Example 2

Let X be a compact subset of $\mathbf{C}$, and $\mathscr{C}(X)$ the Banach space of continuous complex-valued functions on X. Let $u$ denote the endomorphism of $\mathscr{C}(X)$ which to $f\in \mathscr{C}(X)$ associates the function $z\mapsto zf(z)$ of $\mathscr{C}(X)$. The spectrum of $u$ is equal to X, its points of the discrete spectrum are the isolated points of X, and their spectral multiplicities are equal to 1.