Proposition 4

Let E be a complete normed space, $E'$ the dual space of E and $u$ an endomorphism of E.

a) One has Sp$_s(u) =$ Sp$_s(^tu)$, Sp$_n(u) =$ Sp$_{-n}(^tu)$ for every $n\in \overline{\mathbf{Z}}$ and Sp$_{\omega}(u) = Sp_{\omega}(^tu)$;

b) Every point of Sp$_s(u)$ has the same spectral multiplicity for $u$ and for $^tu$.

Assertion a) of prop. 3 of I, p. 131 proves that Sp($u$) $=$ Sp($^tu$), and assertion c) of loc. cit. implies that Sp$_s(u) =$ Sp$_s(^tu)$ and that assertion b) is valid.

For every $\lambda \in \mathbf{C}$, lemma 4 of III, p. 69 implies that $u-\lambda 1_E$ is a strict morphism if and only if $^tu-\lambda 1_{E'}$ is one, and that

dim Coker($^t(u-\lambda 1_E)$) $=$ dim Ker($u-\lambda 1_E$)

dim Ker($^t(u-\lambda 1_E)$) $=$ dim Coker($u-\lambda 1_E$)

in $\overline{\mathbf{Z}}$. Assertion a) follows from this, taking account of the definitions of the various parts of the spectrum (def. 3 of III, p. 85).