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 $n\to +\infty$.

16 pages, 2 figures