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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 Trace inequalities ： ウィキペディア英語版 Trace inequalities In mathematics, there are many kinds of inequalities involving matrices and linear operators on Hilbert spaces. This article covers some important operator inequalities connected with traces of matrices.〔E. Carlen, Trace Inequalities and Quantum Entropy: An Introductory Course, Contemp. Math. 529 (2009).〕〔R. Bhatia, Matrix Analysis, Springer, (1997).〕〔B. Simon, Trace Ideals and their Applications, Cambridge Univ. Press, (1979); Second edition. Amer. Math. Soc., Providence, RI, (2005).〕〔M. Ohya, D. Petz, Quantum Entropy and Its Use, Springer, (1993).〕 ==Basic definitions== Let H''n'' denote the space of Hermitian × matrices, H''n''+ denote the set consisting of positive semi-definite × Hermitian matrices and H''n''++ denote the set of positive definite Hermitian matrices. For operators on an infinite dimensional Hilbert space we require that they be trace class and self-adjoint, in which case similar definitions apply, but we discuss only matrices, for simplicity. For any real-valued function on an interval ⊂ ℝ, one may define a matrix function for any operator with eigenvalues in by defining it on the eigenvalues and corresponding projectors as :$f(A)\backslash equiv\; \backslash sum\_j\; f(\backslash lambda\_j)P\_j\; ~,$ given the spectral decomposition $A=\backslash sum\_j\backslash lambda\_j\; P\_j.$ 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「Trace inequalities」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.