Remark

Let $\pi$ denote the canonical homomorphism of $\mathscr{L}(E)$ onto the Calkin algebra $\mathscr{C}$alk(E) of E (cf. No. 4 of III, p. 75). The spectrum of $\pi (u)$ relative to the algebra $\mathscr{C}$alk(E) is sometimes called the stable spectrum of $u$. Its elements are the complex numbers $\lambda$ such that $u-\lambda 1_E$ is not a Fredholm endomorphism of E (cor. 1 of theorem 3 of III, p. 73). It is therefore equal to Sp$_{\omega}(u)\cup$ Sp$_{-\infty}(u)\cup$ Sp$_{+\infty}(u)$.

Let A be the set of endomorphisms of E which commute with $u$. The closure B of $\pi (A)$ in $\mathscr{C}$alk(E) is a complete normable algebra and the spectrum of $\pi (u)$ relative to B is the essential spectrum Sp$_e(u)$ of $u$ (III, p. 77, prop. 3). By the cor. of prop. 6 of I, p. 28, applied to the subalgebra B of $\mathscr{C}$alk(E), the set Sp$_e(u)$ is the union of Sp$_{\omega}(u)\cup$ Sp$_{-\infty}(u)\cup$ Sp$_{+\infty}(u)$ and of certain bounded connected components of its complement.