Finite-rank commutators of projections onto shift-invariant subspaces

Abstract

In this article, using Halmos' two projections theorem, we completely characterize finite-rank commutators of orthogonal projections onto shift-invariant subspaces ${\varphi }_1{H}^2(\mathbb{D})$ and ${\varphi }_2{H}^2(\mathbb{D})$ of the Hardy space $H^2(\mathbb{D})$ corresponding to inner functions ${\varphi }_1,{\varphi }_2$ on the unit disc $\mathbb{D}$. Using our methods, we connect the finite-rank commutator with the Fredholmness of the projections $(P_{{\varphi }_1},P_{{\varphi }_2})$ as introduced by Avron, Seiler and Simon. We conclude with several characterizations on the polydisc.

The results on compactness of commutators of projections in the earlier version contained errors, particularly Theorem 2.6 and its consequences. The article has been thoroughly revised, with the errors and corresponding references removed. The present version establishes correct results on finite-rank commutators of projections and compactness of products of projections