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\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., ( $\|T\|\leq 1$ ), 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:

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

\oplus _{n\geq 0}H

given by

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:

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

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.

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