JournalsrmiVol. 20, No. 2pp. 563–610

Approximation and symbolic calculus for Toeplitz algebras on the Bergman space

  • Daniel Suárez

    Universitat Autònoma de Barcelona, Bellaterra, Spain
Approximation and symbolic calculus for Toeplitz algebras on the Bergman space cover
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Abstract

If fL(D)f\in L^\infty(\mathbb{D}) let TfT_f be the Toeplitz operator on the Bergman space La2L^2_a of the unit disk D\mathbb{D}. For a CC^\ast-algebra AL(D)A\subset L^\infty(\mathbb{D}) let T(A)\mathfrak{T}(A) denote the closed operator algebra generated by {Tf:fA}\{ T_f : f\in A \}. We characterize its commutator ideal \comm(A)\comm(A) and the quotient T(A)/C(A)\mathfrak{T}(A)/ \mathfrak{C}(A) for a wide class of algebras AA. Also, for n0n\geq 0 integer, we define the nn-Berezin transform BnSB_nS of a bounded operator SS, and prove that if fL(D)f\in L^\infty(\mathbb{D}) and fn=BnTff_n = B_n T_f then TfnTfT_{f_n} \rightarrow T_f.

Cite this article

Daniel Suárez, Approximation and symbolic calculus for Toeplitz algebras on the Bergman space. Rev. Mat. Iberoam. 20 (2004), no. 2, pp. 563–610

DOI 10.4171/RMI/401