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      關(guān)于Kadison算子型不等式的推廣及其經(jīng)濟應(yīng)用

      2015-08-21 21:03:46李筍
      經(jīng)濟數(shù)學(xué) 2015年2期
      關(guān)鍵詞:標(biāo)識碼分類號算子

      李筍

      摘 要 設(shè)A是Hilbert空間H上的嚴(yán)格正算子,Φ是保持單位元的正線性映射.利用已知的算子不等式對Kadison算子型不等式進行非對稱形式的推廣,得到更加廣泛的一些算子不等式,同時給出其中兩種特殊不等式的經(jīng)濟學(xué)解釋, 并指出在一定條件下, 企業(yè)成本與利潤、產(chǎn)出與利潤之間存在對偶關(guān)系.

      關(guān)鍵詞 正線性映射;Kadison算子型不等式;成本;利潤

      中圖分類號 FO241 文獻標(biāo)識碼 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論中的一個重要分支,也是目前數(shù)理經(jīng)濟學(xué)的熱點問題之一, 已發(fā)現(xiàn)不少結(jié)論在數(shù)理經(jīng)濟學(xué)、統(tǒng)計學(xué)、優(yōu)化理論等領(lǐng)域以及相關(guān)學(xué)科中有著廣泛的應(yīng)用[1-4]. R.Kadison(1952)[5]給出 Kadison不等式, 該不等式在數(shù)理經(jīng)濟算子理論中處于重要地位. 隨后J.C. Bourin 與 E.Ricard [6]對上述的結(jié)果進行推廣及完善, 為后來研究工作及本文研究提供了方法. T.Furuta(2011 [7]利用Furuta不等式推廣J.C. Bourin與E. Ricard的工作. 原江濤(2012)[8]進一步改進T. Furuta 的工作, 并給出更加精細的估計. 本文正是借鑒以上工作, 得到了一些新的結(jié)果, 并利用共軛算子及凸性算子的性質(zhì)給出兩種經(jīng)濟模型的解釋.

      2 預(yù)備知識

      參考文獻

      [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] 劉衛(wèi)鋒. 三參數(shù)區(qū)間數(shù)集成算子及決策應(yīng)用[J].經(jīng)濟數(shù)學(xué),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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