Exercise 22, § 1

Let E be a complex Hilbert space. Let $u$ and $v$ be positive endomorphisms of E.

a) For every real number $p$ such that $0< p <1$ and every $x\in \mathbf{R}_+$, one has

$x^p=$ sin($\pi p\pi$ )$\int_0^{+\infty}t^p(\frac{1}{t}-\frac{1}{t + x})dt$.

b) Suppose that $u\leqslant v$. For every real number $p$ such that $0< p <1$, one has $u^p\leqslant v^p$. (Use Lemma 14 of I, p. 119, b) and the previous question.)

c) Let $n\in \mathbf{N}$. Put $a=v^{n/2}\circ u^n\circ v^{n/2}$ and $b= (v^{1/2}\circ u\circ v^{1/2})^n$. If $a\leqslant 1_E$, then $b\leqslant 1_E$.

d) Suppose that $u$ and $v$ are of finite trace. For every integer $n\in \mathbf{N}$, one has

Tr(($v^{1/2}\circ u\circ v^{1/2}$)$^n$)$\leqslant$ Tr($v^{n/2}\circ u^n\circ v^{n/2}$). (Let $a$ and $b$ be as in question c); using Lemma 5 of IV, p. 161, reduce oneself to proving that the greatest eigenvalue of $\widehat{\wedge}^nb$ is less than that of $\widehat{\wedge}^na$; replace E by $\widehat{\wedge}^nE$ and $u,v$ by $\widehat{\wedge}^nu$ and $\widehat{\wedge}^nv$, respectively, then apply the previous question.)