I wish to know if there exists some kind of generalization of the theorem that states that a matrix $M$ is convergent iff its spectral radius $\rho(M)<1$, that is:
$$\lim_{n\to\infty} M^n=0$$
Specifically, given a sequence $T_1, ..., T_n$ of linear operators such that $\rho(T_i)<1$ $\forall i$ I wish to show, if possible, that their infinite left product converges, that is:
$$\lim_{n\to\infty} T_n\dots T_1=0$$
In my case $T_1, ..., T_n$ are actually drawn from a finite set of linear mappings. For context, the domain of application is that of time-varying linear dynamical systems.
It's false. Choose couples that are transposed from one another. For example $T_1=\begin{pmatrix}0&a\\0&0\end{pmatrix},T_2=T_1^T$. Then $\rho(T_1)=\rho(T_2)=0$ and $\rho(T_1T_2)=a^2$. Choose $T_3,T_4$ in the same way and so on.