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In this paper we study the properties of multiplication invariant (MI)operators acting on subspaces of the vector-valued space $L^2(X;\mathcal H)$.We characterize such operators in terms of range functions by showing thatthere is an isomorphism between the category of MI spaces (with MI operators asmorphisms) and the category of measurable range functions whose morphisms aremeasurable range operators. We investigate how global properties of an MIoperator are reflected by local pointwise properties of its corresponding rangeoperator. We also establish several results about frames generated bymultiplications in $L^2(X;\mathcal H)$. This includes the classification offrames of multiplications with respect to unitary equivalence by measurablefields of Gramians. Finally, we show applications of our results in the studyof abelian group frames and translation-invariant (TI) operators acting onsubspaces of $L^2(\mathcal G)$, where $\mathcal G$ is a locally compact group.

Multiplication-invariant operators and the classification of LCA group frames