  We consider the Dolbeault operator of $K^{1/2}$ -- the square root of the
canonical line bundle which determines the spin structure of a compact
Hermitian spin surface (M,g,J). We prove that the Dolbeault cohomology groups
of $K^{1/2}$ vanish if the scalar curvature of g is non-negative and
non-identically zero. Moreover, we estimate the first eigenvalue of the
Dolbeault operator when the conformal scalar curvature k is non-negative and
when k is positive. In the first case we give a complete list of limiting
manifolds and in the second one we give non-K\"ahler examples of limiting
manifolds.
