U-cross Gram matrices and their invertibility

TL;DR

This paper studies U-cross Gram matrices, exploring their invertibility, Schatten class properties, and stability under perturbations, with implications for operator representation using frames.

Contribution

It introduces conditions for invertibility and Schatten class membership of U-cross Gram matrices and links their properties to approximate duals and stability analysis.

Findings

Pseudo-inverse of U-cross Gram matrices can be represented as U-cross Gram matrices with dual frames.

Invertibility is preserved under small perturbations.

Conditions for Schatten p-class properties are established.

Abstract

The Gram matrix is defined for Bessel sequences by combining synthesis with subsequent analysis operators. If different sequences are used and an operator U is inserted we reach so called U-cross Gram matrices. This can be seen as reinterpretation of the matrix representation of operators using frames. In this paper we investigate some necessary or sufficient conditions for Schatten p-class properties and the invertibility of U-cross Gram matrices. In particular, we show that under mild conditions the pseudo-inverse of a U-cross Gram matrix can always be represented as a U-cross Gram matrix with dual frames of the given ones. We link some properties of U-cross Gram matrices to approximate duals. Finally, we state several stability results. More precisely, it is shown that the invertibility of U-cross Gram matrices is preserved under small perturbations.