Hankel operators on the Bergman space of bounded symmetric domains

Abstract:

Let $\Omega$ be a bounded symmetric domain in ${\mathbb {C}^n}$ with normalized volume measure $dV$. Let $P$ be the orthogonal projection from ${L^2}(\Omega ,dV)$ onto the Bergman space $L_a^2(\Omega )$ of holomorphic functions in ${L^2}(\Omega ,dV)$. Let $\overline P$ be the orthogonal projection from ${L^2}(\Omega ,dV)$ onto the closed subspace of antiholomorphic functions in ${L^2}(\Omega ,dV)$. The "little" Hankel operator ${h_f}$ with symbol $f$ is the operator from $L_a^2(\Omega )$ into ${L^2}(\Omega ,dV)$ defined by ${h_f}g = \overline P (fg)$. We characterize the boundedness, compactness, and membership in the Schatten classes of the Hankel operators ${h_f}$ in terms of a certain integral transform of the symbol $f$. These characterizations are further studied in the special cases of the open unit ball and the poly-disc in ${\mathbb {C}^n}$.