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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 Sz.-Nagy's dilation theorem ： ウィキペディア英語版 Sz.-Nagy's dilation theorem The Sz.-Nagy dilation theorem (proved by Béla Szőkefalvi-Nagy) states that every contraction ''T'' on a Hilbert space ''H'' has a unitary dilation ''U'' to a Hilbert space ''K'', containing ''H'', with :$T^n\; =\; P\_H\; U^n\; \backslash vert\_H,\backslash quad\; n\backslash ge\; 0.$ Moreover, such a dilation is unique (up to unitary equivalence) when one assumes ''K'' is minimal, in the sense that the linear span of ∪''n''''UnK'' is dense in ''K''. When this minimality condition holds, ''U'' is called the minimal unitary dilation of ''T''. == Proof == For a contraction ''T'' (i.e., ($\backslash |T\backslash |\backslash le1$), its defect operator ''DT'' is defined to be the (unique) positive square root ''DT'' = (''I - T *T'')½. In the special case that ''S'' is an isometry, the following is an Sz. Nagy unitary dilation of ''S'' with the required polynomial functional calculus property: :$U\; =$ \begin S & D_ \\ 0 & -S^ * \end. Also, every contraction ''T'' on a Hilbert space ''H'' has an isometric dilation, again with the calculus property, on :$\backslash oplus\_\; H$ given by :$S\; =$ \begin T & 0 & 0 & \cdots & \\ D_T & 0 & 0 & & \\ 0 & I & 0 & \ddots \\ 0 & 0 & I & \ddots \\ \vdots & & \ddots & \ddots \end . Applying the above two constructions successively gives a unitary dilation for a contraction ''T'': :$$ T^n = P_H S^n \vert_H = P_H (Q_ U \vert_)^n \vert_H = P_H U^n \vert_H. 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「Sz.-Nagy's dilation theorem」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.