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关于Kadison算子型不等式的推广及其经济应用

2015-08-21李笋

经济数学 2015年2期
关键词:标识码分类号算子

李笋

摘 要 设A是Hilbert空间H上的严格正算子,Φ是保持单位元的正线性映射.利用已知的算子不等式对Kadison算子型不等式进行非对称形式的推广,得到更加广泛的一些算子不等式,同时给出其中两种特殊不等式的经济学解释, 并指出在一定条件下, 企业成本与利润、产出与利润之间存在对偶关系.

关键词 正线性映射;Kadison算子型不等式;成本;利润

中图分类号 FO241 文献标识码 A

Some Extensions and Economic Applications

of the Kadison Type Inequality

LI Sun

(College of Mathematics and Economics, Hunan University, Changsha, Hunan 410082,China)

Abstract

Let A be a strictly positive operator on a Hilbert space H, and Φ be a unit positive linear map. We discussed some asymmetrical extended forms of Kadison type inequality via some related theorems and explained two of these special inequalities from the perspective of economics. The result shows that there is a dual relationship between cost and profit going with output and profit.

Key words Postive linear map; Kadison type inequality; cost; profit

1 引 言

算子不等式是算子理论中的一个重要分支,也是目前数理经济学的热点问题之一, 已发现不少结论在数理经济学、统计学、优化理论等领域以及相关学科中有着广泛的应用[1-4]. R.Kadison(1952)[5]给出 Kadison不等式, 该不等式在数理经济算子理论中处于重要地位. 随后J.C. Bourin 与 E.Ricard [6]对上述的结果进行推广及完善, 为后来研究工作及本文研究提供了方法. T.Furuta(2011 [7]利用Furuta不等式推广J.C. Bourin与E. Ricard的工作. 原江涛(2012)[8]进一步改进T. Furuta 的工作, 并给出更加精细的估计. 本文正是借鉴以上工作, 得到了一些新的结果, 并利用共轭算子及凸性算子的性质给出两种经济模型的解释.

2 预备知识

参考文献

[1] M FUJII, Y O KIM, R NAKAMOTO. A characterization of convex functions and its application to operator monotone functions[J].Banach Journal of Mathematical Analysis,2014,8(2): 118-123.

[2] 刘卫锋. 三参数区间数集成算子及决策应用[J].经济数学,2014,31(4):96-101.

[3] T ANDO. Concavity of certain maps on positive definite matrices and applications to hadamard products[J]. Linear Algebra and its Applications, 1979, 26(4):203-241.

[4] L ZHOU, H CHEN, J LIU. Generalized multiple averaging operators and their applications to group decision making[J]. Group Decision and Negotiation, 2013, 22(2): 331-358.

[5] R. KADISON. A generalized Schwarz inequality and algebraic invariants for operator algebras[J]. Annals of Mathematics, 1952, 56(3):494-503.

[6] J BOURIN, E RICHARD. An asymmetric Kadisons inequality[J]. Linear Algebra and its Applications, 2010, 433(3):499-510.

[7] T FURUTA. Around choi inequalities for positive linear maps[J]. Linear Algebra and its Applications, 2011, 434(1):14-17.

[8] J YUAN, G JI. Extensions of Kadisons inequality on positive linear maps[J]. Linear Algebra and its Applications, 2012, 436(3):747-752.

[9] M CHOI. Some assorted inequalities for positive linear map onCalgebras[J]. Journal of Operator Theory, 1980, 4(2):271-285.

[10]R BHATIA. Positive Definite Matrices[M]. Princeton: Princeton University Press, 2007.

[11]F HENSEN, J PECARIC, I. PERIC. Jensens operator inequality and its converses[J]. Mathematica Scandinavica, 2007, 100(1):61-73.

[12]G PEDERSEN. Some operator monotone functions[J]. Proceedings of the Americal Mathematical Society, 1972, 36(1):309-310.

[13]T FURUTA.A≥B≥0assures(BrApBr)1q≥Bp+2rqforr≥0,p≥0,q≥1With(1+2r)q≥p+2r[J]. Proceedings of the Americal Mathematical Society, 1987, 101(1):85-88.

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