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RESEARCH ON GEOMETRY OF BANACH AND FUNCTION SPACES AND INEQUALITIES

Research Project

Project/Area Number 16540163
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKYUSHU INSTITUTE OF TECHNOLOGY

Principal Investigator

KATO Mikio  KYUSHU INSTITUTE OF TECHNOLOGY, FACULTY OF ENGINEERING, PROFESSOR, 工学部, 教授 (50090551)

Co-Investigator(Kenkyū-buntansha) SUZUKI Tomonari  KYUSHU INSTITUTE OF TECHNOLOGY, FACULTY OF ENGINEERING, ASSOCIATE PROFESSOR, 工学部, 助教授 (00303173)
TAKAHASHI Yasuji  OKAYAMA PREFECTURAL UNIVERSITY, FACULTY OF COMPUTER SCIENCE AND SYSTEM ENGINEERING, PROFESSOR, 情報工学部, 教授 (30001853)
SAITO Kichi-suke  NIIGATA UNIVERSITY, FACULTY OF SCIENCE, PROFESSOR, 理学部, 教授 (30018949)
Project Period (FY) 2004 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2005: ¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 2004: ¥1,800,000 (Direct Cost: ¥1,800,000)
KeywordsHanner-type inequality / random Clarkson inequality / p-uniform smoothness / sharp triangle inequality / ψ-direct sum of Banach spaces / weak nearly uniform smoothness / B_n-convexity / fixed point property / バナッハ空間のψ直和 / uniform non-l^n_1-ness / q-uniform convexity / strong type p / strong random Clarkson不等式 / weak nearly uniform smoothness
Research Abstract

Geometric structures of Banach and function spaces were investigated especially in relation with some norm inequalities. Also ψ-direct sums of Banach spaces were investigated, which seem important as we can easily construct many examples of Banach spaces with a non l_p-type norm from a convex function ψ. Major results are as follows.
[1]On geometric structures of Banach and function spaces and norm inequalities :
(1)We extended the strong random Clarkson inequality, and characterized the Banach spaces in which the resulting inequality is valid are those of p-uniform smooth, or equivalently those of strong type p with the same constants involved them.
(2)We introduced Hanner-type inequalities with a weight and characterized 2-uniformly smooth, and 2-uniformly convex spaces with these inequalities. By changing the place of the weight, the resulting inequalities characterize p-uniformly smooth, and q-uniformly convex spaces. The best value of the weight in these inequalities were determined for L_p-spaces. A duality theorem for these Hanner-type inequalities was presented as well.
[2]We presented a sharp triangle inequality and its reverse in a Banach space.
[3]On ψ-direct sums of Banach spaces :
(1)We characterized the following properties of ψ-direct sums of Banach spaces :
smoothness, weak nearly uniform smoothness, WORTH property, Schur property etc. As an application we presented several examples of Banach spaces which are not uniformly non-square, but have the fixed point property.
(2)We characterized uniform non l^n_1-ness (B_n-convexity) particularly for the l_1-sum and l_∞-sum of Banach spaces.
[4]Some resuts were obatined on the James constant, on the relation between Property M and the fixed point property, and on the Banach-Mazur distance in relation with super-reflexivity etc.

Report

(3 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • Research Products

    (58 results)

All 2006 2005 2004 Other

All Journal Article (56 results) Book (2 results)

  • [Journal Article] The James constant of 2-dimensional Lorentz sequence spaces2006

    • Author(s)
      Tomonari Suzuki, Akiko Yamano, Mikio Kato
    • Journal Title

      Bulletin of the Kyushu Institute of Technology Pure and Applied Mathematics 53

      Pages: 15-24

    • NAID

      120002441844

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The James constant of 2-dimensional Lorentz sequence spaces2006

    • Author(s)
      T.Suzuki, A.Yamano, M.Kato
    • Journal Title

      Bulletin of the Kyushu Institute of Technology Pure and Applied Mathematics 53

      Pages: 15-24

    • NAID

      120002441844

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On some inequalities related with p-uniform smoothness and q-uniform convexity2005

    • Author(s)
      山田康隆, 加藤幹雄, 高橋泰嗣
    • Journal Title

      京都大学数理解析研究所講究録 1415

      Pages: 64-69

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Remarks on strong random Clarkson Inequality2005

    • Author(s)
      Yasutaka Yamada, Yasuji Takahashi, Mikio Kato
    • Journal Title

      Hokkaido university Technical Report Series in Mathematics 91

      Pages: 48-52

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Extension of random Clarkson Inequalities2005

