Dilation of bounded linear operators

Question

Let $H$ be a Hilbert space, and let $A$ be a contraction (bounded linear operator of norm $\leq 1$) on $H$. I heard in a recent talk that there is a (apparently famous) result due to Sz-Nagy which roughly states that there exists a Hilbert space $\tilde{H}$ containing $H$ as a subspace and a unitary operator $U: \tilde{H} \to \tilde{H}$ such that $T = U|_{H}$. There also seems to be an explicit formula for the dilation due to Julia-Halmos.

I am trying to get some perspective on this result; in particular, I would be very happy to learn some non-trivial applications or consequences of this result. I apologize for the soft question.

Answer 1

See e.g. Dilation theory: a guided tour by O. M. Shalit.

From the abstract there:

For example, every contraction can be dilated to [...] a unitary operator, and on this simple fact a penetrating theory of non-normal operators has been developed. In the first part of this survey, I will leisurely review key classical results on dilation theory for a single operator or for several commuting operators, and sample applications of dilation theory in operator theory and in function theory. Then, in the second part, I will give a rapid account of a plethora of variants of dilation theory and their applications. In particular, I will discuss dilation theory of completely positive maps and semigroups, as well as the operator algebraic approach to dilation theory.

Answer 2

The famous application is proving von Neumann's inequality $$\|p(A)\| \leq \|p\|_{\mathbb T, \infty}$$ for every complex polynomial. This is achieved by letting $$ U = \left[\begin{matrix}\ddots&\ddots \\ & I_H & 0 \\ &&(I_H - AA^*)^{1/2} &A \\ && A^* &(I_H - A^*A)^{1/2}& 0 \\ &&&& I_H & 0 \\ &&&&&\ddots&\ddots \end{matrix}\right]$$ which was proved by Schafer. This has the happy property that $P_Hp(U)|_H = p(A)$ and so $$\|p(A)\| \leq \|p(U)\| = \|p\|_{\sigma(U),\infty} \leq \|p\|_{\mathbb T,\infty} $$ This is all rather old stuff and dilation theory has generated a lot of research over the years. A good place to look is "Completely Bounded Maps and Operator Algebras" by Paulsen or "Dilation Theory: a guided tour" by Shalit.