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# Injective Tauberian Operators on $L_1$ and Operators with Dense Range on $\ell_\infty$

Published:2014-11-03
Printed: Jun 2015
• William Johnson,
Department of Mathematics, Texas A&M University, College Station, TX 77843, USA
• Amir Bahman Nasseri,
Fakultät für Mathematik, Technische Universität Dortmund, D-44221 Dortmund, Germany
• Gideon Schechtman,
Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel
• Tomasz Tkocz,
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
 Format: LaTeX MathJax PDF

## Abstract

There exist injective Tauberian operators on $L_1(0,1)$ 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: $L_1$, Tauberian operator, $\ell_\infty$
 MSC Classifications: 46E30 - Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Kothe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46B08 - Ultraproduct techniques in Banach space theory [See also 46M07] 47A53 - (Semi-) Fredholm operators; index theories [See also 58B15, 58J20]

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