Seminar 7. The Strong Law of Large Numbers for Random Semigroups on Uniformly Smooth Banach Spaces

We consider random linear bounded operators $\Omega \to L(X,X)$ acting on a Banach space $X$. Examples of such random operators include random quantum channels. The Strong Law of Large Numbers is well-known for the case where $X$ is a Hilbert space, in the typical form of a standard LLN for random operators, and in several other special cases. Instead of sums of independent, identically distributed random variables, one can consider the composition of random semigroups $e^{A_i t/n}$. We present a Strong Law of Large Numbers in the strong operator topology for random semigroups of linear bounded operators on uniformly smooth Banach spaces. Furthermore, we develop an alternative approach that yields the Strong Law of Large Numbers in the weak operator topology on arbitrary Banach spaces.