Revista Matemática Iberoamericana


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Volume 20, Issue 2, 2004, pp. 563–610
DOI: 10.4171/RMI/401

Approximation and symbolic calculus for Toeplitz algebras on the Bergman space

Daniel Suárez[1]

(1) Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, BELLATERRA, SPAIN

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 $\comm(A)$ 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 $T_{f_n} \rightarrow T_f$.

Keywords: Bergman space, Toeplitz operator, commutator ideal and abelianization

Suárez Daniel: Approximation and symbolic calculus for Toeplitz algebras on the Bergman space. Rev. Mat. Iberoamericana 20 (2004), 563-610. doi: 10.4171/RMI/401