Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-13T04:48:15.796Z Has data issue: false hasContentIssue false
  • English
  • Français

Injective Tauberian Operators on L1 and Operators with Dense Range on ℓ∞

Published online by Cambridge University Press:  20 November 2018

William Johnson ,
Amir Bahman Nasseri ,
Gideon Schechtman  and
Tomasz Tkocz
William Johnson
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843, USA. e-mail: johnson@math.tamu.edu
Amir Bahman Nasseri
Affiliation:
Fakultät für Mathematik, Technische Universität Dortmund, D-44221 Dortmund, Germany. e-mail: amirbahman@hotmail.de
Gideon Schechtman
Affiliation:
Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel. e-mail: gideon@weizmann.ac.il
Tomasz Tkocz
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK. e-mail: t.tkocz@warwick.ac.uk
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.

There exist injective Tauberian operators on ${{L}_{1}}\left( 0,\,1 \right)$ that have dense, nonclosed range. This gives injective nonsurjective operators on ${{\ell }_{\infty }}$ that have dense range. Consequently, there are two quasi-complementary noncomplementary subspaces of ${{\ell }_{\infty }}$ that are isometric to ${{\ell }_{\infty }}$.

Keywords

46E3046B0847A53L1Tauberian operatorℓ∞
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 2 , 01 June 2015 , pp. 276 - 280
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Albiac, F. and Kalton, N. J., Topics in Banach space theory. Graduate Texts in Mathematics, 233, Springer, New York, 2006.Google Scholar
[2] Argyros, S. A., Arvanitakis, A. D., and Tolias, A. G., Saturated extensions, the attractors method and hereditarily James tree spaces. In: Methods in Banach space theory, London Math. Soc. Lecture Note Ser., 337, Cambridge University Press, Cambridge, 2006, pp. 1–90.Google Scholar
[3] Berinde, R., Gilbert, A. C., Indyk, P., Karloff, H., and Strauss, M. J., Combining geometry and combinatorics: a unified approach to sparse signal recovery. In: 46th Annual Allerton Conference on Communication, Control, and Computing, 2008, pp. 798–805.Google Scholar
[4] Diestel, J., Jarchow, H., and Tonge, A., Absolutely summing operators. Cambridge Studies in Advanced Mathematics, 43, Cambridge University Press, Cambridge, 1995.Google Scholar
[5] Gonzalez, M. and Martinez-Abejon, A., Tauberian operators. Operator Theory: Advances and Applications, 194, Birkhâuser Verlag, Basel, 2010.Google Scholar
[6] Gonzalez, M. and Onieva, V. M., On the instability of non-semi-Fredholm operators under compact perturbations. J. Math. Anal. Appl. 114 (1986), no. 2, 450–457. http://dx.doi.Org/10.1016/0022-247X(86)90098-3 Google Scholar
[7] Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces. II. Function spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, 97, Springer-Verlag, Berlin-New York, 1979.Google Scholar
[8] Mackey, G. W., Note on a theorem of Murray. Bull. Amer. Math. Soc. 52 (1946), 322–325. http://dx.doi.org/10.1090/S0002–9904-1946-08570-5 Google Scholar
[9] Nasseri, A. B., The spectrum of operators on C(K) with the Grothendieck property and characterization of -Class Operators which are adjoints. arxiv:1406.3815Google Scholar
[10] Nasseri, A. B., Surjectivity of operators on t00. http://mathoverflow.net/questions/101253Google Scholar
[11] Royden, H. L., Real analysis. Third ed., Macmillan Publishing Company, New York, 1988.Google Scholar