# Dilation (operator theory)

In operator theory, a **dilation** of an operator *T* on a Hilbert space *H* is an operator on a larger Hilbert space *K*, whose restriction to *H* composed with the orthogonal projection onto *H* is *T*.

More formally, let *T* be a bounded operator on some Hilbert space *H*, and *H* be a subspace of a larger Hilbert space * H' *. A bounded operator *V* on * H' * is a dilation of T if

*V* is said to be a **unitary dilation** (respectively, normal, isometric, etc.) if *V* is unitary (respectively, normal, isometric, etc.). *T* is said to be a **compression** of *V*. If an operator *T* has a spectral set , we say that *V* is a **normal boundary dilation** or a **normal dilation** if *V* is a normal dilation of *T* and .

Some texts impose an additional condition. Namely, that a dilation satisfy the following (calculus) property:

where *f(T)* is some specified functional calculus (for example, the polynomial or *H*^{∞} calculus). The utility of a dilation is that it allows the "lifting" of objects associated to *T* to the level of *V*, where the lifted objects may have nicer properties. See, for example, the commutant lifting theorem.

## Applications

We can show that every contraction on Hilbert spaces has a unitary dilation. A possible construction of this dilation is as follows. For a contraction *T*, the operator

is positive, where the continuous functional calculus is used to define the square root. The operator *D _{T}* is called the

**defect operator**of

*T*. Let

*V*be the operator on

defined by the matrix

*V* is clearly a dilation of *T*. Also, *T*(*I - T*T*) = (*I - TT**)*T* implies

Using this one can show, by calculating directly, that *V* is unitary, therefore an unitary dilation of *T*. This operator *V* is sometimes called the **Julia operator** of *T*.

Notice that when *T* is a real scalar, say , we have

which is just the unitary matrix describing rotation by θ. For this reason, the Julia operator *V(T)* is sometimes called the *elementary rotation* of *T*.

We note here that in the above discussion we have not required the calculus property for a dilation. Indeed, direct calculation shows the Julia operator fails to be a "degree-2" dilation in general, i.e. it need not be true that

However, it can also be shown that any contraction has a unitary dilation which **does** have the calculus property above. This is Sz.-Nagy's dilation theorem. More generally, if is a Dirichlet algebra, any operator *T* with as a spectral set will have a normal dilation with this property. This generalises Sz.-Nagy's dilation theorem as all contractions have the unit disc as a spectral set.

## References

- T. Constantinescu,
*Schur Parameters, Dilation and Factorization Problems*, Birkhauser Verlag, Vol. 82, ISBN 3-7643-5285-X, 1996. - Vern Paulsen,
*Completely Bounded Maps and Operator Algebras*2002, ISBN 0-521-81669-6