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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.

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I found the solution this morning. This is not very difficult by using the unitary dilation of a contraction. It's maybe why I can not find a proof of the theorem.

Let $T$ be a contraction on a Hilbert space $\mathfrak{H}$. Then there exist a Hilbert space $\mathfrak{K}\supset \mathfrak{H}$ and a unitary operator $U\in B(\mathfrak{K})$ such that $T^n = P_{\mathfrak{H}}U^n|_{\mathfrak{H}}$ where $P_{\mathfrak{H}}$ is the projection. This $U$ is called the (unitary) dilation of $T$. We can find a proof in a book of Sz-Nagy and Foias.

Returning to the above problem, let $T = \pmatrix{A&B\\C&D}$, then there exist a Hilbert space K' and a unitary operator U\in \mathcal{B}(H\oplus K \oplus K') such that $U$ is of the form U =\pmatrix{T&* \\ 0 & *} = \pmatrix{A' & B' \\ C' & D'} where A' = A. Then the above transfer function $f(z) = P_{K}F(z)|K$, where F(z) = D' + C'z (1-zA')^{-1}B'. We have $\|F(z)\|\leq 1$ as we said in the unitary case. Therefore, it finishes the proof.