Invertibility of quasiconformal operators

The global homeomorphism theorem for quasiconformal maps describes the following specifically higher-dimensional phenomenon: {\em Locally invertible quasiconformal mapping $f: {\R}^{n} \to {\R}^{n}$ 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.}