Definition 10

Let $u$ be a normal partial operator on E. A complex number $\lambda \in$ Sp($u$) belongs to the discrete spectrum of $u$ if $\lambda$ is isolated in Sp($u$) and if $\lambda$ is an eigenvalue of finite multiplicity.

We denote by Sp$_s(u)$ the discrete spectrum of $u$. Its complement in Sp($u$) is called the essential spectrum of $u$ and denoted by Sp$_e(u)$.

The essential spectrum of a normal partial operator $u$ on E is closed in $\mathbf{C}$, since Sp($u$) is closed in $\mathbf{C}$ and the complement of the essential spectrum is open in Sp($u$). The discrete spectrum of $u$ is not necessarily closed in $\mathbf{C}$ (exercise 36 of IV, p. 369).