Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the failure of close-to-normal
structure type conditions
and pathological Kannan maps

Author: Michael A. Smyth
Journal: Proc. Amer. Math. Soc. 124 (1996), 3063-3069
MSC (1991): Primary 46B20, 47H09
MathSciNet review: 1342047
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the failure of close-to-normal structure type conditions and show that a Banach space can be renormed to fail close-to-weak normal structure exactly when it contains a norm inseparable weakly compact subset. Included is an example of a particularly pathological fixed point free Kannan map.

References [Enhancements On Off] (What's this?)

  • 1. D. van Dulst, Equivalent norms and the fixed point property for nonexpansive mappings, J. London Math. Soc. 25 (1982), 139--144. MR 83e:47040
  • 2. K. Goebel and W. A. Kirk, Topics in metric fixed point theory, Cambridge University Press, 1990. MR 92c:47070
  • 3. S. Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Mat. 313 (1980), 72--104. MR 82b:46013
  • 4. A. Lau and P. Mah, Quasi-normal structures for certain spaces of operators on a Hilbert space, Pacific J. Math. 121 (1986), 109--118. MR 87f:47065
  • 5. C. Lennard, $\mathcal {C}_1$ is uniformly Kadec-Klee, Proc. Amer. Math. Soc. 109 (1990), 71--77. MR 90h:46029
  • 6. D. Roux and C. Zanco, Kannan maps in normed spaces, Rend. Sci. Fis. Mat. Nat. 65 (1978), 252--258. MR 81i:47062
  • 7. B. Sims, Ultra-techniques in Banach space theory, Queen's Papers in Pure and Applied Mathematics, Vol. 60, 1982. MR 86h:46032
  • 8. I. Singer, Bases in Banach spaces. II, Springer-Verlag, Berlin, Heidelberg, and New York, 1981. MR 82k:46024
  • 9. M. Smyth, Remarks on the weak star fixed point property in the dual of $C(\Omega )$, J. Math. Anal. Appl. 195 (1995), 294--306. CMP 96:01
  • 10. K.-K. Tan, A note on asymptotic normal structure and close-to-normal structure, Canad. Math. Bull. 25 (1982), 339--343. MR 83i:46022
  • 11. C. Wong, Close-to-normal structure and its applications, J. Functional Anal. 16 (1974), 353--358. MR 50:950
  • 12. ------, On Kannan maps, Proc. Amer. Math. Soc. 47 (1975), 105--111. MR 50:10929

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 46B20, 47H09

Retrieve articles in all journals with MSC (1991): 46B20, 47H09

Additional Information

Michael A. Smyth
Affiliation: Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand
Address at time of publication: 1 Frost Rd., Mt. Roskill, Auckland, New Zealand

Received by editor(s): February 23, 1995
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society