In this paper we discuss the relationship of local spectra of Toeplitz operators on the Bergman space. A sufficient condition is found for the quasicommutator to be compact. Sufficient conditions are given for a class of Toeplitz operators to be Fredholm operators.
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Properties of Locality and Fredholm of Toeplitz Operators on the Bergman Space. Acta Mathematica Sinica, Chinese Series, 1997, 40(5) https://doi.org/10.12386/A1997sxxb0087