Definition 9

Let E be a complex Banach space and let $u$ be a closed partial operator on E. Let $\varepsilon$ be a strictly positive real number. The $\varepsilon$-pseudospectrum of $u$ is by definition the union of the spectrum of $u$ and of the set of complex numbers $\lambda$ belonging to the resolvent set of $u$ such that $\|R(u, \lambda )\|> \varepsilon^{-1}$. This set is denoted by PSp$_{\varepsilon}(u)$.

Some authors define the $\varepsilon$-pseudospectrum of $u$ as the set $T_{\varepsilon}(u)$ equal to the union of Sp($u$) and of the set of $\lambda \in \mathbf{C}-$ Sp($u$) such that $\|R(u, \lambda )\|\geqslant \varepsilon^{-1}$. The closure of PSp$_{\varepsilon}(u)$ is contained in $T_{\varepsilon}(u)$, but this inclusion may be strict, even if E is a Hilbert space (cf. exercises 18 of IV, p. 348 and 19 of IV, p. 349).