Commutativity of Toeplitz Operators with Trigonometric Polynomial Symbols on Beurling Subspaces of Hardy–Sobolev Spaces

We investigate the commutativity of Toeplitz operators with trigonometric polynomial symbols on the Beurling subspaces zkHs2 of the Hardy–Sobolev spaces. An explicit formula for the commutator of two Toeplitz operators is first derived in terms of weighted shift operators. As a consequence, every such commutator is shown to have finite rank. It is then proved that if the degrees of the symbols are at most k, then the corresponding Toeplitz operators commute on the Beurling subspace zkHs2. Moreover, this result is shown to be sharp by proving that, in general, the Beurling subspace zkHs2 cannot be replaced by zk−1Hs2. Finally, these results generalize the corresponding commutativity theorem for Toeplitz operators on the classical Hardy space.