Study on operator means and related topics

• Project/Area Number: 26400120
• 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, 数理学研究院, 学術研究者 (20186612)
• Co-Investigator(Kenkyū-buntansha): 綿谷 安男 九州大学, 数理学研究院, 教授 (00175077)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000); Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 作用素平均 / 作用素ノルム不等式 / 正定値関数 / 並列和 / 非有界作用素 / 絶対連続性 / ノルム不等式 / Heinz不等式

Outline of Final Research Achievements

Some comparison results (in strong sense such as positive definiteness and/or infinite divisibility) for various means were proved. For instance monotonicity (in parameters) for binomial and more generally Stolarsky means was obtained. As a consequence many new norm inequalities for related operator means were established.
Several notions of parallel sums for unbounded positive self-adjoint operators have been studied by some groups with certain additional domain conditions. A satisfactory theory free from any domain conditions was developed in our study. This theory seems to be a useful device for study of
absolute continuity for general positive self-adjoint operators.

Research Products

• [Journal Article] Absolute continuity for unbounded positive self-adjoint operators (2018)
  • Author(s): H. Kosaki
  • Journal Title: to appear in Kyushu J. Math Volume: 印刷中
  • NAID: 130007521540
  • Peer Reviewed / Open Access
• [Journal Article] Maximal abelian C*-sublagebras of C*-algebras associated with self-similar maps and complex dynamical systems (2017)
  • Author(s): T. Kajiwara and Y. Watatani
  • Journal Title: Jounal of Mathematical Analysis and Application Volume: 455 Issue: 2 Pages: 1383-1400
  • DOI: 10.1016/j.jmaa.2017.06.044
  • Peer Reviewed
• [Journal Article] Parallel sum of unbounded positive operators (2017)
  • Author(s): H. Kosaki
  • Journal Title: to appear in Kyushu J. Math. Volume: 印刷中
  • NAID: 130006334235
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Ideals of the core of C*-algebras associated with self-similar maps (2016)
  • Author(s): T. Kajiwara and Y. Watatani
  • Journal Title: J. Operator Theory Volume: 75 Pages: 225-255
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] A certain generalization of the Heinz inequality for positive operators (2016)
  • Author(s): Hideki Kosaki
  • Journal Title: Internat. J. Math. Volume: 27 Issue: 02 Pages: 1650008-1650008
  • DOI: 10.1142/s0129167x16500087
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Strongly irreducible operators and indecomposable representations of quivers on infinite-dimensional Hilbert spaces (2015)
  • Author(s): M. Enomoto and Y. Watatani
  • Journal Title: Integral Equations Operator Theory Volume: 83 Issue: 4 Pages: 563-587
  • DOI: 10.1007/s00020-015-2228-3
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Strong monotonicity for various means (2014)
  • Author(s): H. Kosaki
  • Journal Title: J. Funct. Anal. Volume: 267 Issue: 7 Pages: 1917-1958
  • DOI: 10.1016/j.jfa.2014.07.021
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Traces on cores of C*-algebras associated with self-similar maps (2014)
  • Author(s): Tsuyoshi Kajiwara and Yasuo Watatani
  • Journal Title: Ergodic Theory and Dynamical Systems Volume: 34 Issue: 6 Pages: 1964-1989
  • DOI: 10.1017/etds.2013.26
  • Peer Reviewed / Acknowledgement Compliant