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Let \(X\) be a Frèchet space. A family of operators \((T(t))_{t\geq0}\subseteq L(X)\) is a \(C_0\)-semigroup if
it satisfies the evolution property \(T(t)T(s)=T(t+s)\) for \(s\), \(t\geq0\) and \(T(0)=\operatorname{id}_X\), and if all its orbits are continuous. Its generator \(A\colon D(A)\rightarrow X\) is defined as the differential of the orbit \(t\mapsto T(t)x\) at \(t=0\) on those \(x\in X\) for which this limit exists.
In the talk, we define the \emph{growth bound} of \(T\) as the infimum over those real \(\omega\) for which \(\{e^{-\omega t}T(t)\:;\:t\geq0\}\subseteq L(X)\) is equicontinuous. Then we use classical and recent approaches for non Banach spectral theories to define the spectral bound \(\operatorname{s}(A)\) as the supremum over the realparts of all complex points of the spectrum of \(A\). We study the two bounds and show that the Banach space inequality \(\operatorname{s}(A)\leq\omega_0(T)\) extends to Fr\'{e}chet spaces. In addition, we discuss several concrete examples.
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