Unitary equivalence and reduced minimum modulus preservers

Let \(\mathscr {L}({\mathscr {H}})\) $\mathcal{L}(\mathcal{H})$ be the algebra of all bounded linear operators acting on an infinite-dimensional complex Hilbert space \({\mathscr {H}}\) $\mathcal{H}$ , and denote by \(\gamma (T)\) $\gamma (T)$ the reduced minimum modulus of any operator \(T\in \mathscr {L}({\mathscr {H}})\) $T\in \mathcal{L}(\mathcal{H})$ . We obtain the form of all bijective linear maps \(\Phi \) $\mathrm{\Phi }$ on \(\mathscr {L}({\mathscr {H}})\) $\mathcal{L}(\mathcal{H})$ for which \(\gamma (\Phi (T))=\gamma (\Phi (S))\) $\gamma (\mathrm{\Phi }(T))=\gamma (\mathrm{\Phi }(S))$ whenever \(T,~S\in \mathscr {L}({\mathscr {H}})\) $T,S\in \mathcal{L}(\mathcal{H})$ are two operators equivalent by unitaries. We also obtain similar results when the reduced minimum modulus is replaced by the minimum modulus or the surjectivity modulus.