Lemma 2

Let $S\subset \mathbf{C}$ be an infinite, discrete, bounded and closed set in $\mathbf{C}-\{0\}$. Then S is the set of values of a sequence $(\lambda_n)_{n\in\mathbf{N}}$ of non-zero complex numbers, pairwise distinct, such that the sequence $(|\lambda_n|)_{n\in\mathbf{N}}$ is decreasing and converges to 0.

For every integer $i\geqslant 1$, the set $A_i$ of complex numbers $\lambda \in S$ such that $|\lambda |\geqslant \frac{1}{i}$ is compact and discrete in $\mathbf{C}$, hence finite. Let $a_i$ be its cardinal. Since S is infinite, the sequence $(a_i)$ tends to $+\infty$. Put $A_0=\emptyset$ and $a_0= 0$. For every $i\geqslant 1$, choose a bijection $n\mapsto \lambda_n$ of the interval $[a_{i-1}, a_i[$ of $\mathbf{N}$ onto $A_i-A_{i-1}$ such that the mapping $n\mapsto |\lambda_n|$ is decreasing on $[a_{i-1}, a_i[$. The sequence $(\lambda_n)_{n\in\mathbf{N}}$ satisfies the required properties.

Let E be a complete normable space of infinite dimension. The algebra $\mathscr{L}^c(E)$ is a non-unital subalgebra of $\mathscr{L}(E)$. Recall that for every compact endomorphism $u\in \mathscr{L}^c$(E), the spectrum Sp$'_{\mathscr{L}^c(E)}(u)$ is the spectrum of $u$ relative to the unital subalgebra $\mathscr{L}^c(E)\oplus \mathbf{C}1_E$ of $\mathscr{L}(E) ($I, p. 4, No.$^o4$).