we know from the theory of ODE that $\left\|\exp(tA)\right\|\leq Ke^{-\delta t}$ for $K,\delta >0$ and $t\in\mathbb{R}^+$ if the real part of all eigenvalues are strict non-positiv. My question is: can we generalize that for Banach algebras?
Suppose $a\in A$ with $A$ a Banachalgebra and let $\sigma(a)\subseteq\left\{\lambda\in\mathbb{C}:\Re(\lambda)<0\right\}$, can we find $\delta,K>0$ such that $\left\|\exp(tA)\right\|\leq Ke^{-\delta t}$?
From my point of view we can use the spectral mapping theory but i don't see how to go further. Can someone help me?
Thanks
Remark $\sigma(a)$ is the spectrum of $a\in A$.