Revista Matemática Iberoamericana

Approximation and symbolic calculus for Toeplitz algebras on the Bergman space

Daniel Suárez

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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 $\comm(A)$ 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

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

Dates
First available in Project Euclid: 17 June 2004

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1087482027

Mathematical Reviews number (MathSciNet)
MR2073132

Zentralblatt MATH identifier
1057.32005

Subjects
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]

Keywords
Bergman space Toeplitz operator commutator ideal and abelianization

Citation

Suárez, Daniel. Approximation and symbolic calculus for Toeplitz algebras on the Bergman space. Rev. Mat. Iberoamericana 20 (2004), no. 2, 563--610. https://projecteuclid.org/euclid.rmi/1087482027


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