Adjoint integral equation

Remarks

Example: for $K(x, s) = i x$ on $[0, 1]$, the adjoint kernel is $K^*(x, s) = \overline{i s} = -i s$. The check: put $x$ and $s$ in reverse order and conjugate.

Non-example: the equation with kernel $K(s, x)$ and parameter $\lambda$, without conjugation, is the adjoint only when $K$ is real and $\lambda$ is real. For $K(x, s) = i x$ it gives the kernel $i s$, not $-i s$.

With the inner product $\ip{u}{v} = \int_a^b u(x) \overline{v(x)} \, dx$, the integral operators satisfy $\ip{K u}{v} = \ip{u}{K^* v}$ when $K$ is continuous or square-integrable. For real kernels and real $\lambda$, the adjoint equation is often called the transposed or associated equation. A kernel equal to its adjoint kernel is called Hermitian.