A note on matrix-valued orthonormal bases over LCA groups

It is well known that orthonormal bases for a separable Hilbert space H are precisely collections of the form {Theta e(k)}(k is an element of I), where Theta is a linear unitary operator acting on H and {e(k)}(k subset of I) is a given orthonormal basis for H. We show that this is not true for the matrix-valued signal space L-2(G, C-sxr), G is a locally compact abelian group which is s-compact and metrizable, and s and r are positive integers. This problem is related to the adjointability of bounded linear operators on L-2(G, C-sxr). We show that any orthonormal basis of the space L-2(G, C-sxr) is precisely of the form {UEk}(k is an element of I), where U is a linear unitary operator acting on L-2(G, C-sxr) which is adjointable with respect to the matrix-valued inner product and {E-k}(k is an element of I) is a matrix-valued orthonormal basis for L-2(G, C-sxr).