Dilation (operator Theory)

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 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 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, 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.