The study of eigenvalue distribution of operator function and polynomials

• Project/Area Number: 21540181
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Shimane University
• Principal Investigator: UCHIYAMA Mitsuru 島根大学, 総合理工学研究科, 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, 数理科学研究院, 教授 (00175077) ; SETO Michio 島根大学, 総合理工学研究科, 准教授 (30398953)
• Project Period (FY): 2009 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2012: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2010: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2009: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 作用素関数 / 固有値 / 作用素不等式 / 直交多項式 / ガンマ関数 / Pick 関数 / 作用素単調関数 / 核関数 / 積分核 / Majorization / 作用素凸関数

Research Abstract

We got some result on eigenvalue for an operator function f(X), where f is a concave function: precisely, the sum of eigenvalues of f(X+Y) from the largest through to k-th is not less than one for f(X)+f(Y) for every k. We have also shown that the principal inverse of the gamma function is a Pick function.

Research Products

• [Journal Article] A converse of Loewner-Heinz inequality, geometric mean and spectral order (2013)
  • Author(s): M. Uchiyama
  • Journal Title: Proc. Edinburgh Math. Soc. Volume: -
  • Peer Reviewed
• [Journal Article] Operator monotone functions, Jacobi operators and orthogonal polynomials (2013)
  • Author(s): M. Uchiyama
  • Journal Title: J. Mathematical Analysis and applications Volume: -
  • Peer Reviewed
• [Journal Article] A normal family of operator monotone functions (2013)
  • Author(s): M. S. Moslehian
  • Journal Title: Hokkaido Math. J. Volume: -
  • Peer Reviewed
• [Journal Article] Some inequalities for power series of two operators in Hilbert space (2013)
  • Author(s): S. S. Dragomir
  • Journal Title: Tokyo J. Mathematics Volume: -
  • Peer Reviewed
• [Journal Article] The principal inverse of the gamma function (2012)
  • Author(s): M. Uchiyama
  • Journal Title: Proc. Amer. Math. Soc Volume: 140 Pages: 1343-1348
  • Peer Reviewed
• [Journal Article] The principal inverse of the gamma function, (2012)
  • Author(s): M. Uchiyama
  • Journal Title: Proc. Amer. Math. Soc. Volume: 140 Pages: 1343-1348
  • Peer Reviewed
• [Journal Article] The principal inverse of the gamma function (2012)
  • Author(s): M.Uchiyama
  • Journal Title: Proceeding of Amer.Math.Soc. Volume: 140 Pages: 1343-1348
  • Peer Reviewed
• [Journal Article] A new look at Gamma function, 数理解析研究所講究録 (2011)
  • Author(s): M.Uchiyama
  • Journal Title: 作用素論における非可換構造の研究とその応用 Volume: 1731 Pages: 133-139
• [Journal Article] Positive definiteness of functions with applications to operator norm inequalities (2011)
  • Author(s): H. Kosaki
  • Journal Title: Mem. Amer.Math. Soc Volume: 997
  • Peer Reviewed
• [Journal Article] Operator monotone functions, positive definite kernel andmajorization (2010)
  • Author(s): M. Uchiyama
  • Journal Title: Proc. Amer. Math. Soc Volume: 138 Pages: 3985-3996
  • Peer Reviewed
• [Journal Article] Majorization and some operator monotone functions (2010)
  • Author(s): M. Uchiyama
  • Journal Title: Linear Algebra Appl Volume: 432 Pages: 1867-1872
  • Peer Reviewed
• [Journal Article] Operator monotone functions, positive definite kernels and majorization (2010)
  • Author(s): M.Uchiyama
  • Journal Title: Proceeding of Amer.Math.Soc. Volume: 138 Pages: 3985-3996
  • Peer Reviewed
• [Journal Article] Majorization and some operator monotone functions, (2010)
  • Author(s): M.Uchiyama
  • Journal Title: Linear Alg.Appl. 432
  • Peer Reviewed
• [Journal Article] Positive definite kernels and majorization (2009)
  • Author(s): M.Uchiyama
  • Journal Title: 作用素論における非可換解析学の展望 Volume: 1678 Pages: 97-105
• [Journal Article] A new majorization induced by matrix order (2009)
  • Author(s): M. Uchiyama
  • Journal Title: Oper. Theory Adv. Appl Volume: 187 Pages: 211-216
  • Peer Reviewed
• [Journal Article] The operator equation ニAn-iXBi=Y (2009)
  • Author(s): R. Bhatia, M. Uchiyama
  • Journal Title: Expo. Math Volume: 27 Pages: 251-255
  • Peer Reviewed
• [Journal Article] Infinite sequences of inner functions and submodules in H2(D2), J (2009)
  • Author(s): M. Seto
  • Journal Title: Operator Theory Volume: 61 Pages: 75-86
  • Peer Reviewed
• [Journal Article] Indecomposable representations of quivers on infinite dimensional Hilbert spaces (2009)
  • Author(s): M.Enomoto,Y.Watatani
  • Journal Title: J. Funct. Anal Volume: 256 Pages: 959-991
  • NAID: 110002548905
  • Peer Reviewed
• [Journal Article] The operator equation ΣA^<n-i> X B^i=Y (2009)
  • Author(s): R.Bhatia
