Proposition 3

Let E be a Banach space and let $E'$ be its dual. Let $u$ be an endomorphism of E.

a) We have Sp($u$) $=$ Sp($^tu$) ;

b) For every $f\in \mathscr{O}$(Sp($u$)), we have $f(^tu) =^tf(u)$ ;

c) We have $e_H(^tu) =^te_H(u)$ for every subset H of Sp($u$) which is open and closed.

For an endomorphism of E to be an automorphism, it is necessary and sufficient that its transpose be an automorphism of $E'$ (EVT, IV, p. 30, cor. 5), whence assertion a).

The mapping $v\mapsto^tv$ is a continuous unital homomorphism of the Banach algebra $\mathscr{L}(E)$ into the opposite Banach algebra of $\mathscr{L}(E')$ (EVT, IV, p. 7, prop. 8). Since the algebra $\mathscr{O}$(Sp($^tu$)) is commutative, the mapping $f\mapsto^tf(u)$ is a continuous unital homomorphism of the algebra $\mathscr{O}$(Sp($^tu$)) into the algebra $\mathscr{L}(E')$. This homomorphism maps the germ of the identity mapping of $\mathbf{C}$ to $^tu$, and therefore coincides with the homomorphism $f\mapsto f(^tu)$ (th. 5 of I, p. 74). This proves b).

Assertion c) results from b), applied to the function $f_H$ equal to 1 in a neighbourhood of H and to 0 in a neighbourhood of its complement in Sp($u$).