Contained in the Hardy space
H2 on the unit disk in the complex plane are certain Hilbert spaces which are
invariant under the adjoint of the shift. One such space ℋ(b) is associated with each
function b in the closed unit ball of H∞. In the special case where b is an inner
function, ℋ(b) is just the subspace of H2 orthogonal to the shift-invariant subspace
bH2. It is proven here that for any functions b1 and b2 in the closed ball of H∞, the
spaces ℋ(b1) and ℋ(b2) are isometrically isomorphic under a multiplication
operator if and only if there is a disk automorphism τ such that b2= τ ∘ b1.
In this case, the multiplicative isomorphism is determined explicitly and
uniquely. This motivates an investigation of multipliers between ℋ(b1) and
ℋ(b2), that is, multiplication operators acting bijectively but not necessarily
isometrically. Restricting to the case where b1 and b2 are inner functions, it is shown
that a multiplier between given spaces is unique up to multiplication by a
nonzero constant, and several theorems are proven concerning the existence
of such multipliers. Finally, consideration is given to the implications of
these results for the characterization of the invariant subspaces in H2 on an
annulus.
