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.