Operators for matrix-valued Riesz bases over LCA groups

TL;DR

This paper investigates the conditions under which operators can generate matrix-valued Riesz bases from orthonormal bases in LCA groups, revealing that positivity of certain operators is key to preserving Riesz basis properties.

Contribution

It characterizes classes of operators that produce matrix-valued Riesz bases from orthonormal bases in LCA group settings, extending the understanding of basis transformations.

Findings

Operators that are adjointable and positive map Riesz bases to their duals.

Not all bounded, linear, bijective operators preserve Riesz basis structure.

Positivity of operators is necessary and sufficient for basis duality mapping.

Abstract

The image of a given orthonormal basis for a separable Hilbert space $\mathcal{H}$ under a bijective, bounded, and linear operator acting on $\mathcal{H}$ is called a Riesz basis of $\mathcal{H}$. Contrary to what happens with Riesz bases (in the usual sense) in separable Hilbert spaces, it is not true in general that the image of a matrix-valued orthonormal basis under a bounded, linear, and bijective operator on $L^2(G, \mathbb{C}^{s\times r})$ is also a basis and frame for the space $L^2(G, \mathbb{C}^{s\times r})$, where $G$ is a $\sigma$-compact and metrizable locally compact abelian (LCA) group. We give some classes of operators for the construction of matrix-valued Riesz bases from orthonormal bases of the space $L^2(G, \mathbb{C}^{s\times r})$. Motivated by a result due to Holub, we show that a bounded, linear, and bijective operator acting on $L^2(G, \mathbb{C}^{s\times r})$ which is adjointable with respect to the matrix-valued inner product is positive if and only if it maps a matrix-valued Riesz basis of the space $L^2(G, \mathbb{C}^{s\times r})$ to its dual Riesz basis.