A linear algebraic approach to holomorphic reproducing kernels in Cn

Let Ω⊆Cn be a domain and k be a holomorphic reproducing kernel on Ω. By the Moore–Aronszajn characterization, every finite matrix k(Zi, Zj) is positive semidefinite. We show that, as a direct algebraic consequence, k(Z, U) satisfies an infinite 2n-parameter family of differential inequalities of which the classic diagonal dominance inequality for reproducing kernels is the order 0 case. In addition, the mixed hemisymmetric partial derivative of k with respect to any pair of homologous variables yields again a holomorphic reproducing kernel on Ω. These results are interpreted in terms of the general theory of reproducing kernels.