Proposition 3

Let E and F be complete normed spaces, $u: E\rightarrow F$ and $v: F\rightarrow E$ continuous linear mappings.

a) The traces on $\mathbf{C}-\{0\}$ of the sets Sp($v\circ u$) and Sp($u\circ v$) $($resp. Sp$_s(v\circ u)$ and Sp$_s(u\circ v)$, resp. Sp$_n(v\circ u)$ and Sp$_n(u\circ v)$ for $n\in \overline{\mathbf{Z}}$, resp. Sp$_{\omega}(v\circ u)$ and Sp$_{\omega}(u\circ v))$ are equal;

b) Let $\lambda$ be an element of Sp$_s(v\circ u)$ distinct from 0. The spectral multiplicities of $\lambda$ for $v\circ u$ and for $u\circ v$ are equal.

Let $\mu$ be a nonzero complex number and let $n\in \overline{\mathbf{Z}}$. In order that $\mu1_E-v\circ u$ be an automorphism (resp. a Riesz endomorphism, resp. a strict morphism whose kernel or cokernel is of finite dimension and whose index is $n$), it is necessary and sufficient that $\mu1_F-u\circ v$ be one (III, p. 49, prop. 10). Assertion a) then follows from the definitions.

Let $\lambda$ be a point of Sp$_s(v\circ u)$ distinct from 0. One has

dim Ker(($\lambda 1_E-v\circ u$)$^n$) $=$ dim Ker(($\lambda 1_F-u\circ v$)$^n$)

for every $n\geqslant 0 ($loc. cit.), therefore the spectral multiplicities of $\lambda$ for $v\circ u$ and $u\circ v$ are equal.