    • Author(s)
      Yasutaka Yamada, Yasuji Takahashi, Mikio Kato
    • Journal Title

      Journal of Nonlinear and Convex Analysis 6

      Pages: 419-429

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Convex sets and inequalities2005

    • Author(s)
      Sin-Ei Takahashi, Yasuji Takahashi, et al.
    • Journal Title

      Journal of Inequalities and Applications 2005-2

      Pages: 107-117

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Smoothness of -direct sums of Banach spaces2005

    • Author(s)
      Ken-ich Mitani, S.Oshiro, Kichi-Suke Saito
    • Journal Title

      Mathematical Inequalities and Applications 8

      Pages: 147-157

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Characterizations of fixed points of nonexpansive mappings2005

    • Author(s)
      Tomonari Suzuki
    • Journal Title

      International Journal of Mathematics and Mathematical Sciences 2005-11

      Pages: 1723-1735

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Uniform non l^n_1-ness of -direct sums of Banach spaces X Y2005

    • Author(s)
      Takayuki Tamura, Mikio Kato, Kichi-Suke Saito
    • Journal Title

      京都大学数理解析研究所講究録 1452

      Pages: 227-232

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Property M and the fixed point property for nonexpansive mappings2005

    • Author(s)
      Takayuki Tamura, Mikio Kato
    • Journal Title

      京都大学数理解析研究所講究録 1455

      Pages: 58-64

    • NAID

      120000901204

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Functions related to some geometrical properties of Banach spaces2005

    • Author(s)
      Yasuji Takahashi, Mikio Kato
    • Journal Title

      京都大学数理解析研究所講究録 1455

      Pages: 65-69

    • NAID

      110001900999

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Best weight constants for Lp in Hanner type inequalities related with 2-uniform convexity and smoothness2005

    • Author(s)
      高橋泰嗣, 加藤幹雄, 山田康隆
    • Journal Title

      実解析学シンポジウム2005

      Pages: 89-92

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] On some inequalities related with p-uniform smoothness and q-uniform convexity2005

    • Author(s)
      Yasutaka Yamada, Mikio Kato, Yasuji Takahashi
    • Journal Title

      RIMS Kokyuroku (Kyoto University) 1415

      Pages: 64-69

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Remarks on strong random Clarkson inequality2005

    • Author(s)
      Yasutaka Yamada, Ysuji Takahashi, Mikio Kato
    • Journal Title

      Hokkaido University Technical Report Series in Mathematics 91

      Pages: 48-52

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Extension of random Clarkson inequalities2005

    • Author(s)
      Yasutaka Yamada, Yasuji Takahashi, Mikio Kato
    • Journal Title

      Journal of Nonlinear and Convex Analysis 6

      Pages: 419-429

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Convex sets and inequalities2005

    • Author(s)
      Sin-Ei Takahasi, Yasuji Takahashi, et al.
    • Journal Title

      Journal of Inequalities and Applications 2005-2

      Pages: 107-117

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Smoothness of ψ-direct sums of Banach spaces2005

    • Author(s)
      Ken-ichi Mitani, S.Oshiro, Kichi-Suke Saito
    • Journal Title

      Mathematical Inequalities and Applications 8

      Pages: 147-157

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Characterization of fixed points of nonexpansive mappings2005

    • Author(s)
      Tomonari Suzuki
    • Journal Title

      International Journal of Mathematics and Mathematical Sciences 2005-11

      Pages: 1723-1735

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Uniform non-l^n_1-ness of ψ-direct sums of Banach spaces X【symmetry】_ψ Y2005

    • Author(s)
      Takayuki Tamura, Mikio Kato, Kichi-Suke Saito
    • Journal Title

      RIMS Kokyuroku (Kyoto University) 1452

      Pages: 227-232

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Property M and the fixed point property for nonexpansive mappings2005

    • Author(s)
      Takayuki Tamura, Mikio Kato
    • Journal Title

      RIMS Kokyuroku (Kyoto University) 1455

      Pages: 58-64

    • NAID

      120000901204

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Functions related to some geometrical properties of Banach spaces2005

    • Author(s)
      Yasuji Takahashi, Mikio Kato
    • Journal Title

      RIMS Kokyuroku (Kyoto University) 1455

      Pages: 65-69

    • NAID

      110001900999

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Best weight constants for L_p in Hanner type inequalities related with 2-uniform convexity and smoothness2005

    • Author(s)
      Yasuji Takahashi, Mikio Kato, Yasutaka Yamada
    • Journal Title

