Hermitian kernel

Summary

A Hermitian kernel is a complex-valued kernel with $K(x, s) = \overline{K(s, x)}$ for all $x$ and $s$. It is the continuous analogue of a Hermitian matrix. A square-integrable Hermitian kernel gives a self-adjoint operator on $L^2$, so its characteristic values are real and its eigenfunctions can be chosen orthonormal. The Hilbert–Schmidt theory of integral equations is built on such kernels.