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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 Hilbert–Schmidt operator ： ウィキペディア英語版 Hilbert–Schmidt operator In mathematics, a Hilbert–Schmidt operator, named for David Hilbert and Erhard Schmidt, is a bounded operator ''A'' on a Hilbert space ''H'' with finite Hilbert–Schmidt norm :$\backslash |A\backslash |^2\_=\; (A^=\backslash sum\_\; |A\_|^2\; =\; \backslash |A\backslash |^2\_2$ for $A\_=\backslash langle\; e\_i,\; Ae\_j\; \backslash rangle$ and $\backslash |A\backslash |\_2$ the Schatten norm of $A$ for ''p=2''. In Euclidean space $\backslash |\backslash \; \backslash |\_$ is also called Frobenius norm, named for Ferdinand Georg Frobenius. The product of two Hilbert–Schmidt operators has finite trace class norm; therefore, if ''A'' and ''B'' are two Hilbert–Schmidt operators, the Hilbert–Schmidt inner product can be defined as :$\backslash langle\; A,B\; \backslash rangle\_\backslash mathrm\; =\; \backslash operatorname\; (A^$ *B) = \sum_ \langle Ae_i, Be_i \rangle. The Hilbert–Schmidt operators form a two-sided *-ideal in the Banach algebra of bounded operators on ''H''. They also form a Hilbert space, which can be shown to be naturally isometrically isomorphic to the tensor product of Hilbert spaces :$H^$ * \otimes H, \, where ''H *'' is the dual space of ''H''. The set of Hilbert–Schmidt operators is closed in the norm topology if, and only if, ''H'' is finite-dimensional. An important class of examples is provided by Hilbert–Schmidt integral operators. Hilbert–Schmidt operators are nuclear operators of order 2, and are therefore compact. ==References== 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「Hilbert–Schmidt operator」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.