      Real Analysis Symposium 2005

      Pages: 89-92

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Extension of random Clarkson inequalities2005

    • Author(s)
      Y.Yamada, Y.Takahashi, M.Kato
    • Journal Title

      Journal of Nonlinear and Convex Analysis 6

      Pages: 419-429

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Convex sets and inequalities2005

    • Author(s)
      S.Takahasi, Y.Takahashi, S.Miyajima, H.Takagi
    • Journal Title

      Journal of Inequalities and Applications 2005-2

      Pages: 107-117

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Smoothness of ψ-direct sums of Banach spaces2005

    • Author(s)
      K.Mitani, S.Oshiro, K.-S.Saito
    • Journal Title

      Mathematical Inequalities and Applications 8

      Pages: 147-157

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Characterizations of fixed points of nonexpansive mappings2005

    • Author(s)
      T.Suzuki
    • Journal Title

      International Journal of Mathematics and Mathematical Sciences 2005-11

      Pages: 1723-1735

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Uniform non-l^n_1-ness of ψ-direct sums of Banach spaces X【symmetry】_ΨY2005

    • Author(s)
      T.Tamura, M.Kato, K.-S.Saito
    • Journal Title

      京都大学数理解析研究所講究録 1452

      Pages: 227-232

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Property M and the fixed point property for nonexpansive mappings2005

    • Author(s)
      T.Tamura, M.Kato
    • Journal Title

      京都大学数理解析研究所講究録 1455

      Pages: 58-64

    • NAID

      120000901204

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Functions related to some geometrical properties of Banach spaces2005

    • Author(s)
      Y.Takahashi, M.Kato
    • Journal Title

      京都大学数理解析研究所講究録 1455

      Pages: 65-69

    • NAID

      110001900999

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Hanner type inequality and optimal 2-uniform convexity and smoothness inequalities2004

    • Author(s)
      山田康隆, 高橋泰嗣, 加藤幹雄
    • Journal Title

      京都大学数理解析研究所講究録 1365

      Pages: 68-72

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Fixed point property of psi-direct sums of Banach spaces2004

    • Author(s)
      田村高幸, 加藤幹雄
    • Journal Title

      京都大学数理解析研究所講究録 1399

      Pages: 137-143

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Uniform non-squareness of psi-direct sums of Banach spaces X Y2004

    • Author(s)
      Mikio Kato, Kichi-Suke Saito, Takayuki Tamura
    • Journal Title

      Mathematical Inequalities and Applications 7

      Pages: 429-437

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Some geometric conditions related to fixed point property for -direct sums of Banach spaces X Y2004

    • Author(s)
      Mikio Kato, Takayuki Tamura
    • Journal Title

      Proceeding of the international Symposium on Banach and Function Spaces

      Pages: 243-252

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On Hanner-type inequalities for Banach spaces2004

    • Author(s)
      Yasuji Takahashi, Yasutaka Yamada, Mikio Kato
    • Journal Title

      Proceedings of the International Symposium on Banach and Function Spaces

      Pages: 359-365

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Some remarks on the set of common fixed points of one-parameter semi-groups of nonexpansive mappings in Banach spaces with the Opial property2004

    • Author(s)
      Tomonari Suzuki
    • Journal Title

      Nonlinear Analysis, Theory, Methods & Applications 58

      Pages: 441-458

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Strong random Clarkson inequality and its extension2004

    • Author(s)
      Yasutaka Yamada, Mikio Kato, Ysuji Takahashi
    • Journal Title

      京都大学数理解析研究所講究録 1396

      Pages: 143-147

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Hanner type inequality and optimal 2-uniform convexity and smoothness inequalities2004

    • Author(s)
      Yasutaka Yamada, Yasuji Takahashi, Mikio Kato
    • Journal Title

      RIMS Kokyuroku (Kyoto University) 1365

      Pages: 68-72

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Fixed point property of ψ-direct sums of Banach spaces2004

    • Author(s)
      Takayuki Tamura, Mikio Kato
    • Journal Title

      RIMS Kokyuroku (Kyoto University) 1399

      Pages: 137-143

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Uniform non-squareness of ψ-direct sums of Banach spaces X 【symmetry】_ψ Y2004

    • Author(s)
      Mikio Kato, Kichi-Suke Saito, Takayuki Tamura
    • Journal Title

      Mathematical Inequalities and Applications 7

      Pages: 429-437

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Some geometric conditions related to fixed point property for ψ-direct sums of Banach spaces X 【symmetry】_ψ Y2004

