Let \(\mathscr {L}(\mathscr {H})\) be the algebra of all bounded linear operators acting on a complex Hilbert space of dimension greater than 2, \(\mathscr {A}\) and \(\mathscr {B}\) be subsets of \(\mathscr {L}(\mathscr {H})\) which contain all operators of rank at most one. For \(0< \varepsilon < 1\) and \(A\in \mathscr {L}(\mathscr {H}),\) we denote by \(\sigma _{\varepsilon }(A)\) (respectively, by \(r_{\varepsilon }(A)\) ) the \(\varepsilon \) -condition spectrum (respectively, the \(\varepsilon \) -condition spectral radius) of A. We prove that surjective maps \(\phi _1, \phi _2:\mathscr {A} \longrightarrow \mathscr {B}\) satisfy \( \sigma _{\varepsilon }(\phi _1(A)\phi _2(B)) = \sigma _{\varepsilon }(AB) \hspace{0.2cm} \left( A,B\in \mathscr {A} \right) , \) if and only if there exist a bounded invertible linear operator U on \(\mathscr {H}\) and a unitary operator V on \(\mathscr {H}\) such that, for all \(A\in \mathscr {A}, \phi _1(A) = s VAU^{-1}\) and \(\phi _2(A) = r UAV^*\) for some \(r, s\in \mathbb {C}\) with \(rs=1.\) We also characterize all surjective maps \(\phi _1, \phi _2:\mathscr {A} \longrightarrow \mathscr {B}\) which satisfy \( r_{\varepsilon }(\phi _1(A)\phi _2(B)) = r_{\varepsilon }(AB) \hspace{0.2cm} \left( A,B\in \mathscr {A} \right) . \)
