= Solution
Restricting $T^*T=TT^*$ to $V_i$, using (i), gives $T_i^*T_i=T_iT_i^*$, so the global <normal operator> property implies the <normal operator> property of every restriction. Conversely, if the restrictions are normal and the <adjoint operator>[adjoint] preserves each summand, then for $x=\sum_i x_i$,
$$
(T^*T-TT^*)x=\sum_i(T_i^*T_i-T_iT_i^*)x_i=0.
$$
Thus $\boxed{T\text{ is normal}\iff\text{every }T_i\text{ is normal}}$, with condition (i) automatic under the stated decomposition.
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