Invertibility of quasiconformal operators

The global homeomorphism theorem for quasiconformal maps describes the following specifically higher-dimensional phenomenon: 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. 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.