Exercise 10, § 2

It is said that a continuous endomorphism $u$ of E is hyponormal if one has $u^*\circ u\geqslant u\circ u^*$, and that it is paranormal if $\|u^2(x)\|\|x\|\geqslant \|u(x)\|^2$ for all $x\in E$.

(a) If $u$ is paranormal, then $u^n$ is paranormal for all $n\in \mathbf{N}$.

(b) If $u$ is paranormal and invertible, then $u^{-1}$ is paranormal.

(c) A hyponormal endomorphism is paranormal.

(d) Let $u$ be a paranormal endomorphism whose spectrum is contained in the unit circle; then $u$ is unitary. (e) If $u$ is a paranormal endomorphism, its norm $\|u\|$ is equal to its spectral radius $\varrho (u)$.

(f) Suppose that $u$ is paranormal.