For ε > 0 and a bounded linear operator T acting on some Hilbert space, the ε-pseudospectrum of T is σ ε (T) = {z ∈ C : (zI − T) −1 > ε −1 }. This note provides a characterization of those operators T satisfying σ ε (T) = σ(T) + B(0, ε) for all ε > 0. Here B(0, ε) = {z ∈ C : |z| < ε}. In particular, such operators on finite dimensional spaces must be normal