On matrix-valued Riesz bases over LCA groups

A Riesz basis for a separable Hilbert space Formula: see text is the image of an orthonormal basis under a bounded, linear and bijective operator acting on Formula: see text. Equivalently, it is an exact frame that shares the properties of a basis for Formula: see text. Let Formula: see text be a Formula: see text-compact and metrizable locally compact abelian group, and Formula: see text and Formula: see text be positive integers. It is illustrated that the image of a matrix-valued orthonormal basis under a bijective, bounded and linear operator acting on the matrix-valued signal space Formula: see text may not be a frame, hence not a basis of the space Formula: see text. We introduce a notion of matrix-valued Riesz basis in the space Formula: see text, where the adjointability of a bounded linear operator in the definition of Riesz basis with respect to the matrix-valued inner product plays a crucial role. We establish the existence of matrix-valued Riesz bases of the space Formula: see text. Extending results for standard Riesz bases of separable Hilbert spaces, we give necessary and sufficient conditions, and a characterization of matrix-valued Riesz bases of the space Formula: see text.