The Bochner identities for the K\"ahlerian gradients

We discuss algebraic properties for the symbols of geometric first order differential operators on almost Hermitian manifolds and K\"ahler manifolds. Through study on the universal enveloping algebra and higher Casimir elements, we know algebraic relations for the symbols like the Clifford algebra. From the relations, we have all the Bochner identities for the operators. As applications, we have vanishing theorems, the Bochner-Weitzenb\"ock formula, and eigenvalue estimates for the operators on K\"ahler manifolds.