Definition 1

Let E be a topological vector space and let $u$ be an endomorphism of E. The spectrum of $u$, denoted by Sp($u$), is the spectrum of $u$ relative to the unital algebra $\mathscr{L}(E)$.

Let E be a topological vector space and let $u\in \mathscr{L}(E)$. The spectrum of $u$ is the set of complex numbers $\lambda$ such that $u-\lambda 1_E$ is not an automorphism of E. If E is metrisable complete, it is also the set of complex numbers $\lambda$ such that $u-\lambda 1_E$ is not bijective (EVT, I, p. 19, Corollary 1).

Every eigenvalue of $u$ belongs to the spectrum of $u$, but the converse is false in general.

In the remainder of this paragraph, we shall confine ourselves to studying the notion of spectrum in the case where E is a Banach space.