  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.
