Half Dirichlet problem for matrix functions on the unit ball in Hermitian Clifford analysis

The simultaneous null solutions of the two complex Hermitian Dirac operators are focused on in Hermitian Clifford analysis, where the Hermitian Cauchy integral was constructed and will play an important role in the framework of circulant ( 2 × 2 ) matrix functions. Under this setting we will present the half Dirichlet problem for circulant ( 2 × 2 ) matrix functions on the unit ball of even dimensional Euclidean space. We will give the unique solution to it merely by using the Hermitian Cauchy transformation, get the solution to the Dirichlet problem on the unit ball for circulant ( 2 × 2 ) matrix functions and the solution to the classical Dirichlet problem as the special case, derive a decomposition of the Poisson kernel for matrix Laplace operator, and further obtain the decomposition theorems of solution space to the Dirichlet problem for circulant ( 2 × 2 ) matrix functions.