The Strong Law of Large Numbers for random semigroups on uniformly smooth Banach spaces

Abstract

We consider random linear continuous operators $\Omega \to \mathcal{L}(\mathcal{X}, \mathcal{X})$ on a Banach space $\mathcal{X}$. For example, such random operators may be random quantum channels. The Law of Large Numbers is known when $\mathcal{X}$ is a Hilbert space, in the form of the usual Law of Large Numbers for random operators, and in some other particular cases. Instead of the sum of i.i.d. variables, there may be considered the composition of random semigroups $e^{A_it/n}$. We obtain the Strong Law of Large Numbers in Strong Operator Topology for random semigroups of bounded linear operators on a uniformly smooth Banach space. We also develop another approach giving the SLLN in Weak Operator Topology for all Banach spaces.