Theorem 3 (Weyl)

Let $u\in \mathscr{L}(E)$ be a normal endomorphism of $u$. The essential spectrum of $u$ is the intersection of the sets Sp($u+h$), where $h$ ranges over $\mathscr{L}^c(E)$.

Let $h\in \mathscr{L}^c(E)$. Since Sp$_0(u)$ is empty (corollary above), Theorem 2 of III, p. 89 implies that Sp$_e(u+h) =$ Sp$_e(u)$. The intersection of the sets Sp($u+h$) therefore contains Sp$_e(u)$.

Let $\lambda \in$ Sp$_s(u)$. Let $E_{\lambda}$ denote the proper subspace of $u$ relative to $\lambda$, and $F_{\lambda}$ the image of the spectral projector associated with $u$ and $\mathbf{C}-\{\lambda \}$. The space E is the topological direct sum of $E_{\lambda}$ and $F_{\lambda}$. Let $h$ be the finite-rank endomorphism of E which is zero on $F_{\lambda}$ and coincides on $E_{\lambda}$ with the identity. The endomorphism $u+h$ is invertible, therefore $\lambda \notin$ Sp($u+h$). The theorem follows.

Thus, if $u$ is a normal endomorphism of E, and if $h\in \mathscr{L}^c$(E), the spectrum of $u+h$ can differ from the spectrum of $u$ only by isolated points having finite spectral multiplicity.