Subsymmetric weak$^*$ Schauder bases and factorization of the identity

TL;DR

This paper investigates the structure of Banach spaces with subsymmetric weak$^*$ Schauder bases, demonstrating a factorization property of operators and establishing the primarity of certain direct sum spaces.

Contribution

It proves that operators on such Banach spaces either preserve a complemented subspace isomorphic to the space or its complement, and shows that $oldsymbol{ extit{ ext{ell}}^p( extit{X}^*)}$ spaces are primary.

Findings

Operators split the space into complemented subspaces isomorphic to the original.

$oldsymbol{ extit{ ext{ell}}^p( extit{X}^*)}$ spaces are primary.

Subsymmetry and weak$^*$ Schauder basis conditions lead to factorization results.

Abstract

Let $X^*$ denote a Banach space with a subsymmetric weak$^*$ Schauder basis satisfying condition~\eqref{eq:condition-c}. We show that for any operator $T : X^*\to X^*$, either $T(X^*)$ or $(I-T)(X^*)$ contains a subspace that is isomorphic to $X^*$ and complemented in $X^*$. Moreover, we prove that $\ell^p(X^*)$, $1\leq p \leq \infty$ is primary.