Commutators of inner projections are compact on the disk Hardy space precisely when a Douglas algebra condition holds; on the bidisc their product is compact only if finite rank, and on higher polydiscs any compact product must be trivial.
Dixmier, Position relative de deux variétés linéaires fermées dans un espace de Hilbert , Revue Scien- tifique 86 (1948), 387–399
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Compactness of products and commutators of inner projections
Commutators of inner projections are compact on the disk Hardy space precisely when a Douglas algebra condition holds; on the bidisc their product is compact only if finite rank, and on higher polydiscs any compact product must be trivial.