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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 Positive element ： ウィキペディア英語版 Positive element In mathematics, especially functional analysis, a self-adjoint (or Hermitian) element $A$ of a C *-algebra $\backslash mathcal$ is called positive if its spectrum $\backslash sigma(A)$ consists of non-negative real numbers. Moreover, an element $A$ of a C *-algebra $\backslash mathcal$ is positive if and only if there is some $B$ in $\backslash mathcal$ such that $A\; =\; B^$ * B. A positive element is self-adjoint and thus normal. If $T$ is a bounded linear operator on a complex Hilbert space $H$, then this notion coincides with the condition that $\backslash langle\; Tx,x\; \backslash rangle$ is non-negative for every vector $x$ in $H$. Note that $\backslash langle\; Tx,x\; \backslash rangle$ is real for every $x$ in $H$ if and only if $T$ is self-adjoint. Hence, a positive operator on a Hilbert space is always self-adjoint (and a self-adjoint ''everywhere defined'' operator on a Hilbert space is always bounded because of the Hellinger-Toeplitz theorem). The set of positive elements of a C *-algebra forms a convex cone. == Positive and positive definite operators == A bounded linear operator $P$ on an inner product space $V$ is said to be ''positive'' (or ''positive semidefinite'') if $P\; =\; S^$ *S for some bounded operator $S$ on $V$, and is said to be ''positive definite'' if $S$ is also non-singular. (I) The following conditions for a bounded operator $P$ on $V$ to be positive semidefinite are equivalent: * $P\; =\; S^$ *S for some bounded operator $S$ on $V$, * $P\; =\; T^2$ for some self-adjoint operator $T$ on $V$, * $P$ is self-adjoint and $\backslash langle\; Pu,u\; \backslash rangle\; \backslash geq\; 0,\; \backslash forall\; u\; \backslash in\; V$. (II) The following conditions for a bounded operator $P$ on $V$ to be positive definite are equivalent: * $P\; =\; S^$ *S for some non-singular bounded operator $S$ on $V$, * $P\; =\; T^2$ for some non-singular self-adjoint operator $T$ on $V$, * $P$ is self adjoint and $\backslash langle\; Pu,u\; \backslash rangle\; >\; 0,\; \backslash forall\; u\; \backslash neq\; 0$ in $V$. (III) A complex matrix represents a positive (semi)definite operator if and only if $A$ is Hermitian (or self-adjoint) and $a$, $d$ and $\backslash det(A)\; =\; ad\; -\; bc$ are (strictly) positive real numbers. Let the Banach spaces $X$ and $Y$ be ordered vector spaces and let $T\; \backslash colon\; X\; \backslash to\; Y$ be a linear operator. The operator $T$ is called ''positive'' if $Tx\; \backslash geq\; 0$ for all $x\; \backslash geq\; 0$ in $X$. For a positive operator $T$ we write $T\; \backslash geq\; 0$. A positive operator maps the positive cone of $X$ onto a subset of the positive cone of $Y$. If $Y$ is a field then $T$ is called a positive linear functional. Many important operators are positive. For example: * the Laplace operators $-\backslash Delta$ and $-\backslash frac$ are positive, * the limit and Banach limit functionals are positive, * the identity and absolute value operators are positive, * the integral operator with a positive measure is positive. The Laplace operator is an example of an unbounded positive linear operator. Hence, by the Hellinger-Toeplitz theorem it cannot be everywhere defined. 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「Positive element」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.