On the spectral and growth bound of semigroups associated with hyperbolic equations (Q705982)

Let \(X\) be a Banach space, \(A\) a linear operator on \(X\) and \( \{e^{tA}\}_{t\geq 0}\) a strongly continuous semigroup generated by \(A\). Recall that \[ s(A)=\sup \{\text{Re}\lambda ;\;\lambda \in \sigma (A)\} \] is called the spectral bound of \(A\), and \[ \omega (e^{tA}):=\lim_{t\rightarrow \infty }\frac{\log \left\| e^{tA}\right\| }{t} \] the growth bound of \(\{e^{tA}\}_{t\geq 0}\). It is well-known that \(s(A)\leq \omega (e^{tA})\) and equality may fail. The authors give three general criteria (Theorems 2.1, 2.3, and 2.4) for the equality of the above two bounds. Theorem 2.1 generalizes two earlier results of Hieber and Wood. The article ends with a higher order counterexample (Theorem 3.1) where \( \omega (e^{tA})\geq \gamma >s(A)=0\).