Revista Matemática Iberoamericana

Approximation and symbolic calculus for Toeplitz algebras on the Bergman space

Daniel Suárez

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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 $T_{f_n} \rightarrow T_f$.

Article information

Rev. Mat. Iberoamericana, Volume 20, Number 2 (2004), 563-610.

First available in Project Euclid: 17 June 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32A36: Bergman spaces
Secondary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15]

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), no. 2, 563--610.

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