Past exam of the differential equations and applications course of the University of Oxford / 2009 / A / 2 / B / a / ii

Antisymmetry $k(x,t)=-k(t,x)$ instead gives $L^{\ast }=-L$, so $L$ is skew-adjoint. Its quadratic form satisfies $⟨y,Ly⟩=-\overline{⟨y,Ly⟩}$. Hence an eigenpair satisfiesNonzero eigenvalues are purely imaginary in the complexified function space; zero is allowed as well. On a strictly real vector space this means there are no nonzero real eigenvalues.