Podal subspaces on the unit polydisk
Beurling's classical theorem gives a complete characterization of all invariant subspaces in the Hardy space $H^2(D)$. To generalize the theorem to higher dimensions, one is naturally led to determining the structure of each unitary equivalence (resp. similarity) class. This, in turn, requires finding podal (resp. s-podal) points in unitary (resp. similarity) orbits. In this note, we find that H-outer (resp. G-outer) functions play an important role in finding podal (resp. s-podal) points. By the methods developed in this note, we can assess when a unitary (resp. similarity) orbit contains a podal (resp. an s-podal) point, and hence provide examples of orbits without such points.