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].