When the Berezin transform fails to detect compactness: Toeplitz operators with $L^1$ symbols on weighted Bergman spaces

Abstract

For every $n\geq1$ and $\gamma>-1$, we construct $f\in L^1(\mathbb B_n,dv_\gamma)$ whose Toeplitz form extends to a bounded, noncompact operator on the weighted Bergman spaces $A^2_\gamma(\mathbb B_n)$ and whose Berezin transform vanishes at the boundary. Boundary vanishing of the Berezin transform therefore does not imply compactness for Toeplitz operators with integrable symbols, in contrast with operators in the Toeplitz algebra generated by bounded symbols; this answers a question of Bauer and Isralowitz. The construction approximates rank-one operators in norm by Toeplitz operators with smooth, compactly supported symbols and transports suitably separated blocks toward the boundary. On the unweighted disk, a diagonal version produces a real symbol with the same vanishing and noncompactness properties, for which the operator is positive and Zorboska's two-sided localization condition holds at $p=3$. Her hypothesis $p>3$ is shown to be sharp.