2023/08/12 by Malman, Bartosz · 1 citation
#Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2308.06495
This article consists of two connected parts. In the first part, we study the shift invariant subspaces in certain P2(μ)-spaces, which are the closures of analytic polynomials in the Lebesgue spaces L2(μ) defined by a class of measures μ living on the closed unit disk \mathbbD. The measures μ which occur in our study have a part on the open disk \mathbbD which is radial and decreases at least exponentially fast near the boundary. Our focus is on those shift invariant subspaces which are generated by a bounded function in H^∞. In this context, our results are definitive. We give a characterization of the cyclic singular inner functions by an explicit and readily verifiable condition, and we establish certain permanence properties of non-cyclic ones which are important in the applications. The applications take up the second part of the article. We prove that if a function g ∈ L1(\mathbbT) on the unit circle \mathbbT has a Cauchy transform with Taylor coefficients of order O(exp(-c √(n))) for some c > 0, then the set U = \x ∈ \mathbbT : |g(x)| > 0 \ is essentially open and log |g| is locally integrable on U. We establish also a simple characterization of analytic functions b: \mathbbD → \mathbbD with the property that the de Branges-Rovnyak space H(b) contains a dense subset of functions which, in a sense, just barely fail to have an analytic continuation to a disk of radius larger than 1. We indicate how close our results are to being optimal and pose a few questions.