Tangential limits of functions orthogonal to invariant subspaces

Abstract

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