Barbashin-type characterization of uniform exponential stability

Let $X$ be the Banach space underlying a linear discrete-time system, and let $A_m^k$ denote its associated evolution operator from time $k$ to time $m$. The system is uniformly exponentially stable when it satisfies the corresponding uniform exponential decay condition. Barbashin-type characterization. For every linear discrete-time system, the following statements are equivalent:

1. The system is uniformly exponentially stable.
2. There exist constants $B\geq 1$ and $b>0$ such that

$$\sum_{k=0}^{m}e^{b(m-k)}\lVert A_m^k x\rVert\leq B\lVert x\rVert$$

for all $(m,x)\in\mathbb{N}\times X$. 3. There exists a constant $B\geq 1$ such that

$$\sum_{k=0}^{m}\lVert A_m^k x\rVert\leq B\lVert x\rVert$$

for all $(m,x)\in\mathbb{N}\times X$.

The result would extend the preceding operator-norm characterization by replacing $\lVert A_m^k\rVert$ with pointwise estimates on $A_m^k x$. The implication from condition (iii) to condition (i) is presented as an open problem, while the other direction follows from the stated characterization.