    • Author(s)
      Mikio Kato, Takayuki Tamura
    • Journal Title

      Proceedings of the International Symposium on Banach and Function Spaces

      Pages: 243-252

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Strong random Clarkson inequality and its extension2004

    • Author(s)
      Yasutaka Yamada, Mikio Kato, Yasuji Takahashi
    • Journal Title

      RIMS Kokyuroku (Kyoto University) 1396

      Pages: 143-147

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Uniform non-squareness of Ψ-direct sums of Banach spaces X【symmetry】_ΨY2004

    • Author(s)
      M.Kato, K.-S.Saito, T.Tamura
    • Journal Title

      Mathematical Inequalities and Applications 7

      Pages: 429-437

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Some geometric conditions related to fixed point property for Ψ-direct sums of Banach spaces X【symmetry】_ΨY2004

    • Author(s)
      M.Kato, T.Tamura
    • Journal Title

      Proceedings of the International Symposium on Banach and Function Spaces

      Pages: 243-252

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On Hanner-type inequalities for Banach spaces2004

    • Author(s)
      Y.Takahashi, Y.Yamada, M.Kato
    • Journal Title

      Proceedings of the International Symposium on Banach and Function Spaces

      Pages: 359-365

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Some remarks on the set of common fixed points of one-parameter semi-groups of nonexpansive mappings in Banach spaces with the Opial property2004

    • Author(s)
      T.Suzuki
    • Journal Title

      Nonlinear Analysis, Theory, Methods & Applications 58

      Pages: 441-458

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Strong random Clarkson inequality and its extension2004

    • Author(s)
      Y.Yamada, M.Kato, Y.Takahashi
    • Journal Title

      京都大学数理解析研究所講究録 1396

      Pages: 143-147

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Ψ直和バナッハ空間における非拡大写像に対する不動点性について2004

    • Author(s)
      田村高幸, 加藤幹雄
    • Journal Title

      京都大学数理解析研究所講究録 1399

      Pages: 137-143

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On Hanner type inequalities with a weight for Banach spaces

    • Author(s)
      Yasutake Yamada, Yasuji Takahashi, Mikio Kato
    • Journal Title

      Journal of Mathematical Analysis and Applications (掲載決定)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Hanner type inequalities and duality

    • Author(s)
      Mikio Kato, Yasuji Takahashi, Yasutaka Yamada
    • Journal Title

      京都大学数理解析研究所講究録 (掲載予定)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Banach-Mazur distance and super-reflexive Banach spaces

    • Author(s)
      Yasuji Takahashi, Mikio Kato
    • Journal Title

      北海道大学数学講究録 (掲載予定)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On Hanner type inequalities with a weight for Banach spaces

    • Author(s)
      Yasutaka Yamada, Yasuji Takahashi, Mikio Kato
    • Journal Title

      Journal of Mathematical Analysis and Applications (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Hanner type inequalities and duality

    • Author(s)
      Mikio Kato, Yasuji Takahashi, Yasutaka Yamada
    • Journal Title

      RIMS Kokyuroku (Kyoto University) (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Banach-Mazur distance and super-reflexive Banach spaces

    • Author(s)
      Yasuji Takahashi, Mikio Kato
    • Journal Title

      Hokkaido University Technical Report Series in Mathematics (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Hanner type inequalities with a weight for Banach spaces

    • Author(s)
      Y.Yamada, Y.Takahashi, M.Kato
    • Journal Title

      Journal of Mathematical Analysis and Applications 掲載予定

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Hanner type inequalities and duality

    • Author(s)
      M.Kato, Y.Takahashi, Y.Yamada
    • Journal Title

      京都大学数理解析研究所講究録 掲載予定

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Banach-Mazur distance and super-reflexive Banach spaces

    • Author(s)
      Y.Takahashi, M.Kato
    • Journal Title

      北海道大学数学講究録 掲載予定

    • Related Report
      2005 Annual Research Report
  • [Book] Banach and Function Spaces (Proceedings of the International Symposium on Banach and Function Spaces 2003,Kitakyushu,Japan Oct.2-4,2003)2004

    • Author(s)
      Mikio Kato, Lech Maligranda(editors)
    • Total Pages
      393
    • Publisher
      Yokohama Publishers
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Book] Banach and Function Spaces (Proceedings of the International Symposium on Banach and Function Spaces 2003, Kitakyushu, Japan, Oct. 2-4, 2003)2004

    • Author(s)
      Mikio Kato, Lech Maligranda (editors)
    • Total Pages
      393
    • Publisher
      Yokohama Publishers
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary

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Published: 2004-04-01   Modified: 2016-04-21  

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