Given a compact Hermitian manifold $M$, we have three canonical pseudo-differential operators on the sections of complexified de Rham complex, namely

1) (d + d$^*,\Omega^{*})$

2) ($\partial$ + $\partial^*,\Omega^{(*,0)})$

3) $(\overline{\partial} + \overline{\partial}^*,\Omega^{(0,*)})$

**Question 1:** What is the relation between the K-homology classes (see example 3.4) of these three operators? Can we at least say that all three give non-equivalent classes? In general, given two differential operators which are "representatives" of the same K-homology class, will tensoring them each by a line bundle always land in the same class? I.e. does tensoring by a line bundle descent to an operation on K-homology?

Assume in addition $M$ is spin, which according to Atiyah (see this question) means that $\Omega^{(0,n)}$ admits a line bundle. Then we also have an associated Dirac operator, which is the tensor product of the third operator with the square root endowed with a connection.

Question 2: N-lab says that the class of the Dirac is equal to the class of the operator in (1) above.

Question 3: If so, is this operation a permutation?