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

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

where ${\displaystyle P_{H}}$ is an orthogonal projection on H.

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 ${\displaystyle X}$, we say that V is a normal boundary dilation or a normal ${\displaystyle \partial X}$ dilation if V is a normal dilation of T and ${\displaystyle \sigma (V)\subseteq \partial X}$.

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

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

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.