Exercise 25, § 2

Let U be an open subset of $\mathbf{R}$. A universally measurable function $f: U\rightarrow \mathbf{R}$ is called a Loewner function if, for every complex Hilbert space E and for all Hermitian endomorphisms $u$ and $v$ of E such that Sp($u$)$\subset U$ and Sp($v$)$\subset U$, the condition $u\leqslant v$ in $\mathscr{L}(E)$ implies $f(u)\leqslant f(v)$. We denote by Loe(U) the set of Loewner functions on U. a) Let $f\in$ Loe(U). The function $f$ is increasing on U. The constant function 1 and the identity mapping of U belong to Loe(U).

b) Let $f\in$ Loe($\mathbf{R}^*_+$). Let E be a complex Hilbert space and $u$ a positive endomorphism of E. For every orthogonal projector $p$ of E and every $\varepsilon >0$, one has $f(u)_p\leqslant f((1 +\varepsilon )u_p)$, where $v_p$ denotes the endomorphism of Im($p$) defined by $x\mapsto p(v(x))$ for every $v\in \mathscr{L}(E)$. (Identify E with the Hilbert direct sum Im($p$)$\oplus$ Im(1$_E-p$) and represent $u$ as a matrix $(a b)((1 +\varepsilon )a$ 0 )

; note that $u\leqslant -_1$ for every $\varepsilon >0.)$

$c d$ 0 $(1 +\varepsilon )d$

c) Let $f\in$ Loe($\mathbf{R}^*_+$). The function $f$ is concave on $\mathbf{R}_+^*$. (Let $v$ be an endomorphism of $\mathbf{C}^2$ diagonal in the canonical basis and let $g\in \mathbf{S}\mathbf{O}(\mathbf{R}^2)$ viewed as an endomorphism of $\mathbf{C}^2$; apply the preceding question to the orthogonal projector with image $\mathbf{C}e_1$ and to $u=g^*vg.$)

d) The set Loe($\mathbf{R}$) is the set of functions of the form $f(x) =ax+b$ where $(a, b)\in \mathbf{R}_+\times \mathbf{R}$. (Apply the preceding question to $f$ and also to the function $x\mapsto -f(-x).$)

e) Let $t\in \mathbf{R}_+^*$. The function $f:x\mapsto -(t+x)^{-1}$ belongs to Loe($\mathbf{R}_+^*$).

f ) For every bounded positive measure $\nu$ on $\mathbf{R}^*_+$ such that the function $x\mapsto x^{-1}$ is $\nu$-integrable, the function $f:\mathbf{R}^*_+\rightarrow \mathbf{R}$ defined by

belongs to Loe($\mathbf{R}_+^*$); it is differentiable on $\mathbf{R}_+^*$ and moreover satisfies the following conditions:

(i) lim$_{t\rightarrow 0}f(t) = 0$;

(ii) lim$_{t\rightarrow 0}f'(t)$ exists in $\mathbf{R}$;

(iii) $f$ has a finite limit at $+\infty$.