundedness, invertibility, and positivity, and exploring its behavior in relation to frames and Riesz bases.

Contribution

It offers new necessary and sufficient conditions for the boundedness, invertibility, and positivity of the cross-Gram operator based on the sequences' properties.

Findings

Invertibility of G is not possible for two frames unless one is a Riesz basis.

G is positive when one sequence is the canonical dual of the other.

Conditions for boundedness and compactness of G are established.

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