Spectrum preservers on Banach algebras and the stability of their perturbation

• Project/Area Number: 23740097
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Niigata University (2013-2014); Yamagata University (2011-2012)
• Principal Investigator: MIURA TAKESHI 新潟大学, 自然科学系, 教授 (90333989)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: スペクトル保存写像 / 等距離写像 / 関数環 / 可換Banach環 / 摂動の安定性 / スペクトル保存

Outline of Final Research Achievements

I investigated the structure of mappings which preserve spectra or related information between Banach algebras in detail. I gave a characterization of surjective isometries without assuming linearity between function algebras, which need not be unital. Many results on multiplicatively peripheral spectrum preserving surjections are well known. I generalized some of them and proved the structure theorem on spectra radii preserving mappings between unital semi-simple commutative Banach algebras. I gave a necessary and sufficient condition for a system of Banach space valued differential equations to be stable in the sense of Hyers and Ulam. I also gave a sufficient condition for an integral equation to have the Hyers-Ulam stability.

Research Products

• [Journal Article] Real-linear isometries between subspaces of continuous functions (2014)
  • Author(s): H. Koshimizu, T. Miura, H. Takagi and S.-E. Takahasi
  • Journal Title: J. Math. Anal. Appl. Volume: 413 Issue: 1 Pages: 229-241
  • DOI: 10.1016/j.jmaa.2013.11.050
  • Peer Reviewed / Open Access
• [Journal Article] Ulam type stability problems for alternative homomorphisms (2014)
  • Author(s): S.-E. Takahasi, M. Tsukada, T. Miura, H. Takagi and K. Tanahashi
  • Journal Title: J. Inequal. Appl. Volume: 2014:228
  • NAID: 120007101195
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Ulam typw stability of a generalized additive mapping and concrete examples (2013)
  • Author(s): H. Oda, M. Tsukada, T. Miura, Y. Kobayashi and S.-E. Takahasi
  • Journal Title: Int. J. Math. Math. Sci. Volume: 2013 Pages: 1-7
  • Peer Reviewed
• [Journal Article] A characterization of the stability of a system of the Banach space valued differential equations (2013)
  • Author(s): T. Miura, G. Hirasawa S.-E. Takahasi and T. Hayata
  • Journal Title: Math. Inequal. Appl. Volume: 16 Issue: 3 Pages: 717-728
  • DOI: 10.7153/mia-16-54
  • Peer Reviewed
• [Journal Article] Real linear isometries between function algebras. II (2013)
  • Author(s): Osamu Hatori and Takeshi Miura
  • Journal Title: Cent. Eur. J. Math. Volume: 11 Issue: 10 Pages: 1838-1842
  • DOI: 10.2478/s11533-013-0282-0
  • Peer Reviewed
• [Journal Article] Ulam type stability of a generalized additive mapping and concrete examples (2013)
  • Author(s): Hiroyoshi Oda, Makoto Tsukada, Takeshi Miura, Yuji Kobayashi and Sin-Ei Takahasi
  • Journal Title: Int. J. Math. Math. Sci. Volume: 2013
  • Peer Reviewed
• [Journal Article] A note on the stability of an integral equation (2012)
  • Author(s): Takeshi Miura, Go Hirasawa, Sin-Ei Takahasi and Takahiro Hayata
  • Journal Title: Functional Equations in Mathematical Analysis Volume: 52 Pages: 207-222
  • DOI: 10.1007/978-1-4614-0055-4_17
  • ISBN: 9781461400547, 9781461400554
• [Journal Article] The Hyers-Ulam and Ger type stabilities of the first order linear differential equations (2012)
  • Author(s): Takeshi Miura and Go Hirasawa,
  • Journal Title: Functional Equations in Mathematical Analysis Volume: 52 Pages: 191-199
  • DOI: 10.1007/978-1-4614-0055-4_15
  • ISBN: 9781461400547, 9781461400554
• [Journal Article] On the Butler-Rassias functional equation and its generalized Hyers-Ulam stability (2012)
  • Author(s): Takeshi Miura, Go Hirasawa and Takahiro Hayata
  • Journal Title: Functional Equations in Mathematical Analysis Volume: 52 Pages: 201-206
  • DOI: 10.1007/978-1-4614-0055-4_16
  • ISBN: 9781461400547, 9781461400554
• [Journal Article] Reflections and a generalization of the Mazur-Ulam theorem (2012)
  • Author(s): Osamu Hatori, Kiyotaka Kobayashi, Takeshi Miura and Sin-Ei Takahasi
  • Journal Title: Rocky Mountain J. Math. Volume: 42 Issue: 1
  • DOI: 10.1216/rmj-2012-42-1-117
  • Peer Reviewed
