The generalized Day norm. Part II. Applications

Monika Budzyńska; Aleksandra Grzesik; Mariola Kot

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2017)

  • Volume: 71, Issue: 2
  • ISSN: 0365-1029

Abstract

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In this paper we prove that for each 1 < p , p ˜ < ∞ , the Banach space ( l p ˜ , · p ˜ ) can be equivalently renormed in such a way that  the Banach space ( l p ˜ , · L , α , β , p , p ˜ ) is LUR and has a diametrically complete set with empty interior. This result extends the Maluta theorem about existence of such a set in l 2 with the Day norm. We also show that the Banach space ( l p ˜ , · L , α , β , p , p ˜ ) has the weak fixed point property for nonexpansive mappings.

How to cite

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Monika Budzyńska, Aleksandra Grzesik, and Mariola Kot. "The generalized Day norm. Part II. Applications." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 71.2 (2017): null. <http://eudml.org/doc/289770>.

@article{MonikaBudzyńska2017,
abstract = {In this paper we prove that for each $1< p, \tilde\{p\} < \infty $, the Banach space $(l^\{\tilde\{p\}\}, \left\Vert \cdot \right\Vert _\{\tilde\{p\}\})$ can be equivalently renormed in such a way that  the Banach space $(l^\{\tilde\{p\}\},\left\Vert \cdot \right\Vert _\{L,\alpha ,\beta ,p,\tilde\{p\}\})$ is LUR and has a diametrically complete set with empty interior. This result extends the Maluta theorem about existence of such a set in $l^2$ with the Day norm. We also show that the Banach space $(l^\{\tilde\{p\}\},\left\Vert \cdot \right\Vert _\{L,\alpha ,\beta ,p,\tilde\{p\}\})$ has the weak fixed point property for nonexpansive mappings.},
author = {Monika Budzyńska, Aleksandra Grzesik, Mariola Kot},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {Diametrically complete set; Day norm, fixed point; Kadec-Klee property; LUR space; nonexpansive mapping; non-strict Opial property; 1-unconditional Schauder bases},
language = {eng},
number = {2},
pages = {null},
title = {The generalized Day norm. Part II. Applications},
url = {http://eudml.org/doc/289770},
volume = {71},
year = {2017},
}

TY - JOUR
AU - Monika Budzyńska
AU - Aleksandra Grzesik
AU - Mariola Kot
TI - The generalized Day norm. Part II. Applications
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2017
VL - 71
IS - 2
SP - null
AB - In this paper we prove that for each $1< p, \tilde{p} < \infty $, the Banach space $(l^{\tilde{p}}, \left\Vert \cdot \right\Vert _{\tilde{p}})$ can be equivalently renormed in such a way that  the Banach space $(l^{\tilde{p}},\left\Vert \cdot \right\Vert _{L,\alpha ,\beta ,p,\tilde{p}})$ is LUR and has a diametrically complete set with empty interior. This result extends the Maluta theorem about existence of such a set in $l^2$ with the Day norm. We also show that the Banach space $(l^{\tilde{p}},\left\Vert \cdot \right\Vert _{L,\alpha ,\beta ,p,\tilde{p}})$ has the weak fixed point property for nonexpansive mappings.
LA - eng
KW - Diametrically complete set; Day norm, fixed point; Kadec-Klee property; LUR space; nonexpansive mapping; non-strict Opial property; 1-unconditional Schauder bases
UR - http://eudml.org/doc/289770
ER -

References

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  12. Maluta, E., Papini, P. L., Diametrically complete sets and normal structure, J. Math. Anal. Appl. 424 (2015), 1335-1347. 
  13. Mariadoss, S. A., Soardi, P. M., A remark on asymptotic normal structure in Banach spaces, Rend. Sem. Mat. Univ. Politec. Torino 44 (1986), 393-395. 
  14. Moreno, J. P., Papini, P. L., Phelps, R. R., Diametrically maximal and constant width sets in Banach spaces, Canad. J. Math. 58 (2006), 820-842. 
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  17. Smith, M. A., Turett, B., A reflexive LUR Banach space that lacks normal structure, Canad. Math. Bull. 28 (1985), 492-494. 

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