This paper considers the decentralized control of a nested linear time-varying (LTV) discrete-time system using a nested LTV controller for an ℓ2-induced norm performance criteria. An operator theoretic approach is used to obtain exact conditions for the feasibility of structured linear controller synthesis. These include conditions for the completion of scaling matrices obtained through an updated matrix completion lemma. When individual controller dimensions are chosen at least as large as the plant, these conditions reduce to convex inequalities coupled over time. The completion lemma also provides an algorithm to complete the closed loop scaling matrices leading to inequalities for controller synthesis.