On the Dolbeault-Dirac operators on quantum projective spaces

We consider Dolbeault-Dirac operators on quantum projective spaces, following Krahmer and Tucker-Simmons. The main result is an explicit formula for their squares, up to terms in the quantized Levi factor, which can be expressed in terms of some central elements. This computation is completely algebraic. These operators can also be made to act on the corresponding Hilbert spaces. Using the formula mentioned above, we easily find that they have compact resolvent, thus obtaining a result similar to that of D'Andrea and Dabrowski.