On some spectral properties of the weighted ∂¯-Neumann operator

Abstract

We study necessary conditions for compactness of the weighted $\overline{\partial }$-Neumann operator on the space $L^2({\mathbb{C}}^n,e^{-\phi })$ for a plurisubharmonic function $\phi$. Under the assumption that the corresponding weighted Bergman space of entire functions has infinite dimension, a weaker result is obtained by simpler methods. Moreover, we investigate (non)compactness of the $\overline{\partial }$-Neumann operator for decoupled weights, which are of the form $\phi (z)={\phi }_1(z_1)+\cdots +{\phi }_n(z_n)$. More can be said if every $\Delta {\phi }_j$ defines a nontrivial doubling measure.