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 ${\{\mathrm{\Theta }{e}_k\}}_{k\in 𝕀}$, where $\mathrm{\Theta }$ is a linear unitary operator acting on $H$ and ${\{e_k\}}_{k\in 𝕀}$ is a given orthonormal basis for $H$. We show that this is not true for the matrix-valued signal space $L^2(G,{ℂ}^{s\times r})$, $G$ is a locally compact abelian group which is $\sigma$-compact and metrizable, and $s$ and $r$ are positive integers. This problem is related to the adjointability of bounded linear operators on $L^2(G,{ℂ}^{s\times r})$. We show that any orthonormal basis of the space $L^2(G,{ℂ}^{s\times r})$ is precisely of the form ${\{{𝒰}{E}_k\}}_{k\in 𝕀}$, where ${𝒰}$ is a linear unitary operator acting on $L^2(G,{ℂ}^{s\times r})$ which is adjointable with respect to the matrix-valued inner product and ${\{E_k\}}_{k\in 𝕀}$ is a matrix-valued orthonormal basis for $L^2(G,{ℂ}^{s\times r})$.