Definition 2

The isolated points of Sp($u$) of finite spectral multiplicity are called the sensitive points of the spectrum of $u$. The set of sensitive points of Sp($u$) is called the sensitive spectrum of $u$ and is denoted by Sp$_s(u)$.[^1] The complementary set Sp($u$) - Sp$_s(u)$ is called the essential spectrum of $u$ and is denoted by Sp$_e(u)$.

Since the set Sp$_s(u)$ consists of isolated points of Sp($u$), it is open in Sp($u$), discrete, and countable (III, p. 78, lemma 3); it is also bounded. The set Sp$_e(u)$ is closed in Sp($u$), hence compact.

By Prop. 14 of III, p. 54, the sensitive points of the spectrum of $u$ are the complex numbers $\lambda$ such that $u-\lambda 1_E$ is a Riesz endomorphism of E which is not an automorphism of E.

For every complex number $\lambda$, let $N_{\lambda}(u)$ and $I_{\lambda}(u)$ denote the nilspace and the conilspace of the endomorphism $u-\lambda 1_E$ of E. Since E is a Fréchet space, $u-\lambda 1_E$ is a Riesz endomorphism if and only if $N_{\lambda}(u)$ is of finite dimension and $I_{\lambda}(u)$ is of finite codimension (prop. 13 of III, p. 53); when this is so, $u-\lambda 1_E$ is invertible if and only if $N_{\lambda}(u)$ is reduced to 0 (III, p. 47, prop. 6). The sensitive points of the spectrum of $u$ are therefore the complex numbers $\lambda$ such that $N_{\lambda}(u)$ is of nonzero finite dimension and $I_{\lambda}(u)$ is of finite codimension in E (III, p. 54, prop. 14). The spectral multiplicity $m_{\lambda}(u)$ of $\lambda$ is then the dimension of $N_{\lambda}(u)$, the endomorphism $u-\lambda 1_E$ defines by restriction an automorphism of $I_{\lambda}(u)$ and a nilpotent endomorphism of $N_{\lambda}(u)$. Since $N_{\lambda}(u)$ is not reduced to 0, it follows that $\lambda$ is an eigenvalue of $u$. The smallest integer $n\geqslant 1$ such that Ker(($u-\lambda 1_E$)$^n$) is equal to $N_{\lambda}(u)$ is the order of the pole of the resolvent of $u$ at the point $\lambda$ (No.$^o3$ of I, p. 131), which is bounded above by $m_{\lambda}(u)$.

In particular, we have therefore proved: