Sz.-Nagy's dilation theorem

Proof

For a contraction T (i.e., (|T|\le1), its defect operator DT is defined to be the (unique) positive square root DT = (I - TT*)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.

References