consider an insteresting question:

given Banach Space $ \mathcal{B}$, independent identical distribution random operator on $ \mathcal{B}$: $ (T_i)_{i \ge 1} $, where operator space is endowed with operator norm $ || T_i||= \sup_{v \in \mathcal{B},||v||=1}||T_iv||$.

assume $ \sup_i||T_i|| < \infty $, spectrum radius $ \rho(T_i)<1 $ which is i.i.d random variables.

can we show: $ ||\prod_{i=1}^n T_i|| $ has exponential decay almost surely?

take matrix as example: let $T_i$ be

\begin{pmatrix} \lambda_i & 1 \\ 0 & \lambda_i \end{pmatrix}

where $ \lambda_i <1 $ is i.i.d random variable.

so $ \prod_{i=1}^n T_i=$

\begin{pmatrix} \prod_{i=1}^n\lambda_i & (\prod_{i=1}^n\lambda_i)(\prod_{i=1}^n\frac{1}{\lambda_i}) \\ 0 & \prod_{i=1}^n\lambda_i \end{pmatrix}

then by law of large number when $ n \to \infty $, it is almost surely:

\begin{pmatrix} \exp(n\mathbb{E}\log \lambda_1) & \exp(n\mathbb{E}\log \lambda_1)\cdot n \cdot \mathbb{E}\frac{1}{\lambda_1} \\ 0 & \exp(n\mathbb{E}\log \lambda_1) \end{pmatrix}

so $ ||\prod_{i=1}^n T_i|| $ has exponential decay almost surely if $ \mathbb{E}\frac{1}{\lambda_1} < \infty, \mathbb{E}\log \lambda_1 > -\infty$.

From my example, we have to add some conditions to my problem, otherwise it is not true. But If you know some reference about iid random operator and its spectrum, pls let me know, very appreciate!!!