Hermitian kernel

Remarks

Example: $K(x, s) = e^{i (x - s)}$. The check: $\overline{K(s, x)} = \overline{e^{i (s - x)}} = e^{-i (s - x)} = e^{i (x - s)} = K(x, s)$.

Non-example: $K(x, s) = i$, the constant kernel. Here $\overline{K(s, x)} = -i \ne i$.

On the diagonal a Hermitian kernel is real, since $K(x, x) = \overline{K(x, x)}$. A real-valued kernel is Hermitian exactly when it is symmetric. For square-integrable kernels the condition is often required only for almost all $(x, s)$. Then the integral operator $K$ on $L^2(a, b)$ satisfies $\ip{K u}{v} = \ip{u}{K v}$ for the inner product $\ip{u}{v} = \int_a^b u \bar v \, dx$.