in my research on dilations of contractions on Hilbert spaces and manifolds I have come across this nice publication concerning the classic Sz-Nagy theorem on the Arxiv by Levy and Shalit which states the classical Sz-Nagy theorem on the existence of unitary and isometric dilations for contractions and their minimality:


<http://arxiv.org/pdf/1012.4514.pdf>


Right below quoting the famous theorem they mentioned that if the operator is not a unitary contraction then we must have that the larger Hilbert space K is infinite dimensional. Please forgive my ignorance but I cannot seem to understand why and it seems like a nice result I wish to understand. If the contraction to be dilated is not a unitary operator then why is the larger space embedding H is necessarily infinite dimensional? I am sorry if it was trivial and I missed it but I think I need to understand this. I thank all helpers



Help in understanding result from publication on operator theory
There are probably many ways to see this, but here is a quick one. Suppose $T$ is a contractive matrix which dilates to a unitary matrix $U$, we wish to show that $T$ is already unitary. Now, to say that the matrix $U$ dilates $T$ means there is an isometric matrix $V$ so that
$$ T^n = V^\* U^n V $$
for all $n\geq 0$. From this it follows that the minimal polynomial $p$ for $U$ must annihilate $T$, but since all the roots of $p$ are unimodular, it follows that all the eigenvalues of $T$ are unimodular, and thus $T$ is unitary. 


