The Sz.-Nagy dilation theorem (proved by Béla Szőkefalvi-Nagy) states that every contraction
T
H
U
K
H
Tn=PHUn\vertH, n\ge0,
PH
K
H
cup\nolimitsn\inUnH
For a contraction T (i.e., (
\|T\|\le1
U=\begin{bmatrix}S&
D | |
S* |
\ DS&-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
⊕ nH
given by
S=\begin{bmatrix}T&0&0& … &\ DT&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:
Tn=PHSn\vertH=PH(QH'U\vertH')n\vertH=PHUn\vertH.
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.
A generalisation of this theorem, by Berger, Foias and Lebow, shows that if X is a spectral set for T, and
l{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.