The invertibility of U-fusion cross Gram matrices of operators

TL;DR

This paper explores the properties and invertibility conditions of U-fusion cross Gram matrices of operators, extending the theory of fusion frames for operator representation and analysis.

Contribution

It introduces the concept of U-fusion cross Gram matrices, provides invertibility criteria, and characterizes fusion Riesz and orthonormal bases through these matrices.

Findings

Sufficient conditions for invertibility of U-fusion cross Gram matrices.

Explicit formulas for the inverse of these matrices.

Characterization of fusion Riesz and orthonormal bases via Gram matrices.

Abstract

For applications like the numerical solution of physical equations a discretization scheme for operators is necessary. Recently frames have been used for such an operator representation. In this paper, we apply fusion frames for this task. We interpret the operator representation using fusion frames as a generalization of fusion Gram matrices. We present the basic definition of $U$-fusion cross Gram matrices of operators for a bounded operator $U$. We give sufficient conditions for their (pseudo-)invertibility and present explicit formulas for the inverse. In particular, we characterize fusion Riesz bases and fusion orthonormal bases by such matrices. Finally, we look at which perturbations of fusion Bessel sequences preserve the invertibility of the fusion Gram matrix of operators.