Study on operator inequalities based on various analytic methods

• Project/Area Number: 23540215
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kyushu University
• Principal Investigator: KOSAKI Hideki 九州大学, 数理(科)学研究科(研究院), 教授 (20186612)
• Co-Investigator(Kenkyū-buntansha): WATATANI Yasuo 九州大学, 数理学研究院, 教授 (00175077)
• Co-Investigator(Renkei-kenkyūsha): HIAI Fumio 東北大学, 名誉教授 (30092571) ; SANO Takashi 山形大学, 理学部, 教授 (20250912)
• Project Period (FY): 2011 – 2013
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000); Fiscal Year 2013: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2012: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2011: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
• Keywords: 作用素平均 / 正定値関数 / 量子情報理論 / 完全正写像 / Jensen型不等式 / 作用素ノルム不等式 / ノルム不等式

Research Abstract

To get norm comparison results for operator means we have to check positive definiteness of ratios of relevant means. By computing Fourier transforms we have checked positive definiteness of a variety of ratios. We have also made use of the concept of infinite divisibility to establish this for some ratios whose explicit computations of Fourier transforms seem hopeless.
On the other hand, our technique was proved useful for study on completely positive maps appearing naturally in quantum information theory.

Research Products

• [Journal Article] C*-algebras associated with complex dynamical systems and backward orbit structure (2014)
  • Author(s): T.Kajiwara and Y.Watatani
  • Journal Title: Complex Analysis and Operator Theory Volume: 8 Pages: 243-254
  • Peer Reviewed
• [Journal Article] KMS states on finite graph C*-algebras (2013)
  • Author(s): T.Kajiwara and Y.Watatani
  • Journal Title: Kyushu J. Math Volume: 67 Pages: 83-104
  • NAID: 130003364801
  • Peer Reviewed
• [Journal Article] Families of completely positive maps associated with monotone metrics (2013)
  • Author(s): F.Hiai, H.Kosaki, D.Petz and M.B.Ruskai
  • Journal Title: Linear Algebra Appl Volume: 439 Pages: 1749-1791
  • Peer Reviewed
• [Journal Article] Trace Jensen inequality for self-adjoint operators in semi-finite von Neumann algebtras (2013)
  • Author(s): H.Kosaki
  • Journal Title: Internat. J. Math Volume: 24
  • Peer Reviewed
• [Journal Article] Families of completely positive maps associated with monotone metrics (2013)
  • Author(s): F. Hiai, H. Kosaki, D. Petz and M. B. Ruskai
  • Journal Title: Linear Algebra Appl. Volume: 439 Issue: 7 Pages: 1749-1791
  • DOI: 10.1016/j.laa.2013.05.012
  • Peer Reviewed
• [Journal Article] Trace Jensen inequality for self-adjoint operators in semi-finite von Neumann algebras (2013)
  • Author(s): H. Kosaki
  • Journal Title: Internat. J. Math. Volume: 24 Issue: 09 Pages: 1350075-1350075
  • DOI: 10.1142/s0129167x13500754
  • Peer Reviewed
• [Journal Article] KMS STATES ON FINITE-GRAPH <i>C</i>^∗-ALGEBRAS (2013)
  • Author(s): T. Kajiwara and Y. Watatani
  • Journal Title: Kyushu Journal of Mathematics Volume: 67 Issue: 1 Pages: 83-104
  • DOI: 10.2206/kyushujm.67.83
  • NAID: 130003364801
  • ISSN: 1340-6116, 1883-2032
  • Peer Reviewed
• [Journal Article] KMS states on finite graph $C^*$-algebras (2013)
  • Author(s): T. Kajiwara and Y. Watatani
  • Journal Title: Kyushu J. Math. Volume: 67 Pages: 83-104
  • NAID: 130003364801
  • Peer Reviewed
• [Journal Article] $C^*$-algebras associated with complex dynamical systems and backward orbit structure (2013)
  • Author(s): T. Kajiwara and Y. Watatani
  • Journal Title: Complex analysis and Operator Theory Volume: 未定
  • Peer Reviewed
• [Journal Article] Traces on cores of $C^*$-algebras associated with self-similar maps (2013)
  • Author(s): T. Kajiwara and Y. Watatani
  • Journal Title: Ergodic Theory and Dynamical Systems Volume: 未定」
  • Peer Reviewed
• [Journal Article] Loewner matrices of matrix convex and monotone functions (2012)
  • Author(s): F.Hiai and T.Sano
  • Journal Title: J. Math. Soc., Japan Volume: 64 Pages: 343-364
  • NAID: 10030740187
  • Peer Reviewed
• [Journal Article] Positive definiteness of functions with applications to operator norm inequalities (2011)
  • Author(s): H.Kosaki
  • Journal Title: Mem. Amer. Math. Soc Volume: 212
  • Peer Reviewed
• [Journal Article] C*-algebras associated with algebraic correspondences on the Riemann sphere (2011)
  • Author(s): T.Kajiwara and Y.Watatani
  • Journal Title: J. Operator Theory Volume: 65 Pages: 427-449
  • Peer Reviewed
• [Journal Article] Positive definiteness of functions with applications to operator norm inequalities (2011)
  • Author(s): H. Kosaki
  • Journal Title: Mem. Amer. Math. Soc. Volume: 212
  • Peer Reviewed
• [Presentation] Infinite divisibility in study of operator means (2011)
  • Author(s): H. Kosaki
  • Organizer: Seminare D'Anlysis Fonctionnelle, Universite de Franche-Comte
  • Place of Presentation: Besancon, France
• [Presentation] Infinite divisibility in study of operator means (2011)
  • Author(s): H. Kosaki
  • Organizer: Seminare D'Anlysis Fonctionnelle(招待講演)
  • Place of Presentation: Universite de Franche-Comte, Besancon, France