On uniform exponential stability in locally convex spaces (Q2753277)

The well-known result proved by Datko and Pazy showed that the \(C_0\)-semigroup \(e^{tA}\) on a Banach space \(E\) is uniformly exponentially stable iff there exists a \(p\in [1,\infty)\) such that \(\int^\infty_0\|e^{tA}x\|^p dt< \infty\) holds for all \(x\in E\).NEWLINENEWLINENEWLINEHere, the authors generalize such a result for the case of \(\Phi\)-semigroups in locally convex spaces. They understand uniformly exponentially stable as \(\|e^{tA}\|_{\varphi(t,\gamma),\gamma}\leq M(\gamma) e^{-t\omega(\gamma)}\), with \(\|Q\|_{\beta,\gamma}= \sup\{|Qx|_\gamma: x\in E,|x|_\beta\leq 1\}\) for any \(Q\in B(E)\) and \(\{|\cdot|_\gamma:\gamma\in \Gamma\}\) is a family of seminorms. \(\Phi\) is the set of all functions \(\varphi: R_+\times \Gamma\to\Gamma\) with the properties \(\varphi(0,\gamma)= \gamma\) for all \(\gamma\in \Gamma\), \(\varphi(t+ s,\gamma)= \varphi(t,\varphi(s,\gamma)\) for all \(t,s>0\) and \(\gamma\in \Gamma\). The statement involves the inequality NEWLINE\[NEWLINE\int^{t\alpha(\gamma)}_t\|e^{sA}\|^p_{\varphi(s,\gamma), \gamma} ds\leq K^p(\gamma)\quad\text{for all }t,\gamma\in \mathbb{R}_+\times\Gamma.NEWLINE\]