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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 Aleksandrov–Clark measure ： ウィキペディア英語版 Aleksandrov–Clark measure In mathematics, Aleksandrov–Clark (AC) measures are specially constructed measures named after the two mathematicians, A. B. Aleksandrov and Douglas Clark, who discovered some of their deepest properties. The measures are also called either Aleksandrov measures, Clark measures, or occasionally spectral measures. AC measures are used to extract information about self-maps of the unit disc, and have applications in a number of areas of complex analysis, most notably those related to operator theory. Systems of AC measures have also been constructed for higher dimensions, and for the half-plane. ==Construction of the measures== The original construction of Clark relates to one-dimensional perturbations of compressed shift operators on subspaces of the Hardy space: : $H^2(\backslash mathbb,\backslash mathbb).$ By virtue of Beurling's theorem, any shift-invariant subspace of this space is of the form : $\backslash theta\; H^2(\backslash mathbb,\backslash mathbb),$ where $\backslash theta$ is an inner function. As such, any invariant subspace of the adjoint of the shift is of the form : $K\_\backslash theta\; =\; \backslash left(\backslash theta\; H^2(\backslash mathbb,\backslash mathbb)\backslash right)^\backslash perp.$ We now define $S\_\backslash theta$ to be the shift operator compressed to $K\_\backslash theta$, that is : $S\_\backslash theta\; =\; P\_\; S|\_.$ Clark noticed that all the one-dimensional perturbations of $S\_\backslash theta$, which were also unitary maps, were of the form : $U\_\backslash alpha\; (f)\; =\; S\_\backslash theta\; (f)\; +\; \backslash alpha\; \backslash left\backslash langle\; f\; ,\; \backslash frac\; \backslash right\backslash rangle,$ and related each such map to a measure, $\backslash sigma\_\backslash alpha$ on the unit circle, via the Spectral theorem. This collection of measures, one for each $\backslash alpha$ on the unit circle $^\backslash mathbb$, is then called the collection of AC measures associated with $\backslash theta$. 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「Aleksandrov–Clark measure」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.