Multipliers between invariant subspaces of the backward shift

Abstract

Contained in the Hardy space H2 on the unit disk in the complex plane are certain Hubert spaces which are invariant under the adjoint of the shift. One such space H(b) is associated with each function in the closed unit ball of H°°. In the special case where is an inner function, 7ί(b) is just the subspace of H2 orthogonal to the shift-invariant subspace bH2. It is proven here that for any functions b and b2 in the closed ball of H°°, the spaces U(bι) and 7ί(62) are isometrically isomorphic under a multiplication operator if and only if there is a disk automorphism r such that 62 = τobi. In this case, the multiplicative isomorphism is determined explicitly and uniquely. This motivates an investigation of multipliers between Ή,(b\) and 9^(62)9 that is, multiplication operators acting bijectively but not necessarily isometrically. Restricting to the case where b and 62 ar e 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.