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:<math>P_H \; f(V) | _H = f(T)</math>
:<math>P_H \; f(V) | _H = f(T)</math>


, where ''f(T)'' is some specified [[functional calculus]] (for example, the polynomial or ''H''<sup>&infin;</sup> 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]].
where ''f(T)'' is some specified [[functional calculus]] (for example, the polynomial or ''H''<sup>&infin;</sup> 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==
'''Example''' We now 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

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


:<math>D_T = (I - T^* T)^{\frac{1}{2}}</math>
:<math>D_T = (I - T^* T)^{\frac{1}{2}}</math>

Revision as of 13:19, 18 January 2007

In operator theory, a dilation of an operator on a Hilbert space H is an operator on a larger Hilbert space K , whose restriction to H is . 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

where 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:

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 implies

Using this one can show, by calculating directly, that that V is unitary, therefore an unitary dilation of T. This operator V is sometimes called the Julia operator of T.

Notice that when T is a real scalar, say , 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

.

References

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