Exercise 26, § 2

Let E be a complex Hilbert space and $a$ a positive endomorphism of E. We write $\|x\|_a=\langle x|a(x)\rangle$; this is a seminorm on E. We also write $\|u\|_a$ for the seminorm of an endomorphism $u$ of the space E endowed with this seminorm.

An interpolation function is any function $f:\mathbf{R}_+^*\rightarrow \mathbf{R}^*_+$ such that, for every complex Hilbert space E, every positive endomorphism $a$ of E and every $u\in \mathscr{L}$ (E), the conditions $\|u\|\leqslant 1$ and $\|u\|_a\leqslant 1$ imply that $\|f(u)\|_a\leqslant 1$.

a) A function $f:\mathbf{R}^*_+\rightarrow \mathbf{R}_+^*$ is an interpolation function if and only if, for every complex Hilbert space E, every positive endomorphism $a$ of E and every $u\in \mathscr{L}$ (E), the conditions $u^*u\leqslant 1_E$ and $u^*au\leqslant a$ imply $u^*f(a)u\leqslant f(a)$

b) Every Loewner function on $\mathbf{R}^*_+$ is an interpolation function.

(Given E and $u\in \mathscr{L}(E)$ of norm $\leqslant 1$, put $w=\overline{1_E -u^*u}$ and $w'=$ $\surd$

$\overline{1_E -uu^*}$; consider $F = E\oplus E$ and note that the endomorphism $v$ of F defined by $v(x, y) = (u(x) +w'(y),-w(x) +u(y))$ is an isometry; then consider $a': (x, y)\mapsto (a(x),0)$ and apply part b) of the preceding exercise to the orthoprojector with image $E\oplus \{0\}.)$

c) Every interpolation function is a Loewner function on $\mathbf{R}_+^*$. (Let $0\leqslant u\leqslant v$ be endomorphisms of a complex Hilbert space E with $v$ invertible; put $F = E\oplus E$, and consider the endomorphism $a=u\oplus v$

$($ 0 $1_E)$ of F, as well as the endomorphism $w$ represented by the matrix $_{-1/21/2}$ ;

$vu$ 0

prove that $w^*w\leqslant 1_F$ and $w^*aw\leqslant a.)$

d) Let $f$ be an interpolation function. Let E be a complex Hilbert space; we write $\mathscr{R}(u) =\frac{1}{2}(u+u^*)$ for every $u\in \mathscr{L}(E)$. For $u$ and $v$ in $\mathscr{L}$ (E), prove that $e^{-tv^*}\circ u\circ e^{-tv}\leqslant u$ for every $t\in \mathbf{R}_+$ if and only if $\mathscr{R}(u\circ v)\geqslant 0$. (For given $x\in E$, calculate the derivative of the function $t\mapsto \langle e^{-tv}x|(ue^{-tv})x\rangle .$)

Deduce that for every endomorphisms $u$ and $v$ of E such that the conditions

are satisfied, one has $\mathscr{R}(f(u)v)\geqslant 0$.