Kernels of Toeplitz Operators via Bourgainʹs Factorization Theorem

Using Bourgainʹs factorization theorem, we characterize the subspaces of Hp, 1⩽p⩽∞, that coincide with the kernels of Toeplitz operators. This is related to (but entirely independent of) earlier work of E. Hayashi. One consequence of our characterization is that, for some inner functions θ, the class of uniqueness sets for Kpθ depends on p; here Kpθ=defHp∩θzHp is the star-invariant subspace in Hp generated by θ. Moreover, a construction is provided of Blaschke products B and b for which the dimension function p↦dim(KpB∩bHp) has prescribed jumps at prescribed points.