On some spectral properties of the weighted ∂¯-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 ∆φj defines a nontrivial doubling measure.