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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 Fuglede's theorem ： ウィキペディア英語版 Fuglede's theorem In mathematics, Fuglede's theorem is a result in operator theory, named after Bent Fuglede. == The result == Theorem (Fuglede) Let ''T'' and ''N'' be bounded operators on a complex Hilbert space with ''N'' being normal. If ''TN = NT'', then ''TN *'' = ''N *T'', where ''N *'' denotes the adjoint of ''N''. Normality of ''N'' is necessary, as is seen by taking ''T''=''N''. When ''T'' is self-adjoint, the claim is trivial regardless of whether ''N'' is normal: :$TN^$ * = (NT)^ * = (TN)^ * = N^ *T.\, Tentative Proof: If the underlying Hilbert space is finite-dimensional, the spectral theorem says that ''N'' is of the form :$N\; =\; \backslash sum\_i\; \backslash lambda\_i\; P\_i\; \backslash ,$ where ''Pi'' are pairwise orthogonal projections. One expects that ''TN = NT'' if and only if ''TPi = PiT''. Indeed it can be proved to be true by elementary arguments (e.g. it can be shown that all ''Pi'' are representable as polynomials of ''N'' and for this reason, if ''T'' commutes with ''N'', it has to commute with ''Pi''...). Therefore ''T'' must also commute with :$N^$ * = \sum_i P_i. In general, when the Hilbert space is not finite-dimensional, the normal operator ''N'' gives rise to a projection-valued measure ''P'' on its spectrum, ''σ''(''N''), which assigns a projection ''P''Ω to each Borel subset of ''σ''(''N''). ''N'' can be expressed as :$N\; =\; \backslash int\_\; \backslash lambda\; d\; P(\backslash lambda).\; \backslash ,$ Differently from the finite dimensional case, it is by no means obvious that ''TN = NT'' implies ''TP''Ω = ''P''Ω''T''. Thus, it is not so obvious that ''T'' also commutes with any simple function of the form :$\backslash rho\; =\; \backslash sum\_i\; P\_.$ Indeed, following the construction of the spectral decomposition for a bounded, normal, not self-adjoint, operator ''T'', one sees that to verify that ''T'' commutes with $P\_$, the most straightforward way is to assume that ''T'' commutes with both ''N'' and ''N *'', giving rise to a vicious circle! That is the relevance of Fuglede's theorem: The latter hypothesis is not really necessary. 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「Fuglede's 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.