Spectral properties of the Cauchy transform on modified Bergman spaces
Khaled Chbichib, Noureddine Ghiloufi, Safa Snoun
TL;DR
This work analyzes the spectral properties of the Cauchy transform on modified Bergman spaces ${\mathcal A}^2(\mathbb D,\mu_{\alpha,\beta})$ by studying the operators $T_{\alpha,\beta}=\mathcal C\mathbb P_{\alpha,\beta}$ and $R_{\alpha,\beta}=\mathbb P_{\alpha,\beta}\mathcal C\mathbb P_{\alpha,\beta}$. It derives explicit singular-value formulas for both operators in terms of Beta, Gamma, and hypergeometric structures, and establishes sharp asymptotics: $s_n^2(T_{\alpha,\beta}) \sim C_T\,n^{-2\alpha-2}$ and $s_n(R_{\alpha,\beta}) \sim C_R\,n^{-\alpha-1}$ as $n\to\infty$, with $n^{\alpha+1}s_n(\cdot)$ convergent. The analysis leverages reproducing kernels, orthonormal bases, and Mellin-transform techniques to connect spectral data to hypergeometric series via the incomplete beta function $h$ and its moments. The results extend prior Bergman-space spectral studies to the modified measure setting, provide precise compactness and Schatten-class conclusions, and highlight potential applications to magnetic Schrödinger-type operators in quantum mechanics. Overall, the paper delivers explicit, computable spectral information for these two natural compressions of the Cauchy transform in a weighted holomorphic setting.
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 $n\to+\infty$.