翻訳と辞書 Words near each other ・ Stone wall in Takanawa seashore ・ Stone wall trees in Hong Kong ・ Stone washing ・ Stone Wave ・ Stone Windmill ・ Stone wrist-guard ・ Stone Zoo ・ Stone'd Records ・ Stone's Been Rolled Away ・ Stone's Fall ・ Stone's Ferry, Nevada ・ Stone's Justices' Manual ・ Stone's Reach ・ Stone's representation theorem for Boolean algebras ・ Stone's theorem ・ Stone's theorem on one-parameter unitary groups ・ Stone, Brick, Glass, Wood, Wire ・ Stone, Buckinghamshire ・ Stone, California ・ Stone, Carpenter & Sheldon ・ Stone, Carpenter & Willson ・ Stone, Gloucestershire ・ Stone, Idaho ・ Stone, Indiana ・ Stone, Kent ・ Stone, Kentucky ・ Stone, Staffordshire ・ Stone, Time, Touch ・ Stone, Wisconsin ・ Stone, Worcestershire Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Stone's theorem on one-parameter unitary groups ： ウィキペディア英語版 Stone's theorem on one-parameter unitary groups In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space and one-parameter families :$(U\_)\_\; \backslash in\; \backslash mathbf,\; ~\; \backslash xi\; \backslash in\; H:\; \backslash qquad\; \backslash lim\_\; \backslash xi\; =\; U\_:\; \backslash qquad\; U\_\; =\; U\_\; U\_.$ Such one-parameter families are ordinarily referred to as strongly continuous one-parameter unitary groups. The theorem was proved by , and showed that the requirement that $(U\_)\_\{t\; \backslash in\; \backslash mathbf\{R\}\}$ be strongly continuous can be relaxed to say that it is merely weakly measurable, at least when the Hilbert space is separable. This is a very stunning theorem, as it allows to define the derivative of the mapping , which is only supposed to be continuous. It is also related to the theory of Lie groups and Lie algebras. == Formal statement == Let $(U\_)\_:\; \backslash qquad\; U\_\; =\; e^.$ Conversely, let be a (not necessarily bounded) self-adjoint operator on a Hilbert space . Then the one-parameter family $(U\_)\_:\; \backslash qquad\; U\_\; :=\; e^$ is a strongly continuous one-parameter group. The infinitesimal generator of $(U\_)\_)\_):\; \backslash qquad\; \backslash rho(f)\; :=\; \backslash int\_\; \backslash ,\; dt,$ :and then extending to all of by continuity. * Use the Fourier transform to obtain a non-degenerate *-representation of on . * By the Riesz-Markov Theorem, gives rise to a projection-valued measure on that is the resolution of the identity of a unique self-adjoint operator , which may be unbounded. * Then is the infinitesimal generator of $(U\_)\_),\backslash star)$. Then is defined to be the enveloping C *-algebra of $(L^1(\backslash mathbf),\backslash star)$, i.e., its completion with respect to the largest possible C *-norm. It is a non-trivial fact that, via the Fourier transform, is isomorphic to . A result in this direction is the Riemann-Lebesgue Lemma, which says that the Fourier transform maps to . 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「Stone's theorem on one-parameter unitary groups」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.