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' .
is an orthogonal projection on H. V is said to be a unitary dilation (respectively, normal, isometric, etc.)
We can show that every contraction on Hilbert spaces has a unitary dilation.
For a contraction T, the operator is positive, where the continuous functional calculus is used to define the square root.
, 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 generalises Sz.-Nagy's dilation theorem as all contractions have the unit disc as a spectral set.