On the commutation properties of finite convolution and differential operators I: commutation

The commutation relation $KL=LK$ between finite convolution integral operator $K$ and differential operator $L$ has implications for spectral properties of $K$. We characterize all operators $K$ admitting this commutation relation. Our analysis places no symmetry constraints on the kernel of $K$ extending the well-known results of Morrison for real self-adjoint finite convolution integral operators.