On the pseudospectra of Berezin-Toeplitz operators

We estimate the norm of the resolvent of non-selfadjoint Berezin Toeplitz operators in the semi-classical limit, under various assumptions on the Poisson bracket of the real and imaginary parts of the symbol. In case this bracket is negative, we symbolically construct pseudomodes well-localized in phase space. We also show that the numerical range converges to the convex hull of the image of the symbol. Our techniques involve the theory of Hermite distributions and classical subelliptic results of H\"ormander.