Good morning,
I have searched, by using google for a time, a proof of the following theorem :
Let $\pmatrix{A&B \\ C&D}\colon H \oplus K \to H\oplus K$ be a contractive operator of a Hilbert space where H and K are Hilbert subspaces. Then we have the function $f\colon z\in\mathbb{D}\mapsto D + Cz(1-zA)^{-1}B \in \mathcal{B}(K)$, from the open unit disc to the space of bounded operators on $K$, is a holomorphic function such that $\|f\|_{\infty} = \sup_{z\in\mathbb{D}} \|f(z)\|\leq 1$, where $\|f(z)\|$ is the operator norm.
But I have not found yet. This theorem is called the realization theorem for functions of the Schur class. Does anyone have a proof of it? Thanks in advance.
Duc Anh
EDIT : in the case $\pmatrix{A&B \\ C&D}$ is unitary, everything is simple, so the difficult case is when it is a contractive operator in general.