Exercise 12, § 2

Tag 02ZV, permanent. Cite it as [Bourbaki, Tag 02ZV].

Spectral Theories, IV, p. 321

a) Let $u$ and $v$ be positive continuous endomorphisms of E. Suppose that for all $t >0$ , one has

2tu^2\circ (u^2+t^21_E)^{-1}\leqslant v\leqslant \frac{1}{2t}(u^2+t^21_E)

Prove that $u=v$ .

b) For $u$ to be a normal endomorphism, it is necessary and sufficient that $u$ and $u^*$ be paranormal and have the same kernel.

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