Dilation (operator theory)

In operator theory, a unitary dilation is a dilation which is also a unitary operator. Let T be a bounded operator on some Hilbert space H, and H be a subspace of a larger Hilber space H' . A bounded operator V on H' is a dilation of T if

$P_{H}\;V|_{H}=T$

,where $P_{H}$ is projection on H. V is said to be a unitary dilation (respectively, isometric, etc) if V is unitary (respectively, isometric, etc). V is said to be a compression of T. We note here that, in the literature, a more restrictive definition is sometimes used. Namely it is required that a dilation satisfies the following (calculus) property:

$P_{H}\;f(V)|_{H}=f(T)$

, where f(T) is some specified functional calculus (for example, the polynomial or $H^{\infty }$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.

Example We now show that every contraction on Hilbert spaces has a unitary, and also an isometric, dilation. A possible construction of this dilation is as follows. For a contraction T, the operator $D_{T}=(I-T^{*}T)^{\frac {1}{2}}$ is positive, where the continuous functional calculus is used to define the square root. $D_{T}$ is called the defect operator of T. Let V be the operator on $H\oplus H$ defined by the matrix

$V={\begin{bmatrix}T&D_{T^{*}}\\\ D_{T}&-T^{*}\end{bmatrix}}.$

V is clearly a dilation of T. Also, $T(I-T^{*}T)=(1-TT^{*})T$ implies $TD_{T}=D_{T^{*}}T$. Using this one can show, by calculating directly, that that V is an isometry, therefore an isometric dilation of T. This operator V is sometimes called the Julia operator of T. The Julia operator is not necessarily unitary because, for one thing, it may not be onto. With suitable identification of spaces, V, defined by the same matrix, can be made unitary. Namely, if we define the defect space $G_{T}$ as the closure of range of the defect operator $D_{T}$, then

$V:H\oplus G_{T^{*}}\rightarrow H\oplus G_{T}$

is now unitary.

Notice that when T is a real scalar, say $T=\cos \theta$, we have

$V={\begin{bmatrix}\cos \theta &\sin \theta \\\ \sin \theta &-\cos \theta \end{bmatrix}}.$

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

$T^{2}=P_{H}\;V^{2}|_{H}$.

References

T. Constantinescu, Schur Parameters, Dilation and Factorization Problems, Birkhauser Verlag, Vol. 82, ISBN 3-7643-5285-X, 1996.