Constructions of some families of smooth Cauchy transforms

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.Comment: This is an updated version of a previous preprint, with some topics added in and some topics moved to a different preprint. Any comments from the community would be highly appreciate