Commutativity of Toeplitz and Hankel Operators on Beurling Subspaces of the Hardy Space
DOI:
https://doi.org/10.56532/mjsat.v6i3.841Keywords:
Toeplitz operators, Hankel operators, Beurling subspaces, Hardy space, CommutativityAbstract
We study the commutativity of Toeplitz and Hankel operators on Beurling subspaces of the Hardy space, allowing bounded symbols with polynomial coanalytic parts. Suppose that the Toeplitz symbol has a nonzero coanalytic part of degree p and that the Hankel symbol has coanalytic degree m. We prove that the commutator vanishes on the monomial Beurling subspace zkH2 if and only if k ≥ p + m, and show that this threshold is sharp. When the Toeplitz symbol is analytic and nonconstant, the corresponding threshold is m. We introduce the Toeplitz–Hankel commutativity depth as the first level of the Beurling chain on which the restricted commutator vanishes, assigning infinite depth when no such level exists. We also determine the rank of the restricted commutator as a function of k and establish an infinite-depth result for nonconstant analytic Toeplitz symbols when the Hankel symbol has infinitely many nonzero negative Fourier coefficients. Finally, under the hypothesis that the kernel of the Hankel operator is generated by a finite Blaschke product, we obtain a sharp vanishing threshold on the corresponding Beurling subspaces.
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