Exercise 3, § 6

Let E be a complex Hilbert space. Let U be an open connected subset of $\mathbf{C}$. Let $f$ be a holomorphic function from U into the space $\mathscr{L}^c(E)$. Suppose that there exists $z\in U$ such that 1 belongs to the resolvent set of $f(z)$.

a) The set D of the $z\in U$ such that $1\in$ Sp($f(z)$) is a discrete subset of U.

b) The mapping from U - D into $\mathscr{L}(E)$ defined by $z\mapsto R(f(z),1)$ is holomorphic.

We fix a point $z_0\in D$.

c) There exists an integer $k\geqslant 1$ such that the mapping from U - D into $\mathscr{L}(E)$ defined by $z\mapsto (z-z_0)^kR(f(z),1)$ extends to a holomorphic mapping on (U - $D$)$\cup \{z_0\}$.

d) There exists a unique sequence $(u_n)_{n\in\mathbf{N}}$ of endomorphisms of E such that $u_n= 0$ for $n$ sufficiently large and a unique holomorphic function S from (U- $D$)$\cup \{z_0\}$ into $\mathscr{L}(E)$ such that for every $z\in U$ - D, one has

e) The endomorphism $u_{-1}$ is the spectral projector of $f(z_0)$ relative to the open and closed subset $\{1\}$ of the spectrum of $f(z_0)$.