Sz.-Nagy's dilation theorem

The Sz.-Nagy dilation theorem (proved by Béla Szőkefalvi-Nagy) states that every contraction $$T$$ on a Hilbert space $$H$$ has a unitary dilation $$U$$ to a Hilbert space $$K$$, containing $$H$$, with
 * $$T^n = P_H U^n \vert_H,\quad n\ge 0,$$

where $$P_H$$ is the projection from $$K$$ onto $$H$$. Moreover, such a dilation is unique (up to unitary equivalence) when one assumes K is minimal, in the sense that the linear span of $$\bigcup\nolimits_{n\in \mathbb N} \,U^n H$$ is dense in K. When this minimality condition holds, U is called the minimal unitary dilation of T.

Proof
For a contraction T (i.e., ($$\|T\|\le1$$), its defect operator DT is defined to be the (unique) positive square root DT = (I - T*T)½. In the special case that S is an isometry, DS* is a projector and DS=0, hence the following is an Sz. Nagy unitary dilation of S with the required polynomial functional calculus property:


 * $$U =

\begin{bmatrix} S & D_{S^*} \\ D_S & -S^* \end{bmatrix}. $$

Returning to the general case of a contraction T, every contraction T on a Hilbert space H has an isometric dilation, again with the calculus property, on


 * $$\oplus_{n \geq 0} H$$

given by


 * $$S =

\begin{bmatrix} T & 0 & 0 & \cdots & \\ D_T & 0 & 0 & & \\ 0 & I & 0 & \ddots \\ 0 & 0 & I & \ddots \\ \vdots & & \ddots & \ddots \end{bmatrix} .$$

Substituting the S thus constructed into the previous Sz.-Nagy unitary dilation for an isometry S, one obtains a unitary dilation for a contraction T:



T^n = P_H S^n \vert_H = P_H (Q_{H'} U \vert_{H'})^n \vert_H = P_H U^n \vert_H. $$

Schaffer form
The Schaffer form of a unitary Sz. Nagy dilation can be viewed as a beginning point for the characterization of all unitary dilations, with the required property, for a given contraction.

Remarks
A generalisation of this theorem, by Berger, Foias and Lebow, shows that if X is a spectral set for T, and


 * $$\mathcal{R}(X)$$

is a Dirichlet algebra, then T has a minimal normal δX dilation, of the form above. A consequence of this is that any operator with a simply connected spectral set X has a minimal normal δX dilation.

To see that this generalises Sz.-Nagy's theorem, note that contraction operators have the unit disc D as a spectral set, and that normal operators with spectrum in the unit circle δD are unitary.