On pseudospectral radii of operators on Hilbert spaces

Abstract

For $\epsilon >0$ and a bounded linear operator $T$ acting on some Hilbert space, the $\epsilon$-pseudospectrum of $T$ is ${\sigma }_{\epsilon }\left(T\right)=\{z\in \mathbb{C}:\Vert (zI-T{)}^{-1}\Vert >1/\epsilon \}$ and the $\epsilon$-pseudospectral radius of $T$ is $r_{\epsilon }\left(T\right)=sup\hspace{0.17em}\left\{\right|z|:z\in {\sigma }_{\epsilon }(T\left)\right\}$. In this article, we provide a characterization of those operators $T$ satisfying $r_{\epsilon }\left(T\right)=r\left(T\right)+\epsilon$ for all $\epsilon >0$. Here $r\left(T\right)$ denotes the spectral radius of $T$.