Compact Toeplitz operators for weighted Bergman spaces on bounded symmetric domains
For an irreducible bounded symmetric domain \(\Omega \) in \({\mathbb C}^d\) the author studies Toeplitz operators \(T_f^{\nu }\) acting on the weighted Bergman space \(H_{\nu } ^2(\Omega )\) with weight \(\nu \), i.e. the space of holomorphic functions in \(L^2(\Omega ,\mu _{\nu })\). The symbol \(f\) is a bounded function on \(\Omega \) and \(T_f^{\nu }=P_{\nu }M_f\), where \(M_f\) is the multiplication by \(f\) and \(P_{\nu }\) the orthogonal projection onto \(H_{\nu } ^2(\Omega )\). The first result is about a comparison of the norm \(\|T_f^{\nu }\|\) with the \(L^{\infty }(\Omega )\)-norm of the Berezin transform of \(f\). As a consequence it is proven that, for \(\nu \) large enough, the compactness of the operator \(T_f^{\nu }\) is independent of the weight \(\nu \).
scientific article; zbMATH DE number 5969179