Exercise 17, § 4

a) Let U be a connected open set of $\mathbf{C}$ and let $V\subset U$ be an open set. Let E be a complex Banach space. Let $f: U\rightarrow E$ be a holomorphic mapping, $z_0\in U$ and $M>0$ such that $\|f(z)\|\leqslant M$ for all $z\in V$ and such that $\|f(z_0)\|<M$. Then $\|f(z)\|<M$ for all $z\in U$.

b) Let E be a complex Banach space and let $u\in \mathscr{L}(E)$. Let $U\subset \mathbf{C}$ be an open subset of the unbounded connected component of $\mathbf{C}-$ Sp($u$). If $M>0$ is such that $\|R(u, \lambda )\|\leqslant M$ for all $\lambda \in U$, then $\|R(u, \lambda )\|<M$ for all $\lambda \in U$. In particular, if $\mathbf{C}-$ Sp($u$) is connected, then the norm of the resolvent of $u$ cannot be constant on a nonempty open subset of $\mathbf{C}-$ Sp($u$).