A bounded linear operator \(T\) on a Banach space \(X\) is said to be dissipative if \(\| \exp(tT)\| \leq 1\) for all \(t \geq 0\), and is said to be Hermitian if \(\| \exp(itT)\| = 1\) for all \(t \in \mathbb R\). For a dissipative operator, it is proved that \[ \lim_{t \to \infty}\| \exp(tT)T\| = \sup\{| \lambda| : \lambda\in\sigma(T)\cap i\mathbb R \}, \] and in the case of \(\sigma(T) \cap i \mathbb R \subset [-i \pi /2, i \pi /2]\), \[ \lim_{t \to \infty}\| \exp(tT) \sin T\| = \sup\{| \sin \lambda| : \lambda \in \sigma(T) \cap i \mathbb R \}. \] The proof is based on reduction to local spectrum of some Hermitian operator \(S\) on an auxiliary Banach space \(Y\), associated with \(T\).