Sz.-Nagy's dilation theorem

Short description: Dilation theorem

The Sz.-Nagy dilation theorem (proved by Béla Szőkefalvi-Nagy) states that every contraction [math]\displaystyle{ T }[/math] on a Hilbert space [math]\displaystyle{ H }[/math] has a unitary dilation [math]\displaystyle{ U }[/math] to a Hilbert space [math]\displaystyle{ K }[/math], containing [math]\displaystyle{ H }[/math], with

[math]\displaystyle{ T^n = P_H U^n \vert_H,\quad n\ge 0, }[/math]

where [math]\displaystyle{ P_H }[/math] is the projection from [math]\displaystyle{ K }[/math] onto [math]\displaystyle{ H }[/math]. Moreover, such a dilation is unique (up to unitary equivalence) when one assumes K is minimal, in the sense that the linear span of [math]\displaystyle{ \bigcup\nolimits_{n\in \mathbb N} \,U^n H }[/math] 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., ([math]\displaystyle{ \|T\|\le1 }[/math]), its defect operator D_T is defined to be the (unique) positive square root D_T = (I - T*T)^½. In the special case that S is an isometry, D_S* is a projector and D_S=0, hence the following is an Sz. Nagy unitary dilation of S with the required polynomial functional calculus property:

[math]\displaystyle{ U = \begin{bmatrix} S & D_{S^*} \\ D_S & -S^* \end{bmatrix}. }[/math]

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

[math]\displaystyle{ \oplus_{n \geq 0} H }[/math]

given by

[math]\displaystyle{ S = \begin{bmatrix} T & 0 & 0 & \cdots & \\ D_T & 0 & 0 & & \\ 0 & I & 0 & \ddots \\ 0 & 0 & I & \ddots \\ \vdots & & \ddots & \ddots \end{bmatrix} . }[/math]

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:

[math]\displaystyle{ T^n = P_H S^n \vert_H = P_H (Q_{H'} U \vert_{H'})^n \vert_H = P_H U^n \vert_H. }[/math]

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

[math]\displaystyle{ \mathcal{R}(X) }[/math]

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

• Paulsen, V. (2003). Completely Bounded Maps and Operator Algebras. Cambridge University Press. 
• Schaffer, J. J. (1955). "On unitary dilations of contractions". Proceedings of the American Mathematical Society 6 (2): 322. doi:10.2307/2032368.