Let $A$ be a unital algebra, and $a\in A$. Assume that $\sigma(a)\subset \{\lambda\in \mathbb{C}: Re\lambda < 0\}$.
Show there exists $M,\omega >0$ such that $$\left\|e^{ta}\right\|\leq Me^{-\omega t} $$ for all $t>0$.
Even if $\sigma(a)$ contains an eigenvalue $\lambda$ i find this difficult since then (for an eigenvector $x$) $$\left\|e^{ta(x)}\right\|= \left\|e^{t\lambda x}\right\| =\left\|\sum_{k=0}^{\infty}t^k (\lambda x)^k/k!\right\|... $$
But i cant seem to reduce this further to something of the above form, let alone if $\sigma(a)$ contains no eigenvalues. Is there some handy trick involved? The inequality does remind me of inequalities I have seen in theory for dynamical systems where $a$ is instead some functions $g$.
Any hints of suggestions?
Hint: For a suitable contour $\Gamma$, $$ e^{ta} = \dfrac{1}{2\pi i} \oint_\Gamma e^{tz} (z - a)^{-1}\ dz$$