Concerning q-summable Szlenk index

الأصل · EN

For each ordinal ξ and each 1 q<∞, we define the notion of ξ-q-summable Szlenk index. When ξ=0 and q=1, this recovers the usual notion of summable Szlenk index. We define for an arbitrary weak*-compact set a transfinite, asymptotic analogue αξ,ₚ of the martingale type norm of an operator. We prove that this quantity is determined by norming sets and determines ξ-Szlenk power type and ξ-q-summability of Szlenk index. This fact allows us to prove that the behavior of operators under the αξ,ₚ seminorms passes in the strongest way to injective tensor products of Banach spaces. Furthermore, we combine this fact with a result of Schlumprecht to prove that a separable Banach space with good behavior with respect to the αξ,ₚ seminorm can be embedded into a Banach space with a shrinking basis and the same behavior under αξ,ₚ, and in particular it can be embedded into a Banach space with a shrinking basis and the same ξ-Szlenk power type. Finally, we completely elucidate the behavior of the αξ,ₚ seminorms under ℓᵣ direct sums. This allows us to give an alternative proof of a result of Brooker regarding Szlenk indices of ℓₚ and c₀ direct sums of operators.