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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 Dissipative operator ： ウィキペディア英語版 Dissipative operator In mathematics, a dissipative operator is a linear operator ''A'' defined on a linear subspace ''D''(''A'') of Banach space ''X'', taking values in ''X'' such that for all ''λ'' > 0 and all ''x'' ∈ ''D''(''A'') :$\backslash |(\backslash lambda\; I-A)x\backslash |\backslash geq\backslash lambda\backslash |x\backslash |.$ A couple of equivalent definitions are given below. A dissipative operator is called maximally dissipative if it is dissipative and for all ''λ'' > 0 the operator ''λI'' − ''A'' is surjective, meaning that the range when applied to the domain ''D'' is the whole of the space ''X''. An operator that obeys a similar condition but with a plus sign instead of a minus sign (that is, the negation of a dissipative operator) is called an accretive operator.〔(【引用サイトリンク】url=http://www.encyclopediaofmath.org/index.php/Dissipative_operator )〕 The main importance of dissipative operators is their appearance in the Lumer–Phillips theorem which characterizes maximally dissipative operators as the generators of contraction semigroups. ==Properties== A dissipative operator has the following properties〔Engel and Nagel Proposition II.3.14〕 * From the inequality given above, we see that for any ''x'' in the domain of ''A'', if ‖''x''‖ ≠ 0 then $\backslash |(\backslash lambda\; I-A)x\backslash |\backslash ne\; 0,$ so the kernel of ''λI'' − ''A'' is just the zero vector and ''λI'' − ''A'' is therefore injective and has an inverse for all ''λ'' > 0. (If we have the strict inequality $\backslash |(\backslash lambda\; I-A)x\backslash |>\backslash lambda\backslash |x\backslash |$ for all non-null ''x'' in the domain, then, by the triangle inequality, $\backslash |\backslash lambda\; x\backslash |+\backslash |Ax\backslash |\backslash ge\backslash |(\backslash lambda\; I-A)x\backslash |>\backslash lambda\backslash |x\backslash |,$ which implies that A itself has an inverse.) We may then state that :::$\backslash |(\backslash lambda\; I-A)^z\backslash |\backslash leq\backslash frac\backslash |z\backslash |$ ::for all ''z'' in the range of ''λI'' − ''A''. This is the same inequality as that given at the beginning of this article, with $z=(\backslash lambda\; I-A)x.$ (We could equally well write these as $\backslash |(I-\backslash kappa\; A)^z\backslash |\backslash leq\backslash |z\backslash |\backslash text\backslash |(I-\backslash kappa\; A)x\backslash |\backslash geq\backslash |x\backslash |$ which must hold for any positive κ.) * ''λI'' − ''A'' is surjective for some ''λ'' > 0 if and only if it is surjective for all ''λ'' > 0. (This is the aforementioned maximally dissipative case.) In that case one has (0, ∞) ⊂ ''ρ''(''A'') (the resolvent set of ''A''). * ''A'' is a closed operator if and only if the range of ''λI'' - ''A'' is closed for some (equivalently: for all) ''λ'' > 0. 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「Dissipative 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.