Quantum law of large numbers for Banach spaces

TL;DR

This paper extends the law of large numbers to random operators on Banach spaces, specifically for    spaces, involving compositions of random semigroups, broadening understanding beyond the classical case.

Contribution

It establishes a law of large numbers for compositions of random semigroups on    Banach spaces, generalizing known results for the case p=2.

Findings

Law of large numbers proven for p    spaces

Extension from sum of i.i.d. variables to compositions of random semigroups

Applicable to random operators    Banach spaces

Abstract

We consider random operators $\Omega \to \mathcal{L}(\ell_p, \ell_p)$ for some $1 \leqslant p < \infty$. The law of large numbers is known in the case $p=2$ in the form of usual law of large numbers. Instead of sum of i.i.d. variables there may be considered the composition of random semigroups $e^{A_i t/n}$. We obtain the law of large numbers for the case $p \leqslant 2$.