• Corpus ID: 248377776

Constructions of some families of smooth Cauchy transforms

@inproceedings{Limani2021ConstructionsOS,
  title={Constructions of some families of smooth Cauchy transforms},
  author={Adem Limani and Bartosz Malman},
  year={2021}
}

• Adem Limani, Bartosz Malman
• Published 28 November 2021
• Mathematics

For a given Beurling-Carleson subset E of the unit circle T which has positive Lebesgue measure, we give explicit formulas for measurable functions supported on E such that their Cauchy transforms have smooth extensions from D to T . The existence of such functions has been previously established by Khrushchev in 1978, in non-constructive ways by the use of duality arguments. We construct several particular families of such Cauchy transforms with a few applications in operator and function theory in mind. In one application, we give a new proof of irreducibility of the shift operator on certain Hilbert spaces of functions. In another application, we establish a permanence principle for inner factors under convergence in certain topologies. The applications lead to a self-contained duality proof of the density of smooth functions in a very large class of de Branges-Rovnyak spaces. This extends the previously known approximation results.