If 𝒦′ is an Hilbert space and ℳ
a subspace which is invariant under a unilateral shift S on 𝒦 one can ask
when a bounded operator T on ℳ which commutes with S can be extended
to a bounded operator on all of 𝒦 which also commutes with S. Here this
problem is considered in the special case that 𝒦 is a Hardy space H2 of
functions analytic in the unit-disk with values in a finite dimensional Hilbert
space. For this situation an easily derived necessary condition is shown to
be sufficient. Further those ℳ for which the extension to 𝒦′ is unique are
characterized.
