翻訳と辞書 Words near each other ・ Projection formula ・ Projection keyboard ・ Projection mapping ・ Projection method (fluid dynamics) ・ Projection micro-stereolithography ・ Projection panel ・ Projection Peak ・ Projection plane ・ Projection Planetarium ・ Projection poi ・ Projection Principle ・ Projection pursuit ・ Projection pursuit regression ・ Projection screen ・ Projection-slice theorem ・ Projection-valued measure ・ Projectional radiography ・ Projectionist ・ Projectionless algebra ・ Projections (album) ・ Projections (John Handy album) ・ Projections (journal) ・ Projections of population growth ・ Projections onto convex sets ・ Projective ・ Projective bundle ・ Projective cone ・ Projective connection ・ Projective cover ・ Projective differential geometry 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 Projection-valued measure ： ウィキペディア英語版 Projection-valued measure In mathematics, particularly in functional analysis, a projection-valued measure (PVM) is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. Projection-valued measures are formally similar to real-valued measures, except that their values are self-adjoint projections rather than real numbers. As in the case of ordinary measures, it is possible to integrate complex-valued functions with respect to a PVM; the result of such an integration is a linear operator on the given Hilbert space. Projection-valued measures are used to express results in spectral theory, such as the important spectral theorem for self-adjoint operators. The Borel functional calculus for self-adjoint operators is constructed using integrals with respect to PVMs. In quantum mechanics, PVMs are the mathematical description of projective measurements. They are generalized by positive operator valued measures (POVMs) in the same sense that a mixed state or density matrix generalizes the notion of a pure state. == Formal definition == A projection-valued measure on a measurable space (''X'', ''M''), where ''M'' is a σ-algebra of subsets of ''X'', is a mapping π from ''M'' to the set of self-adjoint projections on a Hilbert space ''H'' such that : $\backslash pi(X)\; =\; \backslash operatorname\_H\; \backslash quad$ and for every ξ, η ∈ ''H'', the set-function :$E\; \backslash mapsto\; \backslash langle\; \backslash pi(E)\backslash xi\; \backslash mid\; \backslash eta\; \backslash rangle$ is a complex measure on ''M'' (that is, a complex-valued countably additive function). We denote this measure by $\backslash operatorname\_\backslash pi(\backslash xi,\; \backslash eta)$. Note that $\backslash operatorname\_\backslash pi(\backslash xi,\; \backslash xi)$ is a real-valued measure, and a probability measure when $\backslash xi$ has length one. If π is a projection-valued measure and : $E\; \backslash cap\; F\; =\; \backslash emptyset,$ then π(''E''), π(''F'') are orthogonal projections. From this follows that in general, : $\backslash pi(E)\; \backslash pi(F)\; =\; \backslash pi(E\; \backslash cap\; F)\; =\; \backslash pi(F)\; \backslash pi(E),$ and they commute. Example. Suppose (''X'', ''M'', μ) is a measure space. Let π(''E'') be the operator of multiplication by the indicator function 1''E'' on ''L''2(''X''). Then π is a projection-valued measure. 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「Projection-valued measure」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.