We discuss upper bounds on the exponentially rapid resolvent growth of quadratic differential operators under a hypothesis of partial ellipticity. We also obtain much more precise information on the norms of spectral projections for these operators, and discuss how both these quantities are connected to the geometric properties of anisotropic weights for Bargmann-Fock-type spaces. Examples include the complex harmonic oscillator studied by Davies and Kramers-Fokker-Planck operators with quadratic potentials.
