On Some Class of Integral Operators Related to the Bergman Projection

Djordjije Vujadinović

We consider the integral operator $$ C_lpha f(z)=ıt_D\frac{f(\xi)}{(1-z\bar{\xi})^{lpha}}\,dA(\xi),\quad zı D, $$ 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.