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
:$T^n = P_H U^n \vert_H,\quad n\ge 0.$
Moreover, such a dilation is unique (up to unitary equivalence) when one assumes ''K'' is minimal, in the sense that the linear span of ∪_''n''''UⁿH'' 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, ''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:

:$U = \begin S & D_ \\ D_S & -S^* \end.$

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

:$\oplus_ H$

given by

:$S = \begin T & 0 & 0 & \cdots & \\ D_T & 0 & 0 & & \\ 0 & I & 0 & \ddots \\ 0 & 0 & I & \ddots \\ \vdots & & \ddots & \ddots \end .$

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'':

:$T^n = P_H S^n \vert_H = P_H (Q_ U \vert_)^n \vert_H = P_H U^n \vert_H.$

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

:$\mathcal(X)$

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

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{{DEFAULTSORT:Sz.-Nagy's Dilation Theorem
Operator theory
Articles containing proofs
Theorems in functional analysis