Locally spectrally bounded linear maps

summary:Let ${\mathcal L}({\mathcal H})$ be the algebra of all bounded linear operators on a complex Hilbert space ${\mathcal H}$. We characterize locally spectrally bounded linear maps from ${\mathcal L}({\mathcal H})$ onto itself. As a consequence, we describe linear maps from ${\mathcal L}({\mathcal H})$ onto itself that compress the local spectrum