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

TL;DR

This paper proves a Strong Law of Large Numbers for random semigroups of unbounded linear operators on uniformly smooth Banach spaces, extending known results beyond Hilbert spaces.

Contribution

It establishes the Strong Law of Large Numbers for compositions of random semigroups of unbounded operators on a broad class of Banach spaces.

Findings

Proves SLLN for random semigroups in Banach spaces

Extends results from Hilbert spaces to uniformly smooth Banach spaces

Applicable to random quantum channels and similar operators

Abstract

We consider random linear unbounded operators 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 $e^{A_i t/n}$. We obtain the Strong Law of Large Numbers in Strong Operator Topology for random semigroups of unbounded linear operators on a uniformly smooth Banach space.