A note on matrix-valued orthonormal bases over LCA groups

It is well known that orthonormal bases for a separable Hilbert space [Formula: see text] are precisely collections of the form [Formula: see text], where [Formula: see text] is a linear unitary operator acting on [Formula: see text] and [Formula: see text] is a given orthonormal basis for [Formula: see text]. We show that this is not true for the matrix-valued signal space [Formula: see text], [Formula: see text] is a locally compact abelian group which is [Formula: see text]-compact and metrizable, and [Formula: see text] and [Formula: see text] are positive integers. This problem is related to the adjointability of bounded linear operators on [Formula: see text]. We show that any orthonormal basis of the space [Formula: see text] is precisely of the form [Formula: see text], where [Formula: see text] is a linear unitary operator acting on [Formula: see text] which is adjointable with respect to the matrix-valued inner product and [Formula: see text] is a matrix-valued orthonormal basis for [Formula: see text].