Approximation and symbolic calculus for Toeplitz algebras on the Bergman space

Abstract

If f\in L^\infty(\mathbb{D}) let T_f be the Toeplitz operator on the Bergman space L^2_a of the unit disk \mathbb{D} . For a C^\ast -algebra A\subset L^\infty(\mathbb{D}) let \mathfrak{T}(A) denote the closed operator algebra generated by \{ T_f : f\in A \} . We characterize its commutator ideal \mathfrak C(A) and the quotient \mathfrak{T}(A)/ \mathfrak{C}(A) for a wide class of algebras A . Also, for n\geq 0 integer, we define the n -Berezin transform B_nS of a bounded operator S , and prove that if f\in L^\infty(\mathbb{D}) and f_n = B_n T_f then T_{f_n} \rightarrow T_f .