The paper completely characterizes essential commutativity of projections P_φ1 and P_φ2 onto φ1 H²(𝔻) and φ2 H²(𝔻) via local conditions on inner functions φ1 and φ2, with extensions to finite-rank commutators, Fredholm pairs, and applications to truncated Toeplitz operators and the polydisc.
Compactness of products and commutators of inner projections
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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.
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2026 1verdicts
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Essentially commuting projections onto shift-invariant subspaces
The paper completely characterizes essential commutativity of projections P_φ1 and P_φ2 onto φ1 H²(𝔻) and φ2 H²(𝔻) via local conditions on inner functions φ1 and φ2, with extensions to finite-rank commutators, Fredholm pairs, and applications to truncated Toeplitz operators and the polydisc.