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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Dilation (operator theory) ： ウィキペディア英語版 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 :$P\_H\; \backslash ;\; V\; |\; \_H\; =\; T$ where $P\_H$ 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 $X$, we say that ''V'' is a normal boundary dilation or a normal $\backslash partial\; X$ dilation if ''V'' is a normal dilation of ''T'' and $\backslash sigma(V)\backslash in\backslash partial\; X$. Some texts impose an additional condition. Namely, that a dilation satisfy the following (calculus) property: :$P\_H\; \backslash ;\; f(V)\; |\; \_H\; =\; f(T)$ 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. ==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 :$D\_T\; =\; (I\; -\; T^$ * T)^} 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 :$H\; \backslash oplus\; H$ defined by the matrix :$V\; =$ \begin T & D_\\ \ D_T & -T^ * \end. ''V'' is clearly a dilation of ''T''. Also, ''T''(''I - T *T'') = (''I - TT *'')''T'' implies :$T\; D\_T\; =\; D\_\; T.$ Using this one can show, by calculating directly, 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 $T\; =\; \backslash cos\; \backslash theta$, we have :$V\; =$ \begin \cos \theta & \sin \theta \\ \ \sin \theta & - \cos \theta \end. 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 :$T^2\; =\; P\_H\; \backslash ;\; V^2\; |\; \_H$. 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 $\backslash mathcal(X)$ is a Dirichlet algebra, any operator ''T'' with $X$ as a spectral set will have a normal $\backslash partial\; X$ dilation with this property. This generalises Sz.-Nagy's dilation theorem as all contractions have the unit disc as a spectral set. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dilation (operator theory)」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.