Abstract

Previous constructions of non-type (D) maximal monotone operators were based on the non-type (D) operators introduced by Gossez, and the construction of such operators or the proof that they were non-type (D) were not straightforward. The aim of this paper is to present a very simple non-type (D) linear isometry.

Author information

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Orestes Bueno

Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, RJ 22460-320, Brazil

obueno@impa.br

Benar Fux Svaiter

Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, RJ 22460-320, Brazil

benar@impa.br

Keywords

  • Monotone operators
  • non-reflexive Banach spaces
  • type (D)

Mathematics Subject Classification

47H05, 49J52, 47N10

References

  1. H. H. Bauschke, J. M. Borwein, X. Wang, L. Yao: Construction of pathological maximally monotone operators on non-reflexive Banach spaces, Set-Valued Var. Analysis 20(3) (2012) 387–415.https://doi.org/10.1007/s11228-012-0209-0Google Scholar
  2. J. M. Borwein: Fifty years of maximal monotonicity, Optim. Lett. 4(4) (2010) 473–490.https://doi.org/10.1007/s11590-010-0178-xGoogle Scholar
  3. H. Brézis: Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Mathematics Studies 5, Notas de Matemática 50, North-Holland, Amsterdam (1973).Google Scholar
  4. F. E. Browder: Nonlinear maximal monotone operators in Banach space, Math. Ann. 175 (1968) 89–113.https://doi.org/10.1007/BF01418765Google Scholar
  5. O. Bueno, B. F. Svaiter: A non-type (D) operator in c0, Math. Program., to appear (Mar. 2011).Google Scholar
  6. R. S. Burachik, B. F. Svaiter: Maximal monotone operators, convex functions and a special family of enlargements, Set-Valued Anal. 10(4) (2002) 297–316.https://doi.org/10.1023/A:1020639314056Google Scholar
  7. J.-P. Gossez: Opérateurs monotones non linéaires dans les espaces de Banach non réflexifs, J. Math. Anal. Appl. 34 (1971) 371–395.https://doi.org/10.1016/0022-247X(71)90119-3Google Scholar
  8. J.-P. Gossez: On a convexity property of the range of a maximal monotone operator, Proc. Amer. Math. Soc. 55(2) (1976) 359–360.https://doi.org/10.1090/S0002-9939-1976-0397485-2Google Scholar
  9. J.-P. Gossez: On the extensions to the bidual of a maximal monotone operator, Proc. Amer. Math. Soc. 62(1) (1977) 67–71.https://doi.org/10.1090/S0002-9939-1977-0428121-5Google Scholar
  10. R. C. James: A non-reflexive Banach space isometric with its second conjugate space, Proc. Natl. Acad. Sci. USA 37 (1951) 174–177.https://doi.org/10.1073/pnas.37.3.174Google Scholar
  11. T. Kato: Demicontinuity, hemicontinuity and monotonicity, Bull. Amer. Math. Soc. 70 (1964) 548–550.https://doi.org/10.1090/S0002-9904-1964-11194-0Google Scholar
  12. G. J. Minty: Monotone networks, Proc. R. Soc. Lond., Ser. A 257 (1960) 194–212.https://doi.org/10.1098/rspa.1960.0144Google Scholar
  13. R. T. Rockafellar: Local boundedness of nonlinear, monotone operators, Michigan Math. J. 16 (1969) 397–407.https://doi.org/10.1307/mmj/1029000324Google Scholar
  14. B. F. Svaiter, M. Marques Alves: Bronsted-Rockafellar property and maximality of monotone operators representable by convex functions in non-reflexive Banach spaces, J. Convex Analysis 15 (2008) 693–706.https://doi.org/10.68381/jca15047Google Scholar
  15. E. Zarantonello: Solving functional equations by contractive averaging, MRC Technical Summary Report 160, Mathematics Research Center, United States Army, Univ. of Wisconsin (1960).Google Scholar

Suggested citation

O. Bueno, B. F. Svaiter. “A Non-Type (D) Linear Isometry.” Journal of Convex Analysis 20 (2013), No. 1, 301–305. https://doi.org/10.68381/jca20019

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Published by Heldermann Verlag, 2013. Rights now held by Banach Press LLC.