In operator theory, a dilation of an operator T on a Hilbert spaceH 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 where is an orthogonal projection on H. V is said to be a unitary dilation if V is unitary. 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 property: where f is some specified functional 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 DT 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 = T and a limit argument imply Using this one can show, by calculating directly, that V is unitary, therefore a 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 is sometimes called the elementary rotation of T. We note here that in the above discussion we have not required the calculusproperty 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.
