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

On the Dolbeault-Dirac operators on quantum projective spaces