Definition of the Lyapunov exponents for compact operators

Question

There is the following well-known result by Goldsheid and Margulis (see Proposition 1.3) on the existence of Lyapunov exponents:

Let $H$ be a $\mathbb R$-Hilbert space, $A_n\in\mathfrak L(H)$ be compact and $B_n:=A_n\cdots A_1$ for $n\in\mathbb N$. Let $|B_n|:=\sqrt{B_n^\ast B_n}$ and $\sigma_k(B_n)$ denote the $k$th largest singular value of $B_n$ for $k,n\in\mathbb N$. If $$\limsup_{n\to\infty}\frac{\ln\left\|A_n\right\|_{\mathfrak L(H)}}n\le0\tag1$$ and $$\frac1n\sum_{i=1}^k\ln\sigma_i(B_n)\xrightarrow{n\to\infty}\gamma^{(k)}\;\;\;\text{for all }k\in\mathbb N\tag2,$$ then

1. $$|B_n|^{\frac1n}\xrightarrow{n\to\infty}B$$ for some compact nonnegative and self-adjoint $B\in\mathfrak L(H)$.
2. $$\frac{\ln\sigma_k(B_n)}n\xrightarrow{n\to\infty}\Lambda_k:=\left.\begin{cases}\gamma^{(k)}-\gamma^{(k-1)}&\text{, if }\gamma^{(i)}>-\infty\\-\infty&\text{, otherwise}\end{cases}\right\}\tag2$$ for all $k\in\mathbb N$.

Question 1: I've seen this result in many lecture books, but wondered why it is stated in this way. First of all, isn't $(2)$ clearly equivalent to $$\frac{\sigma_k(B_n)}n\xrightarrow{n\to\infty}\lambda_i\in[-\infty,\infty)\tag3$$ for some $\lambda_i$ for all $k\in\mathbb N$ which in turn is equivalent to $$\sigma_k(B_n)^{\frac1n}\xrightarrow{n\to\infty}\mu_i\ge0\tag4$$ for some $\mu_i\ge0$ for all $k\in\mathbb N$? $(4)$ seems to be way more intuitive than $(3)$, since not $\lambda_i$, but $\mu_i=e^{\lambda_i}$ are precisely the Lyapunov exponents of the limit operator $B$. Am I missing something? The definition of $\Lambda_i$ (which is equal to $\lambda_i$) seems weird to me.

Question 2: What's the interpretation of $B$? Usually, I'm looking at a discrete dynamical system $x_n=B_nx_0$. What does $B$ (or $Bx$) tells us about the asymptotic behavior/evolution of the orbits?

Answer 1

You are right: we really don't care how it is formulated.

In particular, $(2)$ is clearly equivalent to $$\prod_{i=1}^k\sigma_i(B_n)^{1/n}\xrightarrow{n\to\infty}e^{\gamma^{(k)}}\;\;\;\text{for all }k\in\mathbb N$$ (because the logarithm is continuous). In its turn this is equivalent to $$\sigma_k(B_n)^{1/n}\xrightarrow{n\to\infty}e^{\gamma^{(k)}-\gamma^{(k-1)}}\;\;\;\text{for all }k\in\mathbb N$$ with the convention that $\gamma^{(0)}=0$. The notion of "intuitive" depends on each of us but since Lyapunov came much before Galfand, we (in dynamics) don't care about Gelfand in this context besides having been one of the great mathematicians.

The operator $B$ is kind of the limit value that gives the asymptotic rates of contraction and expansion. In this respect this is exactly as Gelfand's formula for the spectral radius but now giving also the eigendirections, not only one of the eigenvalues or even all eigenvalues. In dynamics it is crucial to have some information of the directions on which there is contraction and expansion.

It is very unexpected that $B$ exists.