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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 index ： ウィキペディア英語版 Fredholm operator In mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of integral equations. It is named in honour of Erik Ivar Fredholm. A Fredholm operator is a bounded linear operator between two Banach spaces, with finite-dimensional kernel and cokernel, and with closed range. (The last condition is actually redundant.〔Yuri A. Abramovich and Charalambos D. Aliprantis, "An Invitation to Operator Theory", p.156〕) Equivalently, an operator ''T'' : ''X'' → ''Y'' is Fredholm if it is invertible modulo compact operators, i.e., if there exists a bounded linear operator :$S:\; Y\backslash to\; X$ such that :$\backslash mathrm\_X\; -\; ST\; \backslash quad\backslash text\backslash quad\; \backslash mathrm\_Y\; -\; TS$ are compact operators on ''X'' and ''Y'' respectively. The ''index'' of a Fredholm operator is :$\backslash mathrm\backslash ,T\; :=\; \backslash dim\; \backslash ker\; T\; -\; \backslash mathrm\backslash ,\backslash mathrm\backslash ,T$ or in other words, :$\backslash mathrm\backslash ,T\; :=\; \backslash dim\; \backslash ker\; T\; -\; \backslash mathrm\backslash ,\backslash mathrm\backslash ,T;$ see dimension, kernel, codimension, range, and cokernel. ==Properties== The set of Fredholm operators from ''X'' to ''Y'' is open in the Banach space L(''X'', ''Y'') of bounded linear operators, equipped with the operator norm. More precisely, when ''T''0 is Fredholm from ''X'' to ''Y'', there exists ''ε'' > 0 such that every ''T'' in L(''X'', ''Y'') with ||''T'' − ''T''0|| < ''ε'' is Fredholm, with the same index as that of ''T''0. When ''T'' is Fredholm from ''X'' to ''Y'' and ''U'' Fredholm from ''Y'' to ''Z'', then the composition $U\; \backslash circ\; T$ is Fredholm from ''X'' to ''Z'' and :$\backslash mathrm\; (U\; \backslash circ\; T)\; =\; \backslash mathrm(U)\; +\; \backslash mathrm(T).$ When ''T'' is Fredholm, the transpose (or adjoint) operator is Fredholm from to , and . When ''X'' and ''Y'' are Hilbert spaces, the same conclusion holds for the Hermitian adjoint ''T''∗. When ''T'' is Fredholm and ''K'' a compact operator, then ''T'' + ''K'' is Fredholm. The index of ''T'' remains unchanged under compact perturbations of ''T''. This follows from the fact that the index ''i''(''s'') of is an integer defined for every ''s'' in (), and ''i''(''s'') is locally constant, hence ''i''(1) = ''i''(0). Invariance by perturbation is true for larger classes than the class of compact operators. For example, when ''T'' is Fredholm and ''S'' a strictly singular operator, then ''T'' + ''S'' is Fredholm with the same index.〔T. Kato, "Perturbation theory for the nullity deficiency and other quantities of linear operators", ''J. d'Analyse Math''. 6 (1958), 273–322.〕 A bounded linear operator ''S'' from ''X'' to ''Y'' is strictly singular when its restriction to any infinite dimensional subspace ''X''0 of ''X'' fails to be an into isomorphism, that is: :$\backslash inf\; \backslash \; =\; 0.\; \backslash ,$ 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 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.