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Quantum law of large numbers for Banach spaces

S. Dzhenzher, V. Sakbaev

Abstract

We consider random operators $Ω\to \mathcal{L}(\ell_p, \ell_p)$ 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 $p \leqslant 2$.

Quantum law of large numbers for Banach spaces

Abstract

We consider random operators for some . The law of large numbers is known in the case 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 . We obtain the law of large numbers for the case .

Paper Structure

This paper contains 5 sections, 9 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Formal definitions and notations
  3. Proof of Theorem \ref{['t:lln-2']}
  4. Appropriate operators
  5. Examples for another operators

Key Result

Lemma 2.1

Let $1 \leqslant p < \infty$ and $1 \leqslant q \leqslant \infty$. Let $A\colon\Omega\to\mathcal{L}(\ell_p, \ell_q)$ be a weakly measurable operator. Then random variables $a_{ij}$ for any $i,j$, $\left\lVert Ax\right\rVert_q$ for any $x \in \ell_p$, and $\left\lVert A\right\rVert_{\mathcal{L}(\ell_

Theorems & Definitions (24)

  • Lemma 2.1: on measurability
  • proof
  • Lemma 2.2: on integrability
  • proof
  • Lemma 2.3: on independence
  • proof
  • Theorem 2.4: proved in §\ref{['s:proof']}
  • Conjecture 2.5
  • Lemma 3.1: Chebyshev's inequality
  • proof
  • ...and 14 more