a^* & 1
\end{array}\right)$. This proves our claim.

Observe also that for any selfadjoint $x\in\mathcal A$, we have $x>0$ if and only if for any invertible $a\in\mathcal A$, we have $a^*xa>0$. Indeed, one implication is obvious. Conversely, if $a$ is invertible and $a^*xa>0$, then there is an $\varepsilon_a\in(0,+\infty)$ such that $a^*xa>\varepsilon_a\cdot1_\mathcal A$. For any $\xi\in\mathcal H$, $\langle x\xi,\xi\rangle=\langle x
a(a^{-1}\xi),a(a^{-1}\xi)\rangle=\langle a^*x
a(a^{-1}\xi),(a^{-1}\xi)\rangle>\varepsilon_a\|(a^{-1}\xi)\|_2^2\ge
\varepsilon_a\|a\|^{-2}\|\xi\|_2^2,$ independently of $\xi$, so that $x\ge\varepsilon_a\|a\|^{-2}\cdot1_\mathcal A>0.$ We use these last two results to conclude that $$\left(\begin{array}{cc}
u & v\\
v^* & w
\end{array}\right)>0\text{ in }M_2(\mathcal A)\iff u,w>0\text{ in }\mathcal A\text{ and }
\left\{\begin{array}{c}
u>vw^{-1}v^* \\
\text{and/or}\\
w>v^*u^{-1}v 
\end{array}\right..$$ This follows from the above by writing $$\left(\begin{array}{cc}
u & v\\
v^* & w
\end{array}\right)=
\left(\begin{array}{cc}
u^\frac12 & 0\\
0 & w^\frac12
\end{array}\right)
\left(\begin{array}{cc}
1 & u^{-\frac12}vw^{-\frac12}\\
w^{-\frac12}v^*u^{-\frac12} & 1
\end{array}\right)
\left(\begin{array}{cc}
u^\frac12 & 0\\
0 & w^\frac12
\end{array}\right)$$ and recalling the chain of equivalences $\|u^{-1/2}vw^{-1/2}(u^{-1/2}vw^{-1/2})^*\|<1\iff
\|(u^{-1/2}vw^{-1/2})^*u^{-1/2}vw^{-1/2}\|<1\iff
u^{-1/2}vw^{-1/2}(u^{-1/2}vw^{-1/2})^*<1\iff (u^{-1/2}vw^{-1/2})^*u^{-1/2}vw^{-1/2}<1.$ We shall use these facts below without further referencing to them.

The Julia-Wolff-Carathéodory Theorem, classical and noncommutative
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We state the classical Julia-Wolff-Carathéodory Theorem for analytic self-maps of the upper half-plane $\mathbb C^+$ at a point of the real line $\mathbb R$. In the following we denote by $\displaystyle
