Compactness of products and commutators of inner projections

TL;DR

This paper characterizes the conditions under which products and commutators of inner projections are compact on Hardy spaces, revealing a rigidity phenomenon in multivariable cases.

Contribution

It provides a complete characterization for the single-variable case and uncovers a rigidity phenomenon in multivariable settings regarding compactness.

Findings

The commutator of two inner projections is compact iff characterized by Douglas algebra in single-variable case.

On the bidisc, the product of two inner projections is compact iff it has finite rank.

On polydiscs of dimension greater than two, any compact product of inner projections is trivial.

Abstract

In this paper, we study the compactness of the product and the commutator of two inner projections on the Hardy spaces over the unit disk and the polydisc. For the single-variable case, we provide a complete characterization of the compactness of the commutator of two inner projections by means of Douglas algebra. In the multivariable setting, we discover a rigidity phenomenon: on the bidisc, the product of two inner projections is compact if and only if it has finite rank, whereas on the polydisc of dimension strictly greater than two, any such compact product must be trivial.