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On S$ _1$-strictly singular operators


Authors: Edward Odell and Ricardo V. Teixeira
Journal: Proc. Amer. Math. Soc. 143 (2015), 4745-4757
MSC (2010): Primary 46B03, 46B25, 46B28, 46B45, 47L20
DOI: https://doi.org/10.1090/proc/12452
Published electronically: July 30, 2015
MathSciNet review: 3391033
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Abstract: Let $ X$ be a Banach space and denote by $ SS_1(X)$ the set of all S$ _1$-strictly singular operators $ T: X \longrightarrow X$. We prove that there is a Banach space $ X$ such that $ SS_1(X)$ is not an ideal. More specifically, we construct spaces $ X$ and operators $ T_1, T_2 \in SS_1(X)$ such that $ T_1 + T_2 \notin SS_1(X)$. We show one example where the space $ X$ is reflexive and another where it is $ c_0$-saturated.


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Additional Information

Ricardo V. Teixeira
Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas
Address at time of publication: Department of Mathematics, University of Houston-Victoria, Victoria, Texas
Email: teixeirar@uhv.edu

DOI: https://doi.org/10.1090/proc/12452
Received by editor(s): September 25, 2013
Received by editor(s) in revised form: January 15, 2014
Published electronically: July 30, 2015
Additional Notes: Edward Odell (1947–2013). The author passed away during the production of this paper.
This work is the main result of the second author’s Ph.D. thesis which was written under the supervision of the first author at the University of Texas at Austin
Dedicated: In Memoriam of Professor Ted Odell
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2015 American Mathematical Society

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