Tangential Limits of Functions Orthogonal to Invariant Subspaces

Abstract

For any inner function $\varphi$, let ${M^ \bot }$ be the orthogonal complement of $\varphi {H^2}$, in ${H^2}$, where ${H^2}$ is the usual Hardy space. The relationship between the tangential convergence of all functions in ${M^ \bot }$ and the finiteness of certain sums and integrals involving $\varphi$ is studied. In particular, it is shown that the tangential convergence of all functions in ${M^ \bot }$ is a stronger condition than the tangential convergence of $\varphi$, itself.