Asymptotic smoothness in Banach spaces, three-space properties and applications

<p>We study four asymptotic smoothness properties of Banach spaces, denoted <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="sans-serif upper T Subscript p Baseline comma sans-serif upper A Subscript p Baseline comma sans-serif upper N Subscript p Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">T</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">A</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">N</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {T}_p,\mathsf {A}_p, \mathsf {N}_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="sans-serif upper P Subscript p"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">P</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathsf {P}_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We complete their description by proving the missing renorming characterization for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="sans-serif upper A Subscript p"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">A</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathsf {A}_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that asymptotic uniform flattenability (property <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper T Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">T</mml:mi> </mml:mrow> <mml:mi mathvariant="normal">∞<! -- ∞ --></mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathsf {T}_\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) and summable Szlenk index (property <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper A Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">A</mml:mi> </mml:mrow> <mml:mi mathvariant="normal">∞<! -- ∞ --></mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathsf {A}_\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) are three-space properties. Combined with the positive results of the first-named author, Draga, and Kochanek, and with the counterexamples we provide, this completely solves the three-space problem for this family of properties. We also derive from our characterizations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="sans-serif upper A Subscript p"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">A</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathsf {A}_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="sans-serif upper N Subscript p"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">N</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathsf {N}_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in terms of equivalent renormings, new coarse Lipschitz rigidity results for these classes. </p>