The essential norm of operators in the toeplitz algebra on A ^p(double-struck B sign<inf>n</inf>)

Let Ap be the Bergman space on the unit ball double-struck B signn of ℂn for 1 < p < ∞, and T-fraktur signP be the corresponding Toeplitz algebra. We show that every S ∈T-fraktur signp can be approximated by operators that are specially suited for the study of local behavior. This is used to obtain several estimates for the essential norm of S ∈T-fraktur signp, an estimate for the essential spectral radius of S ∈T-fraktur sign p, and a localization result for its essential spectrum. Finally, we characterize compactness in terms of the Berezin transform for operators in T-fraktur signP. Indiana University Mathematics Journal ©,.