Operators for matrix-valued Riesz bases over LCA groups

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.