Optimal Remainder Estimates in the Quantization of Complex Projective Spaces

We study Berezin-Toeplitz quantization of complex projective spaces ${\mathbb{C}\mathbb{P}}^{d-1}$ and obtain full asymptotic expansions of the Berezin transformation and of products of Toeplitz operators. In each case, the remainder is controlled by the next term of the expansion, either through a positivity-preserving transformation or via an operator inequality. This leads to bounds which are optimal in terms of the required regularity and feature sharp or asymptotically sharp constants.