Sz.-Nagy's dilation theorem

Remarks

Proof idea: the defect operator $D = (I - T^*T)^{1/2}$ satisfies $\norm{Tx}^2 + \norm{Dx}^2 = \norm{x}^2$. On $\bigoplus_{n \ge 0} H$, the operator $(x_0, x_1, \ldots) \mapsto (T x_0, D x_0, x_1, x_2, \ldots)$ is an isometry, and it compresses to the powers of $T$ on the first copy of $H$. Every isometry extends to a unitary operator on a larger space, which gives $U$.

Example: the unilateral shift on $\ell^2(\N)$ is a contraction, and the bilateral shift on $\ell^2(\Z)$ is a unitary dilation of it.

The compression property holds only for powers $n \ge 0$; for negative $n$ one has $T^{*\abs{n}} = P_H U^n|_H$ instead.

Going deeper

For a student

The dilation turns questions about a contraction into questions about a unitary operator, where the spectral theorem is available. An immediate consequence is von Neumann's inequality. The minimal unitary dilation is the starting point of the Sz.-Nagy–Foias model theory of contractions and of the commutant lifting theorem.