If $A_n$ is a sequence of self-adjoint nonnegative operators with $\limsup_{n\to\infty}\frac{\ln\left\|A_n\right\|}n\le0$, does $|A_n|^{-n}$ converge?

Question

Let $H$ be a $\mathbb R$-Hilbert space. Note that if $A\in\mathfrak L(H)$, then $A^\ast A$ is nonnegative and self-adjoint and hence $$|A|:=\sqrt{A^\ast A}$$ is well-defined.

Now let $(A_n)_{n\in\mathbb N}\subseteq\mathfrak L(H)$ with $$\limsup_{n\to\infty}\frac{\ln\left\|A_n\right\|_{\mathfrak L(H)}}n\le0.$$ Are we able to conclude that $\left(|A_n|^{\frac1n}\right)_{n\in\mathbb N}$ is convergent and the limiting operator $A$ is nonnegative and self-adjoint as well? If each $A_n$ is compact, will $A$ be compact as well?

EDIT: I guess we need further assumptions. What I want to achieve is a generalization of Proposition 1.3 in the paper Ergodic theory of differentiable dynamical systems:

$^1$

I guess the "$\wedge q$" is a notation for the wedge product. How can we generalize this condition?