On the Friedrichs operator

Abstract:

Let $\Omega$ be a simply connected domain in ${\mathbb {C}^1}$ with the area measure dA. Let ${\bar P_\Omega }$ be the orthogonal projection from ${L^2}(\Omega ,dA)$ onto the closed subspace of antiholomorphic functions in ${L^2}(\Omega ,dA)$. The Friedrichs operator ${\bar T_\Omega }$ associated to $\Omega$ is the operator from the Bergman space $L_a^2(\Omega )$ into ${L^2}(\Omega ,dA)$ defined by ${\bar T_\Omega }f = {\bar P_\Omega }f$. In this note, some smoothness conditions on the boundary of $\Omega$ are given such that the Friedrichs operator ${\bar T_\Omega }$ belongs to the Schatten classes ${S_p}$.