Corollary 2

Let $u$ be a normal partial operator on the Hilbert space E. It is supposed that E is nonzero.

a) The spectrum of $u$ is nonempty;

b) For every $\lambda \in \mathbf{C}-$ Sp($u$), the norm of the resolvent $R(u, \lambda )$ is equal to $1/\delta$, where $\delta >0$ is the distance in $\mathbf{C}$ from $\lambda$ to the spectrum of $u$.

c) For every $\varepsilon >$ 0, the $\varepsilon$-pseudospectrum PSp$_{\varepsilon}(u)$ is the set of $\lambda \in \mathbf{C}$ at distance $< \varepsilon$ from Sp($u$).

Suppose that the spectrum of $u$ is empty. Then $u$ is injective, and the endomorphism $u^{-1}=-R(u,0)$ is normal (cor. 1, c) and a)). We have Sp($u^{-1}$)$\subset \{0\}$ (prop. 15, a)), hence Sp($u^{-1}$) $=\{0\}($I, p. 26, cor. 1). Since $u^{-1}$ is normal, this implies that $u^{-1}= 0 ($I, p. 110, example 1), which is a contradiction since E is not zero.

To prove b), one may suppose that $u$ is the multiplication operator $m_g$ on $L^2(X, \mu)$, where $g$ is a continuous function on a locally compact topological space X endowed with a positive measure $\mu$ (th. 1 of IV, p. 266). Let $\lambda \in \mathbf{C}-$ Sp($u$) and let $\delta >0$ be the distance from $\lambda$ to the spectrum of $u$. To prove that $\|R(u, \lambda )\|=\delta^{-1}$, one is reduced to the case $\lambda = 0$ by replacing $u$ by $u-\lambda 1_E$. The real number $\delta$ is then the distance from 0 to the spectrum of $m_g$. Since the latter coincides with the $\mu$-essential image of $g$, the result is a consequence of lemma 3 of IV, p. 182.

Finally, assertion c) follows from b) and from the definition of PSp$_{\varepsilon}(u)$ (IV, p. 250, def. 9).