The essential norm of operators in the Toeplitz algebra on $A^p(B_n)$ Daniel Suarez 32A3647B35Bergman spacesToeplitz algebraessential normBerezin transfom Let $A^p$ be the Bergman space on the unit ball $\mathbb{B}_n$ of $\mathbb{C}^n$ for $1 < p < \infty$, and $\mathfrak{T}_p$ be the corresponding Toeplitz algebra. We show that every $S \in \mathfrak{T}_p$ 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 $S \in \mathfrak{T}_p$, an estimate for the essential spectral radius of $S \in \mathfrak{T}_2$, and a localization result for its essential spectrum. Finally, we characterize compactness in terms of the Berezin transform for operators in $\mathfrak{T}_p$. Indiana University Mathematics Journal 2007 text pdf 10.1512/iumj.2007.56.3095 10.1512/iumj.2007.56.3095 en Indiana Univ. Math. J. 56 (2007) 2185 - 2232 state-of-the-art mathematics http://iumj.org/access/