The study of operator monotone functions and its application to polynomials and operator perturbation

• Project/Area Number: 25400116
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Ritsumeikan University (2014-2016); Shimane University (2013)
• Principal Investigator: Uchiyama Mitsuru 立命館大学, 理工学部, 教授 (60112273)
• Co-Investigator(Renkei-kenkyūsha): KOSAKI HIDEKI 九州大学, 数理科学研究院, 教授 (20186612) ; WATATANI YASUO 九州大学, 数理化学研究院, 教授 (00175077) ; SETO MICHIO 島根大学, 総合理工学研究院, 准教授 (30398953)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000); Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2013: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: Operator functions / Operator monotone / Operator convex / Loewner's theorem / Pick functions / Gamma function / Operator inequality / Positive linear maps / Positive linear map / Kadison's inequality / Loewner-Heinz inequality / Orthogonal polynomials / Jacobi operator / 作用素単調関数 / Pick 関数 / 解析接続 / 作用素不等式 / Loewner の定理

Outline of Final Research Achievements

It is known that an inner product of a unit vector and a vector function which is a transformation of the unit vector by the resolvent of a Jacobi operator is an operator monotone function. We had conversely proved that every operator monotone function is represented by using inner product.
We next gave a condition under which the converse of the Loewner-Heinz inequality holds using the perturbation of the identity operator. The paper on this result was published from Proceeding of the Edinburgh Math. Soc. Two relevant papers to this result were published as joint works. Further we have proved the Choi conjecture on positive linear maps on C*-algebras.

Research Products

• [Int'l Joint Research] University of Copenhagen(Denmark)
• [Int'l Joint Research] University of Copenhagen(Denmark)
• [Journal Article] Inverses of operator convex functions (2016)
  • Author(s): Henrik L. Pedersen; Mitsuru Uchiyama
  • Journal Title: Trend in Mathematics (Ordered Structures and Aplications: Positivity VII) Volume: - Pages: 363-370
  • DOI: 10.1007/978-3-319-27842-1_23
  • ISBN: 9783319278407, 9783319278421
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Order of operators determined by operator mean (2015)
  • Author(s): Masaru Nagisa; Mitsuru Uchiyama
  • Journal Title: Tohoku Math. J. (2) Volume: 67 Pages: 39-50
  • NAID: 130007961972
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Operator inequalities: from a general theorem to concrete inequalities (2015)
  • Author(s): Mitsuru Uchiyama
  • Journal Title: Linear Algebra Appl Volume: 465
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Superadditivity and derivative of operator functions (2015)
  • Author(s): Uchiyama, Mitsuru; Uchiyama, Atsushi; Giga, Mariko
  • Journal Title: Linear Algebra Appl Volume: 465
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] A converse of Loewner-Heinz inequality and applications to operator means. (2014)
  • Author(s): Mitsuru Uchiyama; Yamazaki, Takeaki
  • Journal Title: J. Math. Anal. Appl Volume: 413
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] A converse of Loewner-Heinz inequality, geometric mean and spectral order (2014)
  • Author(s): Mitsuru Uchiyama
  • Journal Title: Proceeding of the Edinburgh Math. Soc. Volume: 57
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Operator monotone functions, Jacobi operators and orthogonal polynomials. (2013)
  • Author(s): Uchiyama Mitsuru,
  • Journal Title: J. Math. Anal. Appl Volume: 401 Pages: 501-509
  • Peer Reviewed
• [Journal Article] Inequality between unitary orbits (2013)
  • Author(s): Uchiyama Mitsuru, Seto Michio
  • Journal Title: C. R. Math. Acad. Sci. Paris Volume: 351 Pages: 285-288
  • Peer Reviewed
• [Presentation] Positive linear maps and rigid functions -Choi's conjecture- (2016)
  • Author(s): Mitsuru Uchiyama
  • Organizer: Holomorphic Day (University of Copenhagen)
  • Place of Presentation: Copenhagen (Denmark)
  • Year and Date: 2016-11-18
  • Int'l Joint Research / Invited
• [Presentation] Positive linear maps on C*-algebras and rigid functions (2016)
  • Author(s): 内山 充
  • Organizer: 作用素環-作用素論研究集会
  • Place of Presentation: 前橋県立大学（群馬県・前橋市）
  • Year and Date: 2016-10-11
  • Invited
• [Presentation] Positive linear maps and rigid functions (2016)
  • Author(s): Mitsuru Uchiyama
  • Organizer: ILAS 2016
  • Place of Presentation: Leuven (Bergium)
  • Year and Date: 2016-07-11
  • Int'l Joint Research / Invited
• [Presentation] Some conditions for a positive linear map to be a C*-homomorphism (2015)
  • Author(s): Mitsuru Uchiyama
  • Organizer: Workshop on Quantum Information Theory and related Topics
  • Place of Presentation: Vietnam Academy of Science and Technology（ベトナム）
  • Year and Date: 2015-09-03
  • Int'l Joint Research / Invited
• [Presentation] Some conditions for a linear map to be C*-homomorphic (2015)
  • Author(s): Mitsuru Uchiyama
  • Organizer: 九大数理セミナー
  • Place of Presentation: 九州大学数理棟（福岡市西区）
  • Year and Date: 2015-08-03
• [Presentation] 作用素関数の優加法性とフレッシェ微分 (2015)
  • Author(s): 儀我真理子・内山敦・内山充
  • Organizer: 日本数学会年会
  • Place of Presentation: 明治大学（東京）
  • Year and Date: 2015-03-22
• [Presentation] From a general theory to concrete inequalities (2014)
  • Author(s): Mitsuru Uchiyama
  • Organizer: ICM satellite (ILAS2014)
  • Place of Presentation: Sungkyunkwan Univ (Korea)Seoul
  • Year and Date: 2014-08-07
  • Invited
• [Presentation] 行列関数の解析ー作用素不等式、多項式系、ガンマ関数ー
  • Author(s): Uchiyama Mitsuru
  • Organizer: 日本数学会 - 企画特別講演
  • Place of Presentation: 学習院大学(東京都)
  • Invited
• [Presentation] The principal inverse of the gamma function
  • Author(s): Uchiyama Mitsuru
  • Organizer: Positivity VII
  • Place of Presentation: Leiden University(オランダ)
• [Presentation] A converse of Loewner-Heinz inequality
  • Author(s): Uchiyama Mitsuru
  • Organizer: 日本数学会
  • Place of Presentation: 愛媛大学(愛媛県)