Browsing by Subject "Banach space"
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Item On S₁-strictly singular operators(2010-05) Teixeira, Ricardo Verotti O.; Odell, E. (Edward); Todd, Arbogast; Treisman, Philip U.; Vishik, Mikhail; Schlumprecht, ThomasLet X be a Banach space and denote by SS₁(X) the set of all S₁-strictly singular operators from X to X. We prove that there is a Banach space X such that SS₁(X) is not a closed ideal. More specifically, we construct space X and operators T₁ and T₂ in SS₁(X) such that T₁+T₂ is not in SS₁(X). We show one example where the space X is reflexive and other where it is c₀-saturated. We also develop some results about S_alpha-strictly singular operators for alpha less than omega_1.Item Szlenk Index, Upper Estimates, and Embedding in Banach Spaces(2014-08-07) Causey, Ryan MichaelWe investigate the relationship between two notions, one which refers to a coordinate system and one which does not, of asymptotic domination by subsequences of a fixed basis. We use this relationship to prove the existence of a universal space with a coordinate system satisfying this asymptotic domination condition. Last, we relate this asymptotic domination notion to the Szlenk index and prove a result concerning the existence of a universal space for classes determined by Szlenk index. Each of these results also has a corresponding result for reflexive spaces.