The Sz.-Nagy dilation theorem (proved by Béla Szőkefalvi-Nagy) states that every contraction
Moreover, such a dilation is unique (up to unitary equivalence) when one assumes K is minimal, in the sense that the linear span of
Nagy unitary dilation of S with the required polynomial functional calculus property: 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 given by 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: The Schaffer form of a unitary Sz.
A generalisation of this theorem, by Berger, Foias and Lebow, shows that if X is a spectral set for T, and 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.