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Approximation and symbolic calculus for Toeplitz algebras on the Bergman space. (English) Zbl 1057.32005

If \(f\in L^{\infty}(D)\), let \(T_f\) be the Toeplitz operator on the Bergman space \( L^2_a\) of the unit disk \(D\). For a \(C^*\) -algebra \(A\subset L^{\infty}(D)\), let \(\Sigma (A)\) denote the closed operator algebra generated by \(\{T_f:f\in A\}\). This paper characterizes its commutator ideal \(C(A)\) and the quotient \(\Sigma(A)/C(A)\) for a wide class of algebras \(A\). For \(n\geq 0\) integer, the \(n\)-Berezin transform of a bounded operator \(S\) is defined by \(B_nS\); this paper also proves that if \(f\in L^{\infty}(D)\) and \(f_n=B_nT_f\) then \(T_{f_n}\to T_f.\)

MSC:

32A36 Bergman spaces of functions in several complex variables
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
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References:

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