This note is a study of unitary
equivalence of invariant subspaces of H2 of the polydisk. By definition, this means
joint unitary equivalence of the shift operators restricted to the invariant
subspaces.
In one variable, all invariant subspaces are unitarily equivalent and all can be
represented as inner functions times H2. In several variables, our results suggest that
unitary equivalence and multiplication by inner functions are again related. For
example, all invariant subspaces of a given invariant subspace ℳ which are unitarily
equivalent to ℳ are φℳ, for φ inner; and all invariant subspaces unitarily equivalent
to an invariant subspace ℳ of finite codimension are φℳ. In particular, two
invariant subspaces of finite codimension are unitarily equivalent if and only if they
are equal.
