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*.

While I can't think of an example off the top of my head, I suspect 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. 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
$$\sup_{n \geq 1} \sup_{\omega_1,\ldots,\omega_n \in \mathrm{supp} \mathbb{P}}\|T_{\omega_n}\cdots T_{\omega_1}\|^{\frac{1}{n}}\leq 1-\varepsilon $$
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and those would probably give you some insight into the state of the art in this area as well as a good source of historical references.