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.

Moreover, such a dilation is unique (up to unitary equivalence) when one assumes K is minimal, in the sense that the linear span of ∪nUnK 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, the following is an Sz. Nagy unitary dilation of S with the required polynomial functional calculus property:

U = \begin{bmatrix} S & D_{S^*} \\ 0 & -S^* \end{bmatrix}.

Also, 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

V = \begin{bmatrix} T & 0 & & \\ D_T & 0 & \ddots & \\ 0 & I & 0 & \\ & \ddots & \ddots & \end{bmatrix} .

Applying the above two constructions successively gives 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