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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 Nuclear operator ： ウィキペディア英語版 Nuclear operator In mathematics, a nuclear operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis (at least on well behaved spaces; there are some spaces on which nuclear operators do not have a trace). Nuclear operators are essentially the same as trace class operators, though most authors reserve the term "trace class operator" for the special case of nuclear operators on Hilbert spaces. The general definition for Banach spaces was given by Grothendieck. This article presents both cases concentrates on the general case of nuclear operators on Banach spaces; for more details about the important special case of nuclear (=trace class) operators on Hilbert space see the article on trace class operators. ==Compact operator== An operator $\backslash mathcal$ on a Hilbert space $\backslash mathcal$ :$\backslash mathcal:\backslash mathcal\; \backslash to\; \backslash mathcal$ is compact if it can be written in the form :$\backslash mathcal\; =\; \backslash sum\_^N\; \backslash rho\_n\; \backslash langle\; f\_n,\; \backslash cdot\; \backslash rangle\; g\_n$ where $1\; \backslash le\; N\; \backslash le\; \backslash infty$ and $f\_1,\backslash ldots,f\_N$ and $g\_1,\backslash ldots,g\_N$ are (not necessarily complete) orthonormal sets. Here, $\backslash rho\_1,\backslash ldots,\backslash rho\_N$ are a set of real numbers, the singular values of the operator, obeying $\backslash rho\_n\; \backslash to\; 0$ if $N\; =\; \backslash infty$. The bracket $\backslash langle\backslash cdot,\backslash cdot\backslash rangle$ is the scalar product on the Hilbert space; the sum on the right hand side must converge in norm. An operator that is compact as defined above is said to be nuclear or trace-class if : 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「Nuclear 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.