Composition preserves rigidity

Say that f in H^p (1 ≤ p ≤ ∞) of the unit disk is rigid if it is determined in H^p by its argument modulo 2 on the unit circle. We show that if f is rigid and u is an inner function, then the composition f o u is rigid. The proof uses the disintegration of Lebesgue measure with respect to u to compute the adjoint of the operator of composition with u. This result is a generalization of the work of Younis, who proved the special case p = 1 using operator theoretic methods.