LAUSR

Let $${\mathscr {L}}({\mathscr {H}})$$ be the algebra of all bounded linear operators on a complex Hilbert space $${\mathscr {H}}$$ with $$\dim {\mathscr {H}}\ge 3$$ , and let $$\mathscr {A} $$ and $$\mathscr {B}$$ be two subsets of $${\mathscr {L}}({\mathscr {H}})$$ containing all operators of rank at most one. For $$\varepsilon \in (0,1)$$ the $$\varepsilon $$ -condition spectrum of any $$A\in {\mathscr {L}}({\mathscr {H}})$$ is defined by $$\begin{aligned} \sigma _{\epsilon }(A) := \sigma (A)\cup \left\{ \lambda \in \mathbb {C}\setminus \sigma (A):~\Vert (\lambda I -A)^{-1}\Vert \Vert \lambda I -A\Vert \ge \frac{1}{\varepsilon }\right\} , \end{aligned}$$ where $$\sigma (A)$$ is the spectrum of A. The $$\varepsilon $$ -condition spectral radius of A is given by $$\begin{aligned} r_\varepsilon (A):=\sup \left\{ |z| : z\in \sigma _\varepsilon (A) \right\} . \end{aligned}$$ We compute the $$\varepsilon $$ -condition spectrum of any operator of rank at most one, and give an explicit formula for its $$\varepsilon $$ -condition spectral radius. It is then illustrated that the results can be applied to characterize surjective mappings $$\phi :\mathscr {A} \longrightarrow \mathscr {B}$$ satisfying $$\begin{aligned} \delta (\phi (A)^*\phi (B)) = \delta (A^*B) \quad \text{ for } \text{ all } A,B\in \mathscr {A} \end{aligned}$$ where $$\delta $$ stands for $$\sigma _\varepsilon (\cdot )$$ or $$r_\varepsilon (\cdot ).$$