In this paper we consider Toepliz operators with (locally) integrable symbols acting on Bergman spaces $A^p$ ($1<p<\infty$) of the open unit disc of the complex plane. We give a characterization of compact Toeplitz operators with symbols in $L^1$ under a mild additional condition. Our result is new even in the Hilbert space setting of $A^2$, where it extends the well-known characterization of compact Toeplitz operators with bounded symbols by Stroethoff and Zheng.
"On compactness of Toeplitz operators in Bergman spaces." Funct. Approx. Comment. Math. 59 (2) 305 - 318, December 2018. https://doi.org/10.7169/facm/1727