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
where formula_2 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 formula_3, we say that "V" is a normal boundary dilation or a normal formula_4 dilation if "V" is a normal dilation of "T" and formula_5.
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.
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"("I - T*T") = ("I - TT*")"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 formula_11, 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 is Sz.-Nagy's dilation theorem. More generally, if formula_14 is a Dirichlet algebra, any operator "T" with formula_3 as a spectral set will have a normal formula_4 dilation with this property. This generalises Sz.-Nagy's dilation theorem as all contractions have the unit disc as a spectral set.