Dilation (operator theory): Difference between revisions
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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''. |
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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 |
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:<math>P_H \; V | _H = T</math> |
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where <math>P_H</math> is projection on ''H''. |
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''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]] <math>X</math>, we say that ''V'' is a '''normal boundary dilation''' or a '''normal <math>\partial X</math> dilation''' if ''V'' is a normal dilation of ''T'' and <math>\sigma(V)\in\partial X</math>. |
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It should be noted that some texts impose an additional condition. Namely, that a dilation satisfy the following (calculus) property: |
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:<math>P_H \; f(V) | _H = f(T)</math> |
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where ''f(T)'' is some specified [[functional calculus]] (for example, the polynomial or ''H''<sup>∞</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]]. |
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==Applications== |
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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 |
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:<math>D_T = (I - T^* T)^{\frac{1}{2}}</math> |
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is positive, where the [[continuous functional calculus]] is used to define the square root. The operator ''D<sub>T</sub>'' is called the '''defect operator''' of ''T''. Let ''V'' be the operator on |
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:<math>H \oplus H</math> |
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defined by the matrix |
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:<math> V = |
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\begin{bmatrix} T & D_{T^*}\\ |
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\ D_T & -T^* |
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\end{bmatrix}. </math> |
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''V'' is clearly a dilation of ''T''. Also, ''T''(''I - T*T'') = (''I - TT*'')''T'' implies |
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:<math> T D_T = D_{T^*} T.</math> |
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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''. |
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Notice that when ''T'' is a real scalar, say <math>T = \cos \theta</math>, we have |
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:<math> V = |
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\begin{bmatrix} \cos \theta & \sin \theta \\ |
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\ \sin \theta & - \cos \theta |
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\end{bmatrix}. </math> |
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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''. |
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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 |
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:<math>T^2 = P_H \; V^2 | _H</math>. |
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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 <math>\mathcal{R}(X)</math> is a [[Dirichlet algebra]], any operator ''T'' with <math>X</math> as a spectral set will have a normal <math>\partial X</math> dilation with this property. This generalises Sz.-Nagy's dilation theorem as all contractions have the unit disc as a spectral set. |
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==References== |
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*T. Constantinescu, ''Schur Parameters, Dilation and Factorization Problems'', Birkhauser Verlag, Vol. 82, ISBN 3-7643-5285-X, 1996. |
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*Vern Paulsen, ''Completely Bounded Maps and Operator Algebras'' 2002, ISBN 0-521-81669-6 |
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[[Category:Operator theory]] |
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[[Category:Unitary operators]] |
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Revision as of 21:26, 17 March 2008
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