Positive spectrum of a dissipative operator is adjoint point spectrum ($\alpha \in \sigma (Z)⟺\alpha \in {\sigma }_p(Z^{\ast })(\alpha >0)$)

For a closed densely defined dissipative operator, the estimate $\Vert (\alpha I-Z)f\Vert \ge \alpha \Vert f\Vert$ makes its range closed and its kernel trivial. Thus a positive spectral value means this range is proper, which is equivalent to a nonzero vector in $\ker (\alpha I-Z^{\ast })$. Dissipativity is essential: a real multiplication operator may have positive continuous spectrum with no adjoint eigenvector.

// Target: analysis.bigb