Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T16:09:28.140Z Has data issue: false hasContentIssue false
  • English
  • Français

Almost Disjointness Preservers

Published online by Cambridge University Press:  20 November 2018

Timur Oikhberg  and
Pedro Tradacete
Timur Oikhberg
Affiliation:
Dept. of Mathematics, University of Illinois, Urbana IL 61801, USA e-mail: [email protected]
Pedro Tradacete
Affiliation:
Mathematics Department, Universidad Carlos III de Madrid, E-28911 Leganés, Madrid, Spain e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the stability of disjointness preservers on Banach lattices. In many cases, we prove that an “almost disjointness preserving” operator is well approximable by a disjointness preserving one. However, this approximation is not always possible, as our examples show.

Keywords

47B3846B42Banach latticedisjointness preserving
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 69 , Issue 3 , 01 June 2017 , pp. 650 - 686
Copyright
Copyright © Canadian Mathematical Society 2017

References

[1] Abramovich, Y., Operators preserving disjointness on rearrangement invariant spaces. Pacific J. Math. 148(1991) 201–206. http://dx.doi.org/10.2140/pjm.1991.148.201 Google Scholar
[2] Abramovich, Y. and Aliprantis, C., An invitation to operator theory. Graduate Studies in Mathematics 50. American Mathematical Society, Providence, RI, 2002.http://dx.doi.Org/10.109O/gsm/O5O Google Scholar
[3] Aliprantis, C. and Burkinshaw, O., Positive operators. Springer, Dordrecht, 2006.Google Scholar
[4] Alspach, D., Small into isomorphisms on Lp spaces. Illinois J. Math. 27(1983), 300–314.Google Scholar
[5] Araujo, J., Linear biseparating maps between spaces of vector-valued differentiable functions and automatic continuity. Adv. Math. 187(2004), 488–520.http://dx.doi.Org/10.1016/j.aim.2003.09.007 Google Scholar
[6] Araujo, J. and Font, J., Stability of weighted composition operators between spaces of continuous functions. J. London Math. Soc. 79(2009), 363–376.http://dx.doi.Org/10.1112/jlms/jdnO79 Google Scholar
[7] Araujo, J., Stability of weighted point evaluation functionals. Proc. Amer. Math. Soc. 138(2010), 3163–3170.http://dx.doi.Org/10.1090/S0002-9939-10-10214-7 Google Scholar
[8] Araujo, J., On the stability index for weighted composition operators. J. Approx. Theory 162(2010), 2136–2148.http://dx.doi.Org/10.1016/j.jat.2O10.06.006 Google Scholar
[9] Arendt, W., Spectral properties ofLamperti operators. Indiana Univ. Math. J. 32(1983), 199–215. http://dx.doi.Org/10.1512/iumj.1 983.32.32018 Google Scholar
[10] Bennett, C. and Sharpley, R. Interpolation of operators. Pure and Applied Mathematics 129. Academic Press, Boston, MA, 1988.Google Scholar
[11] Carothers, N. L., A short course on Banach space theory. London Mathematical Society Student Texts 64. Cambridge University Press, Cambridge, 2005.Google Scholar
[12] Diestel, J., Geometry of Banach spaces-selected topics. Lecture Notes in Mathematics 485. Springer-Verlag, Berlin-New York, 1975 Google Scholar
[13] Dolinar, G., Stability of disjointness preserving mappings. Proc. Amer. Math. Soc. 130(2002), 129–138.http://dx.doi.Org/10.1090/S0002-9939-01-06023-3 Google Scholar
[14] L. E.Dor, , On projections in L1. Ann. of Math. (2) 102(1975), no. 3, 463–474.http://dx.doi.Org/10.2307/1971039 Google Scholar
[15] Godefroy, G., Renormings of Banach spaces. In: Handbook of the geometry of Banach spaces, Vol. I. North-Holland, Amsterdam, 2001, pp. 781–835. http://dx.doi.Org/10.1016/S1 874-5849(01)80020-6 Google Scholar
[16] Huijsmans, C.B., Disjointness preserving operators on Banach lattices. In: Operator theory in function spaces and Banach lattices. Oper. Theory Adv. Appl. 75. Birkhäuser, Basel, 1995, pp. 173–189.Google Scholar
[17] Kantrowitz, R. and Neumann, M., Disjointness preserving and local operators on algebras of differentiable functions. Glasg. Math. J. 43(2001), 295–309.http://dx.doi.Org/10.1017/S001 7089501020134 Google Scholar
[18] Kantrowitz, R., Approximation by weighted composition operators on C(X) Math. Proc. R. Ir. Acad. 108(2008), 119–135.http://dx.doi.Org/10.3318/PRIA.2008.108.2.119 Google Scholar
[19] Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces. I. Ergebnisse der Mathematik und ihrer Grenzgebiete 92. Springer-Verlag, Berlin, 1977.Google Scholar
[20] Lindenstrauss, J. Classical Banach spaces. II. Ergebnisse der Mathematik und ihrer Grenzgebiete 97. Springer-Verlag, Berlin, 1979.Google Scholar
[21] 5, Biseparating maps on generalized Lipschitz spaces. Studia Math. 196(2010), 23–40.http://dx.doi.org/10.4064/sm196-1-3 Google Scholar
[22] Lin, Y.-F. and Wong, N.-C., The structure of compact disjointness preserving operators on continuous functions. Math. Nachr. 282(2009), 1009–1021.http://dx.doi.Org/10.1OO2/mana.2OO610786 Google Scholar
[23] Meyer-Nieberg, P., Banach lattices. Springer-Verlag, Berlin, 1991.http://dx.doi.org/10.1007/978-3-642-76724-1 Google Scholar
[24] Milman, V. and Schechtman, G., Asymptotic theory of finite-dimensional normed spaces. Lecture Notes in Mathematics 1200. Springer-Verlag, Berlin, 1986.Google Scholar
[25] Oikhberg, T., Peralta, A., and Puglisi, D., Automatic continuity of orthogonality or disjointness preserving bijections. Rev. Mat. Complut. 26(2013), 57–88.http://dx.doi.Org/10.1007/s13163-011-0089-0 Google Scholar
[26] Schaefer, H. H., Banach lattices and positive operators. Die Grundlehren der mathematischen Wissenschaften 215. Springer-Verlag, Berlin (1974).Google Scholar
[27] Schep, A., Krivine's theorem and the indices of a Banach lattice. In: Positive operators and semigroups on Banach lattice (Curacao, 1990). Acta Appl. Math. 27(1992), 111–121. http://dx.doi.org/10.1007/BF00046642 Google Scholar
[28] Schwarz, H.-U., Banach lattices and operators. Teubner-Texte zur Mathematik 71. Teubner, Leipzig, 1984.Google Scholar
[29] Wickstead, A., Regular operators between Banach lattices. In: Positivity. Trends Math. Birkhäuser, Basel, 2007, pp. 255–279.http://dx.doi.Org/10.1007/978-3-7643-8478-4_9 Google Scholar