Exercise 11, § 2

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

Spectral Theories, IV, p. 321

Let $u$ be a continuous endomorphism of E.

a) For $u$ to be paranormal, it is necessary and sufficient that

(u^*)^2\circ u^2-2t(u^*\circ u) +t^21_E\geqslant 0

for every real number $t >0$ .

b) For $u$ to be paranormal, it is necessary and sufficient that

p\circ q^2\circ p-2tp^2+t1_E\geqslant 0

for every $t >0$ , where $p= (u\circ u^*)^{1/2}$ and $q= (u^*\circ u)^{1/2}$ .

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