Probability matching

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

$T^n=P_H{U}^n{|}_H,n\ge 0.$

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., ($\Vert T\Vert \le 1$), 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, the following is an Sz. Nagy unitary dilation of S with the required polynomial functional calculus property:

$U=\left[\begin{array}{cc}S& D_{S^{*}}\\ 0& -S^{*}\end{array}\right].$

Also, every contraction T on a Hilbert space H has an isometric dilation, again with the calculus property, on

${\oplus }_{n\ge 0}H$

given by

$V=\left[\begin{array}{cccc}T& 0& & \\ D_T& 0& \ddots & \\ 0& I& 0& \\ & \ddots & \ddots & \end{array}\right].$

Applying the above two constructions successively gives a unitary dilation for a contraction T:

$T^n=P_H{S}^n{|}_H=P_H(Q_{H^{\prime }}U{|}_{H^{\prime }}{)}^n{|}_H=P_H{U}^n{|}_H.$

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

• V. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press, 2003.

• J.J. Schaffer, On unitary dilations of contractions, Proc. Amer. Math. Soc. 6, 1955, 322.