en>Limit-theorem: clarifying example

2014-01-25T15:57:36Z

clarifying example

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The '''Sz.-Nagy dilation theorem''' (proved by [[Béla Szőkefalvi-Nagy]]) states that every contraction ''T'' on a [[Hilbert space]] ''H'' has a unitary [[Dilation (operator theory)|dilation]] ''U'' to a Hilbert space ''K'', containing ''H'', with
:<math>T^n = P_H U^n \vert_H,\quad n\ge 0.</math>
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''''UnK'' is dense in ''K''. When this minimality condition holds, ''U'' is called the '''minimal unitary dilation''' of ''T''.

== Proof ==
For a [[contraction (operator theory)|contraction]] ''T'' (i.e., (<math>\|T\|\le1</math>), its '''defect operator''' ''DT'' is defined to be the (unique) positive square root ''DT'' = (''I - T*T'')½. In the special case that ''S'' is an isometry, the following is an Sz. Nagy unitary dilation of ''S'' with the required polynomial functional calculus property:

:<math>U =
\begin{bmatrix} S & D_{S^*} \\ 0 & -S^* \end{bmatrix}.
</math>

Also, every contraction ''T'' on a Hilbert space ''H'' has an isometric dilation, again with the calculus property, on

:<math>\oplus_{n \geq 0} H</math>

given by

:<math>V =

\begin{bmatrix} T & 0 & & \\ D_T & 0 & \ddots & \\ 0 & I & 0 & \\ & \ddots & \ddots & \end{bmatrix}
.</math>

Applying the above two constructions successively gives a unitary dilation for a contraction ''T'':

:<math>
T^n = P_H S^n \vert_H = P_H (Q_{H'} U \vert_{H'})^n \vert_H = P_H U^n \vert_H.
</math>

== Schaffer form ==
{{Expand section|date=June 2008}}

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

:<math>\mathcal{R}(X)</math>

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==

*V. Paulsen, ''Completely Bounded Maps and Operator Algebras'', Cambridge University Press, 2003.

*J.J. Schaffer, On unitary dilations of contractions, ''Proc. Amer. Math. Soc.'' '''6''', 1955, 322.

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

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en>Limit-theorem: clarifying example