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

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 $e greatest singular value of $ B_n $ for $ k, n in mathbb N $. Yes $$ limsup_ {n to infty} frac { ln left | A_n right | _ { mathfrak L (H)}} n le0 tag1 $$ 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

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

**question 1**: I have seen this result in many textbooks, but I have wondered why it is stated this way. First of all, isn't it $ (2) $ clearly equivalent to $$ frac { sigma_k (B_n)} n xrightarrow {n to infty} lambda_i in (- infty, infty) tag3 $$ for some people $ lambda_i $ for everyone $ k in mathbb N $ which in turn is equivalent to $$ sigma_k (B_n) ^ { frac1n} xrightarrow {n to infty} lambda_i ge0 tag4 $$ for some people $ mu_i ge0 $ for everyone $ k in mathbb N $? $ (4) $ seems to be much more intuitive than $ (3) $, since no $ lambda_i $, But $ mu_i = e ^ { lambda_i} $ are precisely Lyapunov's exponents of the limit operator $ B $. Am I missing something? The definition of $ Lambda_i $ (which is equal to $ lambda_i $) seems odd to me.

**question 2**: What is the interpretation of $ B $? Usually I look at a discrete dynamic system $ x_n = B_nx_0 $. What is $ B $ (or $ Bx $) tells us about the asymptotic behavior / evolution of orbits?