We study universality problems in Banach space theory. We show that if A is an analytic class, in the Effros-Borel structure of subspaces of C([0, 1]), of non-universal separable Banach spaces, then there exists a non-universal separable Banach space Y , with a Schauder basis, that contains isomorphs of each member of A with the bounded approximation property. The proof is based on the amalgamation technique of a class C of separable Banach spaces, introduced in the paper. We show, among others, that there exists a separable Banach space R not containing L 1 (0, 1) such that the indices β and r ND are unbounded on the set of Baire-1 elements of the ball of the double dual R * * of R. This answers two questions of H.P. Rosenthal.We also introduce the concept of a strongly bounded class of separable Banach spaces. A class C of separable Banach spaces is strongly bounded if for every analytic subset A of C there exists Y ∈ C that contains all members of A up to isomorphism. We show that several natural classes of separable Banach spaces are strongly bounded, among them the class of non-universal spaces with a Schauder basis, the class of reflexive spaces with a Schauder basis, the class of spaces with a shrinking Schauder basis and the class of spaces with Schauder basis not containing a minimal Banach space X.
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221 222 G. ANDROULAKIS ET AL. Isr. J. Math. ABSTRACT V. D. Milman proved in [20] that the product of two strictly singular operators on Lp[0, 1] (1 p < ∞) or on C[0, 1] is compact. In this note we utilize Schreier families S ξ in order to define the class of S ξ -strictly singular operators, and then we refine the technique of Milman to show that certain products of operators from this class are compact, under the assumption that the underlying Banach space has finitely many equivalence classes of Schreier-spreading sequences. Finally we define the class of S ξ -hereditarily indecomposable Banach spaces and we examine the operators on them.
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