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