The simultaneous null solutions of the two complex Hermitean Dirac operators
are focused on in Hermitean Clifford analysis, where the matrix Hilbert
transform was presented and proved to satisfy the analogous properties of the
Hilbert transform in classical analysis and in orthogonal Clifford analysis.
Under this setting we will introduce the Szeg$\ddot{o}$ projection operator for
the Hardy space of Hermitean monogenic functions defined on a bounded subdomain
of even dimensional Euclidean space, establish the Kerzman-Stein formula which
closely connects the Szeg$\ddot{o}$ projection operator with the Hardy
projection operator onto the Hardy space of Hermitean monogenic functions
defined on a bounded subdomain of even dimensional Euclidean space, and get the
Szeg$\ddot{o}$ projection operator in terms of the Hardy projection operator
and its adjoint. Further we will give the algebraic and geometric
characterizations for the matrix Hilbert transform to be unitary in Hermitean
Clifford analysis.