Constructions of some families of smooth Cauchy transforms

TL;DR

This paper provides explicit formulas for smooth Cauchy transforms supported on Beurling-Carleson sets with positive measure, with applications to operator theory and function space density results.

Contribution

It constructs explicit families of smooth Cauchy transforms supported on certain sets, advancing previous non-constructive existence proofs and applying them to operator and function theory.

Findings

Explicit formulas for smooth Cauchy transforms on Beurling-Carleson sets.

New proof of shift operator irreducibility in specific Hilbert spaces.

Extended density results for smooth functions in de Branges-Rovnyak spaces.

Abstract

For a given Beurling-Carleson subset $E$ of the unit circle $\mathbb{T}$ which has positive Lebesgue measure, we give explicit formulas for measurable functions supported on $E$ such that their Cauchy transforms have smooth extensions from $\mathbb{D}$ to $\mathbb{T}$. 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.