On some spectral properties of the weighted $\overline{\partial}$-Neumann operator

We study necessary conditions for compactness of the weighted ∂¯-Neumann operator on the space L2(Cn,e−φ) for a plurisubharmonic function φ. 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 ∂¯-Neumann operator for decoupled weights, which are of the form φ(z)=φ1(z1)+⋯+φn(zn). More can be said if every Δφj defines a nontrivial doubling measure.