Exercise 8, § 7

Let $u$ be an endomorphism of a complex Hilbert space E. Let $S_u$ be the set of endomorphisms of E of the form $wuw^{-1}$ where $w$ runs through the set of invertible endomorphisms of E.

(a) One has

$\varrho (u) =$ inf$_{v\in S_u}\|v\|$.

(One may suppose that $\varrho (u)\leqslant 1$; apply the preceding exercise to the endomorphisms $u_{\varepsilon}= (1 +\varepsilon )^{-1}u$ for $\varepsilon >0$.) (b) The convex envelope of the spectrum of $u$ is equal to the intersection of the subsets $\iota (v)$ for $v\in S_u$ (“Hildebrandt theorem”).