Linear maps preserving G-quasi-isometry operators

Let \({\mathscr {H}}\) be a complex Hilbert space and \({\mathscr {B}}({\mathscr {H}})\) the algebra of all bounded linear operators on \({\mathscr {H}}\). We give the concrete forms of surjective continue unital linear maps from \({\mathscr {B}}({\mathscr {H}})\) onto itself that preserves G-quasi-isometric operators.