(Sorry, I do hate editing this many many times but let me try the last time) 

Gelfand's formula says that

$$\lim_{k\rightarrow \infty} \|A^k\|^{1/k} = \rho(A)$$

I am wondering whether there is any way to make this non-asymptotic. for example, I would like to have a set $S$ of matrices so that for any matrix $A\in S$, $\|A^k\|$ goes to 0 with some exponential rate. (A candidate might be, for any matrix $A$ with $\rho(A) < 1/2$ and $\|A\|\le T$, $\|A^k\| \le T^{100}(2/3)^k$. Although I haven't thought through whether there is any trivial counterexample for this statement. )