Theorem 2

Let E be a complete normable space, $u$ an endomorphism of E and $h$ a compact endomorphism of E. One has Sp$_{\omega}(u+h) =$ Sp$_{\omega}(u)$ and Sp$_n(u+h) =$ Sp$_n(u)$ for every $n\in \overline{\mathbf{Z}}-\{0\}$.

Let $\lambda \in \mathbf{C}$. In order that $u+h-\lambda 1_E$ be a strict morphism whose kernel (resp. cokernel) is of finite dimension, it is necessary and sufficient that $u-\lambda 1_E$ be so, by Theorem 1 of III, p. 72 (resp. Theorem 2 of III, p. 73).

The equalities

Sp$_{-\infty}(u+h) =$ Sp$_{-\infty}(u)$, Sp$_{+\infty}(u+h) =$ Sp$_{+\infty}(u)$,

Sp$_{\omega}(u+h) =$ Sp$_{\omega}(u)$

follow from this. Moreover, one has Sp$_n(u+h) =$ Sp$_n(u)$ by Theorem 3 of III, p. 73 for every $n\in \mathbf{Z}-\{0\}$.