Invertibility of quasiconformal operators

Abstract

The global homeomorphism theorem for quasiconformal maps describes the following specifically higher-dimensional phenomenon: {\em Locally invertible quasiconformal mapping $f:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is globally invertible provided $n>2$.} We prove the following operator version of the global homeomorphism theorem. {\em If the operator $f:H\to H$ acting in the Hilbert space $H$ is locally invertible and is an operator of bounded distortion, then it is globally invertible.}