Splitting Invariant Subspaces in the Hardy Space Over the Bidisk

Abstract

Let $H^{2}$ be the Hardy space over the bidisk. It is known that Hilbert–Schmidt invariant subspaces of $H^{2}$ have nice properties. An invariant subspace which is unitarily equivalent to some invariant subspace whose continuous spectrum does not coincide with $\overline{\mathbb{D}}$ is Hilbert–Schmidt. We shall introduce the concept of splittingness for invariant subspaces and prove that they are Hilbert–Schmidt.

A Brief Survey of Operator Theory in H 2(𝔻2)

“…For Rudin's Example 2.3 this was proved in [76] with a condition. For the so-called splitting submodules, the fact is proved in [49].…”

Hilbert-Schmidtness of some finitely generated submodules in H2(D2)

“…Hilbert-Schmidt submodules have many good properties and have been studied extensively in the literature, see e.g. [8,[16][17][18][19] and the references therein. In particular, it was shown in [19] that C 2 is unitarily equivalent to…”