翻訳と辞書 Words near each other ・ Borek, Wolsztyn County ・ Borek, Włocławek County ・ Borek, Zielona Góra County ・ Borek, Łęczyca County ・ Borek, Świętokrzyskie Voivodeship ・ Borel ・ Borel & Co. ・ Borel (crater) ・ Borel (surname) ・ Borel Bo.11 ・ Borel conjecture ・ Borel determinacy theorem ・ Borel distribution ・ Borel equivalence relation ・ Borel fixed-point theorem ・ Borel functional calculus ・ Borel hierarchy ・ Borel hydro-monoplane ・ Borel isomorphism ・ Borel measure ・ Borel military monoplane ・ Borel regular measure ・ Borel right process ・ Borel set ・ Borel subgroup ・ Borel summation ・ Borel Torpille ・ Borel transform ・ Borel's lemma ・ Borel's theorem 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 Borel functional calculus ： ウィキペディア英語版 Borel functional calculus In functional analysis, a branch of mathematics, the Borel functional calculus is a ''functional calculus'' (that is, an assignment of operators from commutative algebras to functions defined on their spectra), which has particularly broad scope. Thus for instance if ''T'' is an operator, applying the squaring function ''s'' → ''s''2 to ''T'' yields the operator ''T''2. Using the functional calculus for larger classes of functions, we can for example define rigorously the "square root" of the (negative) Laplacian operator or the exponential : $e^.$ The 'scope' here means the kind of ''function of an operator'' which is allowed. The Borel functional calculus is more general than the continuous functional calculus. More precisely, the Borel functional calculus allows us to apply an arbitrary Borel function to a self-adjoint operator, in a way which generalizes applying a polynomial function. == Motivation == If ''T'' is a self-adjoint operator on a finite-dimensional inner product space ''H'', then ''H'' has an orthonormal basis consisting of eigenvectors of ''T'', that is :$T\; e\_k\; =\; \backslash lambda\_k\; e\_k,\; \backslash qquad\; 1\; \backslash leq\; k\; \backslash leq\; \backslash ell.$ Thus, for any positive integer ''n'', :$T^n\; e\_k\; =\; \backslash lambda\_k^n\; e\_k.$ In this case, given a Borel function ''h'', we can define an operator ''h''(''T'') by specifying its behavior on the basis: :$h(T)\; e\_k\; =\; h(\backslash lambda\_k)\; e\_k.$ In general, any self-adjoint operator ''T'' is unitarily equivalent to a multiplication operator; this means that for many purposes, ''T'' can be considered as an operator :$(\backslash psi)(x)\; =\; f(x)\; \backslash psi(x)$ acting on ''L''2 of some measure space. The domain of ''T'' consists of those functions for which the above expression is in ''L''2. In this case, we can define analogously :$(\backslash psi)(x)\; =\; (\backslash circ\; f)(x)\; \backslash psi(x).$ For many technical purposes, the preceding formulation is good enough. However, it is desirable to formulate the functional calculus in a way in which it is clear that it does not depend on the particular representation of ''T'' as a multiplication operator. This we do in the next section. 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「Borel functional calculus」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.