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

TL;DR

This paper establishes a Strong Law of Large Numbers for random semigroups of bounded linear operators on uniformly smooth Banach spaces, extending known results from Hilbert spaces to a broader class of Banach spaces.

Contribution

It introduces new SLLN results for random semigroups in Banach spaces, including a novel approach for weak operator topology convergence applicable to all Banach spaces.

Findings

Proves SLLN in Strong Operator Topology for uniformly smooth Banach spaces.

Develops an alternative approach for SLLN in Weak Operator Topology for all Banach spaces.

Extends classical results from Hilbert spaces to general 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.