On spectrum preserving maps on Banach spaces and its application

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K03650
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Niigata University
• Principal Investigator: Miura Takeshi 新潟大学, 自然科学系, 教授 (90333989)
• Co-Investigator(Kenkyū-buntansha): 大井 志穂 新潟大学, 自然科学系, 助教 (90891789)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 可換Banach環 / 関数環 / 等距離写像 / Tingley's problem / Mazur-Ulam property

Outline of Final Research Achievements

We have analyzed the structures of surjective, not necessarily linear, isometries between Banach spaces. For example, we have characterized surjective isometries on the Banach space of all continuous complex valued functions on the closed unit disk, which are analytic on the open unit disk with continuous derivatives on the closed unit disk. Motivated by the above result, we attacked a similar problem for a Banach space of bounded analytic functions with continuous derivatives on the open unit disk. We have characterized surjective isometries on the Banach space. A similar result was obtained by Novinger and Oberlin in 1985 for a Banach space of analytic functions on the open unit disk, whose p-th power are integrable with continuous derivatives on the closed unit disk. In addition, we have obtained some results on Tingley's problem for some Banach spaces including function algebras and abstract function spaces with continuous derivatives.

Free Research Field

可換Banach環理論

Academic Significance and Societal Importance of the Research Achievements

1985年にNovinger and Oberlinは，連続な導関数をもつ単位開円板上のHardy空間H^p関数全体に対して，その間の線形等距離写像の形を決定した．この定理は1≦p<∞のに対して示されているが，p=∞のときは対応する結果が知られていなかった．本研究では連続な導関数をもつH^∞関数全体に対して，その間の等距離写像を解明した．その際，Novinger and Oberlinの定理とは異なり，全射性を必要とするが，線形性を仮定することなく，等距離写像の構造を決定することに成功した．
またTingley問題は世界的に活発な研究がなされている未解決問題であるが，その部分的解答も与えている．