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 $L_a^2$ 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 $n\ge 0$ integer, we define the $n$-Berezin transform $B_{n}S$ of a bounded operator $S$, and prove that if $f\in L^{\infty }(\mathbb{D})$ and $f_n=B_n{T}_f$ then $T_{f_n}\to T_f$.