Definition 3

Let E be a complete normable space and $u$ an endomorphism of E. For every $n\in \overline{\mathbf{Z}}$, denote by Sp$_n(u)$ the set of complex numbers $\lambda \in$ Sp$_e(u)$ such that $u-\lambda 1_E$ is a strict morphism whose kernel or cokernel is of finite dimension, and whose index is $n$. Denote by Sp$_{\omega}(u)$ the complement in Sp$_e(u)$ of the union of the subsets Sp$_n(u)$ for $n\in \mathbf{Z}$.

The sets Sp$_s(u)$, Sp$_n(u)$ for $n\in \overline{\mathbf{Z}}$, and Sp$_{\omega}(u)$ form a partition of the spectrum of $u$.

Every endomorphism of E whose cokernel is of finite dimension is strict (III, p. 52, Lemma 6). The Fredholm endomorphisms of E are the endomorphisms of E whose kernel and cokernel are of finite dimension (III, p. 52, Prop. 11). The set $\mathbf{C}-$ Sp$_e(u)$ consists of the $\lambda \in \mathbf{C}$ such that $u-\lambda 1_E$ is a Riesz endomorphism of E, and such an endomorphism is a Fredholm endomorphism of E of index 0.

Consequently, for $\lambda \in \mathbf{C}$ and for $n\in \mathbf{Z}-\{0\}$, we have:

$\lambda \in \mathbf{C}-$ Sp($u$)$\Leftarrow \Rightarrow u-\lambda 1_E$ is an automorphism ;

$\lambda \in$ Sp$_s(u)\Leftarrow \Rightarrow u-\lambda 1_E$ is a Riesz endomorphism, but

is not an automorphism ;

$\lambda \in$ Sp$_0(u)\Leftarrow \Rightarrow u-\lambda 1_E$ is a Fredholm endomorphism

of index 0 of E but is not a Riesz endomorphism of E ;

$\lambda \in$ Sp$_n(u)\Leftarrow \Rightarrow u-\lambda 1_E$ is a Fredholm endomorphism

$(n\not = 0)$ of index $n$ of E ;

$\lambda \in$ Sp$_{-\infty}(u)\Leftarrow \Rightarrow u-\lambda 1_E$ is strict, its kernel is of dimension

infinite and its cokernel is of finite dimension ; $\lambda \in$ Sp$_{+\infty}(u)\Leftarrow \Rightarrow u-\lambda 1_E$ is strict, its kernel is of dimension

finite and its cokernel is of dimension infinite ; $\lambda \in$ Sp$_{\omega}(u)\Leftarrow \Rightarrow$ either $u-\lambda 1_E$ is not strict, or its kernel

and its cokernel are of dimension infinite.