Compact operators on the weighted Bergman space $A^1(\psi )$
Volume 177 / 2006
Studia Mathematica 177 (2006), 277-284
MSC: 47B35, 47A15.
DOI: 10.4064/sm177-3-6
Abstract
We show that a bounded linear operator $S$ on the weighted Bergman space $A^1(\psi )$ is compact and the predual space $A_0(\varphi )$ of $A^1(\psi )$ is invariant under $S^\ast $ if and only if $Sk_z \rightarrow 0$ as $z\rightarrow \partial D$, where $k_z$ is the normalized reproducing kernel of $A^1(\psi )$. As an application, we give conditions for an operator in the Toeplitz algebra to be compact.
