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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 Fredholm determinant ： ウィキペディア英語版 Fredholm determinant In mathematics, the Fredholm determinant is a complex-valued function which generalizes the determinant of a matrix. It is defined for bounded operators on a Hilbert space which differ from the identity operator by a trace-class operator. The function is named after the mathematician Erik Ivar Fredholm. Fredholm determinants have had many applications in mathematical physics, the most celebrated example being Gábor Szegő's limit formula, proved in response to a question raised by Lars Onsager and C. N. Yang on the spontaneous magnetization of the Ising model. ==Definition== Let ''H'' be a Hilbert space and ''G'' the set of bounded invertible operators on ''H'' of the form ''I'' + ''T'', where ''T'' is a trace-class operator. ''G'' is a group because : $(I+T)^\; -\; I\; =\; -\; T(I+T)^,$ so ''(I+T)-1-I'' is trace class if ''T'' is. It has a natural metric given by ''d''(''X'', ''Y'') = ||''X'' - ''Y''||1, where || · ||1 is the trace-class norm. If ''H'' is a Hilbert space with inner product $(\backslash cdot,\backslash cdot)$, then so too is the ''k''th exterior power $\backslash Lambda^k\; H$ with inner product :$(v\_1\; \backslash wedge\; v\_2\; \backslash wedge\; \backslash cdots\; \backslash wedge\; v\_k,\; w\_1\; \backslash wedge\; w\_2\; \backslash wedge\; \backslash cdots\; \backslash wedge\; w\_k)\; =\; \backslash ,\; (v\_i,w\_j).$ In particular : gives an orthonormal basis of $\backslash Lambda^k\; H$ if (''e''''i'') is an orthonormal basis of ''H''. If ''A'' is a bounded operator on ''H'', then ''A'' functorially defines a bounded operator $\backslash Lambda^k(A)$ on $\backslash Lambda^k\; H$ by :$\backslash Lambda^k(A)\; v\_1\; \backslash wedge\; v\_2\; \backslash wedge\; \backslash cdots\; \backslash wedge\; v\_k\; =\; Av\_1\; \backslash wedge\; Av\_2\; \backslash wedge\; \backslash cdots\; \backslash wedge\; Av\_k.$ If ''A'' is trace-class, then $\backslash Lambda^k(A)$ is also trace-class with :$\backslash |\backslash Lambda^k(A)\backslash |\_1\; \backslash le\; \backslash |A\backslash |\_1^k/k!.$ This shows that the definition of the Fredholm determinant given by :$\backslash ,\; (I+\; A)\; =\; \backslash sum\_^\backslash infty\; \backslash Lambda^k(A)$ makes sense. 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「Fredholm determinant」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.