  • Journal Title: Expositiones Math. 27
  • Peer Reviewed
• [Journal Article] converse of Loewner-Heinz inequality
  • Author(s): M.Uchiyama
  • Journal Title: geometric mean and spectral order
  • Peer Reviewed
• [Journal Article] Inequality between unitary orbits
  • Author(s): M. Uchiyama, M. Seto
  • Journal Title: C. R. Sci. Paris, Ser.1
  • Peer Reviewed
• [Journal Article] Someinequality for power series of two operators in Hilbert space
  • Author(s): S.S. Dragomir, M. Uchiyama
  • Journal Title: Tokyo J. Mathematics
  • Peer Reviewed
• [Journal Article] A normal family of operator monotone functions
  • Author(s): M.S.Moslehian, H. Najafi, M. Uchiyama
  • Journal Title: Hokkaido Math. Journal
  • Peer Reviewed
• [Journal Article] Operator monotone functions,Jacobi operators and orthogonal polynomials
  • Author(s): M. Uchiyama
  • Journal Title: J. Math, Anal. Appl
  • Peer Reviewed
• [Presentation] 直交多項式の主逆関数 (2013)
  • Author(s): M. Uchiyama
  • Organizer: 日本数学会
  • Place of Presentation: 京都大学(京都市)
  • Year and Date: 2013-03-22
• [Presentation] 直交多項式の主逆関数 (2013)
  • Author(s): M. Uchiyama
  • Organizer: 日本数学会
  • Place of Presentation: 京都大学（京都市）
• [Presentation] Operator monotone functions and polynomials (2012)
  • Author(s): M. Uchiyama
  • Organizer: Matrices and Operators
  • Place of Presentation: Indian Institute of Science
  • Year and Date: 2012-12-28
• [Presentation] 作用素単調関数Jacobi 作用素,直交多項式 (2012)
  • Author(s): M. Uchiyama
  • Organizer: 作用素論・作用素環論研究集会
  • Place of Presentation: 大阪教育大学(大阪市)
  • Year and Date: 2012-11-24
• [Presentation] The principal inverse of the gamma function (2012)
  • Author(s): M. Uchiyama
  • Organizer: 日本数学会
  • Place of Presentation: 九州大学(福岡市)
  • Year and Date: 2012-09-20
• [Presentation] The principal inverse of the gamma function (2012)
  • Author(s): M. Uchiyama
  • Organizer: 日本数学会
  • Place of Presentation: 九州大学（福岡市）
• [Presentation] Operator monotone functions and polynomials. (2012)
  • Author(s): M. Uchiyama
  • Organizer: Internatinal conference 'Matrices and Operators`
  • Place of Presentation: Indian Institute of Science（インドBangalore）
  • Invited
• [Presentation] The principal inverse of the gamma function (2011)
  • Author(s): M. Uchiyama
  • Organizer: Operator Theory and its applications
  • Place of Presentation: Kyungpook National University
  • Year and Date: 2011-08-26
• [Presentation] The principal inverse of the gamma function (2011)
  • Author(s): M.Uchiyama
  • Organizer: Operator Theory and its Application
  • Place of Presentation: Kyungpok Natinal Univ.Daegu,韓国(招待講演)
  • Year and Date: 2011-08-26
• [Presentation] Loewner の定理と応用ー行列順序、多項式系、ガンマ関数 (2011)
  • Author(s): M. Uchiyama
  • Organizer: Korovkin の定理-日本数学会(企画特別講演)
  • Place of Presentation: 早稲田大学(東京)
  • Year and Date: 2011-03-22
• [Presentation] Loewnerの定理と応用-行列順序、多項式系、ガンマ関数、Korovkinの定理 (2011)
  • Author(s): 内山充
  • Organizer: 日本数学会
  • Place of Presentation: 早稲田大学(招待:企画特別講演)
  • Year and Date: 2011-03-22
• [Presentation] 作用素論における非可換構造の研究とその応用 (2010)
  • Author(s): M.Uchiyama
  • Organizer: A new look at Gamma function
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2010-10-29
• [Presentation] Matrix functions, Majorizations and Orthogonal polynomials (2010)
  • Author(s): M. Uchiyama
  • Organizer: ICM Satellite conference "Functional Analysis and Operator Theory"
  • Place of Presentation: Indian Statistical Institute at Bangalore
  • Year and Date: 2010-08-10
• [Presentation] Matrix functions, orthogonal polynomials and kernel functions (2010)
  • Author(s): M.Uchiyama
  • Organizer: 国際数学者会議satellite conference "Functional Analysis and Operator Theory"
  • Place of Presentation: Indian Statistical Institute Bangalore(招待講演)
  • Year and Date: 2010-08-10
• [Presentation] Operator monotone functions, positive definite kernels and majorization (2010)
  • Author(s): M. Uchiyama
  • Organizer: 16th Internatinal Linear Algebra Society
  • Place of Presentation: Pisa(イタリア)
  • Year and Date: 2010-06-23
• [Presentation] Operator monotone functions, Positive definite kernels and Majorization (2010)
  • Author(s): M.Uchiyama
  • Organizer: International Linear Algebra Society
  • Place of Presentation: Pisa
  • Year and Date: 2010-06-23
• [Presentation] 作用素論における非可換解析学の展望 (2009)
  • Author(s): M.Uchiyama
  • Organizer: Positive definite kernels and majorization
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2009-10-30
• [Presentation] 正定値核とマジョリゼーション (2009)
  • Author(s): M.Uchiyama
  • Organizer: 京都大学数理解析研究所研究集会
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2009-10-30