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:$T^n\; =\; P\_H\; U^n\; vert\_H$Moreover, such a dilation is unique (up to unitary equivalence) when one assumes "K" is minimal, in the sense that the linear span of ∪_"n""UⁿK" 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 "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:

:

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

:$V\; =$

egin{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":

:$T^n\; =\; P\_H\; S^n\; vert\_H\; =\; P\_H\; (Q\_\{H\text{'}\}\; U\; vert\_\{H\text{'}\})^n\; vert\_H\; =\; P\_H\; U^n\; vert\_H.$

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 "δ"D 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.