• [Journal Article] Stability of the Banach space valued Chebyshev differential equation (2012)
  • Author(s): Takeshi Miura, Sin-Ei Takahasi, Takahiro Hayata and Kotaro Tanahashi
  • Journal Title: Appl. Math. Lett. Volume: 25 Issue: 11 Pages: 1976-1979
  • DOI: 10.1016/j.aml.2012.03.012
  • Peer Reviewed
• [Journal Article] Stability problems for generalized additive mappings and Euler-Lagrange type mappings (2012)
  • Author(s): Mamoru Todoroki, Keisaku Kumahara, Takeshi Miura and Sin-Ei Takahasi
  • Journal Title: Aust. J. Math. Anal. Appl. Volume: 9
  • Peer Reviewed
• [Journal Article] Isometries and translations of groups (2012)
  • Author(s): Osamu Hatori, Go Hirasawa, Takeshi Miura and Lajos Molnar
  • Journal Title: Tokyo J. Math. Volume: 35
  • Peer Reviewed
• [Journal Article] An equality condition of Arhippainen-Muller's estimate and its related problem (2011)
  • Author(s): S.-E. Takahasi, T. Miura and T. Hayata
  • Journal Title: Taiwanese J. Math. Volume: 15
  • Peer Reviewed
• [Journal Article] Real-linar isometries between function algebras. (2011)
  • Author(s): T. Miura
  • Journal Title: Cent. Eur. J. Math. Volume: 9
  • Peer Reviewed
• [Journal Article] Divisibly norm-preseving maps betwenn commutative Banach algebras. (2011)
  • Author(s): T. Miura, D. Honma and R. Shindo
  • Journal Title: Rocky Mountain J. Math. Volume: 41
  • Peer Reviewed
• [Journal Article] Spectral preserver problems in commutative Banach algebras. (2011)
  • Author(s): O. Hatori, S. Lambert, A. Luttman, T. Miura, T. Tonev and R. Yates
  • Journal Title: Contemp. Math. Volume: 547
  • Peer Reviewed
• [Journal Article] Spectral radii conditions for isomorphisms between unital semisimple commutative Banach algebras. (2011)
  • Author(s): G. Hirasawa T. Miura and H. Takagi
  • Journal Title: Contemp. Math. Volume: 547
  • Peer Reviewed
• [Presentation] 関数環上のスペクトル保存写像の拡張 (2015)
  • Author(s): 三浦 毅，Thomas Tonev
  • Organizer: 日本数学会
  • Place of Presentation: 明治大学（東京都・千代田区）
  • Year and Date: 2015-03-23
• [Presentation] 関数空間上の等距離写像とその例 (2015)
  • Author(s): 三浦 毅
  • Organizer: つくばセミナー
  • Place of Presentation: 筑波大学（茨城県・つくば市）
  • Year and Date: 2015-03-06
  • Invited
• [Presentation] 関数空間上の全射等距離写像 (2014)
  • Author(s): 三浦 毅
  • Organizer: 集合論的・幾何学的トポロジーと種々の分野の交流
  • Place of Presentation: 京都大学数理解析研究所（京都府・京都市）
  • Year and Date: 2014-10-23
  • Invited
• [Presentation] Surjective isometries between function spaces (2014)
  • Author(s): Takeshi Miura
  • Organizer: The Seventh Conference on Function Spaces
  • Place of Presentation: Southern Illinois University at Edwardsville (Illinois, America)
  • Year and Date: 2014-05-22
  • Invited
• [Presentation] Real linear isometries between function algebras (2011)
  • Author(s): T. Miura
  • Organizer: Reaserch on preserver problems concerning to Banach algebras and its applications
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] 関数環上の実線形等距離写像の構造 (2011)
  • Author(s): 三浦 毅
  • Organizer: 関数環研究集会
  • Place of Presentation: 山形大学
• [Presentation] 関数環上の全射等距離写像
  • Author(s): 三浦 毅
  • Organizer: 日本数学会
  • Place of Presentation: 愛媛大学
• [Presentation] Isometries between function spaces
  • Author(s): H. Koshimizu, T. Ｍｉｕｒａ, H. Takagi and S.-E. Takahasi
  • Organizer: Complex Analysis and Applications '13
  • Place of Presentation: The Institute of Mathematics and Informatics, Bulgaria
  • Invited
• [Presentation] 関数環上のある種のスペクトル保存写像の構造，II
  • Author(s): 三浦 毅
  • Organizer: 関数環研究集会
  • Place of Presentation: クロスパルにいがた
• [Presentation] Surjective Isometries between function spaces
  • Author(s): Takeshi Miura
  • Organizer: Department of Mathematical Sciences, Colloquium Series
  • Place of Presentation: The University of Montana
  • Invited
• [Presentation] Surjective Isometries between function algebras
  • Author(s): Takeshi Miura
  • Organizer: International Conference on Mathematics and Statictics
  • Place of Presentation: University of Memphis
  • Invited
• [Presentation] 関数環上のある種のスペクトル保存写像の構造
  • Author(s): 三浦 毅
  • Organizer: 関数環研究集会
  • Place of Presentation: 日本大学
• [Presentation] 関数空間上の全射等距離写像と端点の性質
  • Author(s): 三浦 毅
  • Organizer: つくばセミナー
  • Place of Presentation: 筑波大学