Two-endpoint symmetric derivative has whole complex spectrum ($A_{0}f=i{f}^{\prime },D(A_0)=H_0^1(0,1),\sigma (A_0)=\mathbb{C}$)

Its adjoint is $i{f}^{\prime }$ on all of $H^1(0,1)$, with no endpoint conditions. Every complex $\lambda$ is an adjoint eigenvalue, with eigenfunction $e^{-i\lambda x}$. Thus every complex $\lambda$ obstructs surjectivity of $\lambda I-A_0$. The original operator has no eigenvalues because its zero initial value kills any solution of $i{f}^{\prime }=\lambda f$.

// Target: analysis.bigb