On dominant contractions and a generalization of the zero-two law

TL;DR

This paper generalizes a result about positive contractions on L^1 spaces, extending the zero-two law to multi-parameter contractions and providing new insights into their behavior.

Contribution

It introduces a multi-parameter extension of the dominant contraction result and generalizes the zero-two law for these operators.

Findings

Extended the dominant contraction result to multi-parameter contractions.

Proved a generalized zero-two law for multi-parameter contractions.

Demonstrated applications of the generalized law in operator theory.

Abstract

Zaharopol proved the following result: let $T,S:L^1(X,{\cf},\m)\to L^1(X,{\cf},\m)$ be two positive contractions such that $T\leq S$. If $\|S-T\|<1$ then $\|S^n-T^n\|<1$ for all $n\in\bn$. In the present paper we generalize this result to multi-parameter contractions acting on $L^1$. As an application of that result we prove a generalization of the "zero-two" law.