Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
I am essentially convinced, but do you mean $e^{-\epsilon n} \lambda^n \|e\|\le \|A^ne\|\le e^{\epsilon n} \lambda^n \|e\|$ and $\|A^nf\|\le \mu^n\|f\|$
