More Differential Operators on Almost Hermitian Manifolds

In 1980 Michelsohn defined a differential operator on sections of the complex Clifford bundle over a compact K\"ahler manifold M . This operator is a differential and its Laplacian agrees with the Laplacian of the Dolbeault operator on forms through a natural identification of differential forms with sections of the Clifford bundle. Relaxing the condition that M be K\"ahler, we introduce two differential operators on sections of the complex Clifford bundle over a compact almost Hermitian manifold which naturally generalize the one introduced by Michelsohn. We show surprising K\"ahler-like symmetries of the kernel of the Laplacians of these operators in the almost Hermitian and almost K\"ahler settings, along with a correspondence of these operators to operators on forms which are of present interest in almost complex geometry.