The Szlenk power type and tensor products of Banach spaces

TL;DR

This paper establishes a formula for the Szlenk power type of tensor products of Banach spaces with small Szlenk index, and shows that these properties are separably determined, extending previous results on direct sums.

Contribution

It provides a new formula for Szlenk power type of tensor products and demonstrates separable determination of Szlenk properties, advancing understanding of Banach space geometry.

Findings

Derived a formula for Szlenk power type of tensor products.

Showed Szlenk power type and summability are separably determined.

Extended results on direct sums of Banach spaces.

Abstract

We prove a formula for the Szlenk power type of the injective tensor product of Banach spaces with Szlenk index at most $\omega$. We also show that the Szlenk power type as well as summability of the Szlenk index are separably determined, and we extend some of our recent results concerning direct sums.