Lemma 2

*Let E be a complex Banach space, u an element of $\mathscr{L}(E)$, S the spectrum of u in $\mathscr{L}(E)$ (Spectral Theories, Chapter I, § 1, no. 2) and f a holomorphic complex function on an open neighbourhood $\Omega$ of S. Suppose that f admits in $\Omega$ only a finite number of distinct zeros $z_1, \ldots, z_n$, of multiplicities $h_1, \ldots, h_n$. Then $\mathrm{Ker}\ f(u)$ is the direct sum of the $\mathrm{Ker}(u - z_i)^{h_i}$ for $1 \leqslant i \leqslant n$.

(For the definition of $f(u)$, see Spectral Theories, Chapter I, § 4, no. 8.)

There exists a holomorphic function g on $\Omega$, everywhere non-zero, such that $f(z) = (z - z_1)^{h_1} \ldots (z - z_n)^{h_n} g(z)$. Then $g(u)g^{-1}(u) = g^{-1}(u)g(u) = 1$ and hence $\mathrm{Ker}\ f(u) = \mathrm{Ker} \prod_{i=1}^n (u - z_i)^{h_i}$. Considering $\mathrm{Ker}\ f(u)$ as a $\mathbf{C}[X]$-module by means of the external law $(h, x) \mapsto h(u)x$ for $h \in \mathbf{C}[X]$, $x \in \mathrm{Ker}\ f(u)$, we see that Ker $f(u)$ is the direct sum of the $\mathrm{Ker}(u - z_i)^{n_i}$, using Algebra, Chapter VII, § 2, no. 1, Proposition 1.