An observation about pseudospectra

Abstract

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