Beurling's classical theorem gives a complete characterization of all invariant subspaces in the Hardy space H²(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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm149-2-2,
author = {Kunyu Guo},
title = {Podal subspaces on the unit polydisk},
journal = {Studia Mathematica},
volume = {151},
year = {2002},
pages = {109-120},
zbl = {1018.46028},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm149-2-2}
}
Kunyu Guo. Podal subspaces on the unit polydisk. Studia Mathematica, Tome 151 (2002) pp. 109-120. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm149-2-2/