Spectral properties of the Cauchy transform on modified Bergman spaces

Abstract

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