On Lipschitz continuity with respect to the Poincar\xe9 metric of linear contractions between M\xf6bius gyrovector spaces

Keiichi Watanabe

doi:10.1186/s13660-021-02700-0

Abstract

For every linear operator between inner product spaces whose operator norm is less than or equal to one, we show that the restriction to the Möbius gyrovector space is Lipschitz continuous with respect to the Poincaré metric. Moreover, the Lipschitz constant is precisely the operator norm.

Highlights

The notion of Lipschitz continuity of mappings between two metric spaces is well known and significant in all fields of mathematics, in geometry and analysis
The Möbius addition on D is defined by the equation a⊕b= a+b 1 + ab for any a, b ∈ D, which appears in various branches of mathematics
This result implies that every linear functional u → u, w with w ≤ 1 is Lipschitz continuous on the Möbius gyrovector space with respect to the Poincaré metric, and that the Lipschitz constant is precisely w

Summary

Introduction

The notion of Lipschitz continuity of mappings between two metric spaces is well known and significant in all fields of mathematics, in geometry and analysis It is one of the most fundamental facts in functional analysis that an arbitrary bounded linear operator T between normed spaces is Lipschitz continuous. No knowledge of general theory of gyrogroups or gyrovector spaces is required to read this paper except for some basic facts on Möbius addition, Möbius scalar multiplication, and (gyro) distance function; it is fundamental for our motivation and background. For any s > 0 and w ∈ H with w ≤ 1, the following identity holds: h( u, w , v, w ) This result implies that every linear functional u → u, w with w ≤ 1 is Lipschitz continuous on the Möbius gyrovector space with respect to the Poincaré metric, and that the Lipschitz constant is precisely w.

Hs into

We have to show

Hence we obtain

Then we have