Toeplitz operators on Bergman spaces (Q1822027)

The author studied Toeplitz operators on Bergman spaces. In particular, he showed that if \(\phi \in H^{\infty}(D)+C(\bar D)\) such that \(\phi^*\) is locally sectorial, then the Toeplitz operator \(T_{\phi}\) defined on the Bergman space \(A^ 2(D)\) is Fredholm. Also, it is proved that if S is an operator on \(A^ 2(D)\) which commutes with the Toeplitz operator \(T_{\phi}\) whose symbol \(\phi\) is a finite Blaschke product, then \(SH^{\infty}(D)\) is contained in \(H^{\infty}(D)\). Moreover, some spectral properties of Toeplitz operators are discussed, and it is shown that the spectrum of a class of Toeplitz operators defined on the Bergman space \(A^ 2(D)\) is not connected. An example of a quasinilpotent Toeplitz operator was given.