On some class of integral operators related to the Bergman projection

Djordjije Vujadinovic

Abstract: We consider the integral operator $$ C_\alpha f(z)=\int_D\frac{f(\xi)}{(1-z\bar{\xi})^{\alpha}} dA(\xi),\quad z\in D, $$ where $0<\alpha<2$ and $D$ is the unit disc in the complex plane. and investigate boundedness of it on the space $L^p(D,d\lambda)$, $1<p<\infty$, where $d\lambda$ is the Möbius invariant measure in $D$. We also consider the spectral properties of $C_\alpha$ when it acts on the Hilbert space $L^2(D,d\lambda)$, i.e., in the case $p=2$, when $C_\alpha$ maps $L^2(D,d\lambda)$ into the Dirichlet space.

Keywords: Bergman projection; singular numbers of a compact operator

Classification (MSC2000): 46E15; 46E20

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