An Observation About Pseudospectra

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