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

${\displaystyle T^{n}=P_{H}U^{n}\vert _{H},\quad n\geq 0.}$

Moreover, such a dilation is unique (up to unitary equivalence) when one assumes K is minimal, in the sense that the linear span of ∪_nUⁿH 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., (${\displaystyle \|T\|\leq 1}$), 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, 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:

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

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

${\displaystyle \oplus _{n\geq 0}H}$

given by

${\displaystyle S={\begin{bmatrix}T&0&0&\cdots &\\D_{T}&0&0&&\\0&I&0&\ddots \\0&0&I&\ddots \\\vdots &&\ddots &\ddots \end{bmatrix}}.}$

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:

${\displaystyle T^{n}=P_{H}S^{n}\vert _{H}=P_{H}(Q_{H'}U\vert _{H'})^{n}\vert _{H}=P_{H}U^{n}\vert _{H}.}$

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

${\displaystyle {\mathcal {R}}(X)}$

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

• 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. JSTOR 2032368.