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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!!!

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Let $\mathbb{P}$ be a Borel probability measure on the space of bounded operators on $\mathcal{B}$, equipped with the operator norm topology. By the subadditive ergodic theorem, the limit $$\lim_{n \to \infty} \frac{1}{n}\log \|T_{\omega_n}\cdots T_{\omega_1}\|$$ exists a.s., where the operators $T_{\omega_i}$ are chosen IID according to the law $\mathbb{P}$. (This is also true if the sequence $(\omega_k)$ is chosen according to a stationary ergodic process.) The limit is a.s. equal to $$\lim_{n \to \infty} \frac{1}{n}\int \log\|T_{\omega_n}\cdots T_{\omega_1}\|d\mathbb{P}(\omega_n)\cdots d\mathbb{P}(\omega_1)$$ in the IID case, and a related property holds in the stationary ergodic case. This limit is conventionally called the (top) Lyapunov exponent.

Here is an example to illustrate that the criterion $\rho(T_i)<1$ a.s. is insufficient to guarantee that the top Lyapunov exponent is negative even if $\mathcal{B}$ is two-dimensional. It is sufficient to find a probability measure supported on a pair of $2 \times 2$ matrices with spectral radius $1$ for which the top Lyapunov exponent is positive, since then we can divide both matrices by $e$ to the power of the top Lyapunov exponent and obtain something with spectral radii $<1$ but with the IID products not exponentially decaying. So, consider $$A_1=\begin{pmatrix}0&-1\\1&0\end{pmatrix},\qquad A_2=\begin{pmatrix}1&-\frac{1}{2}\\2&0\end{pmatrix}$$ each with probability $\frac{1}{2}$. One may check that the two matrices have spectral radius $1$, have determinant $1$, do not preserve a finite union of one-dimensional subspaces of $\mathbb{R}^2$, and have the property that $\rho(A_1A_2)>1$. By Furstenberg's Theorem on random matrix products (see for example Viana's book Lectures on Lyapunov exponents, or Bougerol and Lacroix's Products of Random Matrices with Applications to Schrödinger Operators) it follows from these observations that the top Lyapunov exponent is positive. Thus, the desired implication is false even for $2 \times 2$ matrices.

However, the stronger condition $$\rho(T_{\omega_n}\cdots T_{\omega_1})<(1-\varepsilon)^n$$ for $\mathbb{P}\times \cdots \times \mathbb{P}$ almost every $\omega_1,\ldots,\omega_n$, for all $n \geq 1$, is sufficient if $\mathbb{P}$ has bounded support and $\mathcal{B}$ is finite-dimensional. Indeed, this condition implies $$\limsup_{n \to \infty} \sup_{\omega_1,\ldots,\omega_n \in \mathrm{supp} \mathbb{P}}\|T_{\omega_n}\cdots T_{\omega_1}\|^{\frac{1}{n}}\leq 1-\varepsilon $$ and the almost sure result follows trivially. This result is sometimes referred to as the Berger-Wang formula for the joint spectral radius. This result also holds in infinite dimensions if the operators are compact, or satisfy a more complicated condition involving their approximability by compact operators. A while ago I wrote a paper on this, "The generalised Berger-Wang formula and the spectral radius of linear cocycles", and you might find some of the references therein helpful.

You would probably also be interested in reading about Oseledec's multiplicative ergodic theorem and its infinite-dimensional generalisations. Anthony Quas has (co-)written numerous papers on infinite-dimensional multiplicative ergodic theorems and those would probably give you some insight into the state of the art in this area as well as a good source of contemporary and historical references.

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    $\begingroup$ your example is very helpful!! Thanks for your reference. BTW, my problem is actually from ergodic theory. very appreciate!! $\endgroup$ – jason May 30 '18 at 3:27
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You should have a look on the book "Product of random matrices and application to Schrodinger operators" (Lacroix, Bougerol) http://fr.booksc.org/book/32773481/48bc02 or the paper of Le Page "Théorèmes limites pour les produits de matrices aléatoires" (in French).

I am not sure that the spectral radius condition is very appropriate because we can't estimate $\rho(AB)$ with $\rho(A)$ and $\rho(B)$.

In a nutshell what is done in the literature is the following :

1-Almost surely we have the Lyapunov exponent : $$ \frac{1}{N}\log \|\prod_i T_i \|\rightarrow \gamma$$ 2-Assuming $\det(T_i)=1$, then in many cases you can show that $\gamma>0$ ( $\|\prod T_i \|$ is exponentially increasing).

Therefore, you have exponentially decreasing if $\gamma_1<-\frac{1}{d}\mathbb{E}(\log(\det(T)))$ where $\gamma_1$ is the Lyapunov exponent of $\frac{1}{\det(T_i)^{1/d}}T_i$ and $d$ the size of the matrices.

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  • $\begingroup$ Thank you so much!! I will read the book you recommend( I am sorry I do not know French). Thanks a lot!! $\endgroup$ – jason May 30 '18 at 3:24

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