Let $A_n$ be a compact linear operator on a $\mathbb R$-Hilbert space $H$. What can we infer from the condition $$\limsup_{n\to\infty}\frac{\ln\left\|A_n\right\|_{\mathfrak L(H)}}n\le0\tag1$$ and in which sense is related to the asymptotic stability?
$(1)$ reminds me of the growth bound of a $C^0$-semigroup. Is there a natural relation to it?
I know that $\left\|A_n\right\|_{\mathfrak L(H)}$ is equal to the largest singular value of $A_n$. Moreover, $(1)$ immediately yields that $$\forall\varepsilon>0:\exists N\in\mathbb N:\forall n\ge N:\left\|A_n\right\|_{\mathfrak L(H)}<e^{\varepsilon n}\tag2.$$ But a bound by an exponential seems to be of little use to me (and I guess this is heavily wrong). So, what's the use of it?
If you need more context, $(1)$ is the condition imposed by Goldsheid and Margulis in the paper Lyapunov indices of a product of random matrices, Proposition 1.3, equation 16.
Not exactly the same as for a strongly continuous semigroup only because this is used differently there.
Namely, you should then have a single generator and here you have something depending on $n$, so you would first need to consider an "autonomization" of the dynamics" or say start discussing a generalization of the bounded growth condition, more or less as in A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences $44$, Springer-Verlag, New York, $1983$, specifically in Chapter 5, although I note that this is somewhat outdated by now.
Here it just means that one element does not change the asymtptotic properties of the composition, and that's the usual approach, as in Raghunathan's work. Condition (1) is the usual one (and it is used regularly). More precisely, essentially it allows one to show that the eigenvectors obtained from the singular values converge, thus giving the splitting in the multiplicative ergodic theorem.
It is also hidden in the proof that (1) guarantees that the Lyapunov indices are finite.