An application of the supremum cosine angle between multiplication invariant spaces in \(\varvec{L^2(X; \mathcal H)}\)

In this article, we study the supremum cosine angle between two multiplication invariant (MI) spaces and its connection with the closedness of the sum of those spaces. The results obtained for MI spaces are preserved almost everywhere by the corresponding fiber spaces. Also, we provide equivalent conditions for the injectivity of the sampling operator associated with the multiplication generated Bessel system for the union of the sum of finitely generated MI spaces. Employing the Zak transform, we obtain the results for translation invariant spaces on locally compact groups by the action of its closed abelian subgroup as an application.