Spectral properties of the Cauchy transform on modified Bergman spaces

Abstract

In this paper, we determine the singular values $s_n(T_{α,β})$ and $s_n(R_{α,β})$ of the operators $T_{α,β}=\mathcal C\mathbb P_{α,β}$ and $R_{α,β}=\mathbb P_{α,β}\mathcal C\mathbb P_{α,β}$ where $\mathcal C$ is the integral Cauchy transform and $\mathbb P_{α,β}$ is the orthogonal projection from $L^2(\mathbb D,μ_{α,β})$ onto the modified Bergman space $\mathcal A^2(\mathbb D,μ_{α,β})$. These singular values will be expressed in terms of some series involving hypergeometric functions. We show that in both cases the sequence $n^{α+1}s_n(.)$ has a finite limit as $n\to+\infty$.