Julia's lemma for analytic operator functions (Q1114911)

Let f:D\(\to {\mathbb{C}}\) be analytic in the open unit disc D such that \(| f(z)| <1\) and f(x)\(\to 1\) as \(x\to 1-0\). Then \(\lim (1-f(x))/(1- x)=\alpha\) (x\(\to 1-0)\), where \(0<\alpha \leq +\infty\) is the smallest constant such that \[ \frac{| 1-f(z)|^ 2}{1-| f(z)|^ 2}\leq \alpha \frac{| 1-z|^ 2}{1-| z|^ 2},\quad | z| <1. \] This lemma of Julia and Wolff can be generalized to the setting where f:D\(\to L(H)\), H a complex Hilbert space, is an analytic operator function such that \(f(z_ 1)f(z_ 2)=f(z_ 2)f(z_ 1)\) for all \(z_ 1,z_ 2\in D\), and T, \(\| T\| <1\), is an operator which commutes which each f(z). The generalized Lemma provides estimates for \((I-f(T))(I-f(T)^*f(T))^{-1}(I-f(T)^*)\) and \((I-f(T)^*)(I-f(T)).\)