Non-Asplund Banach spaces and operators

TL;DR

This paper investigates the implications of the adjoint of an operator between Banach spaces having a non-separable range, leading to new insights in basic sequences, universal operators, and dual space separability.

Contribution

It provides new operator-theoretic characterizations of non-separable duals and applications to basic sequences and universal operators in Banach space theory.

Findings

Characterization of Banach spaces with non-separable duals.

Existence of universal operators for classes of operators.

Applications to the structure of basic sequences.

Abstract

Let $W$ and $Z$ be Banach spaces such that $Z$ is separable and let $R:W\longrightarrow Z$ be a (continuous, linear) operator. We study consequences of the adjoint operator $R^\ast$ having non-separable range. From our main technical result we obtain applications to the theory of basic sequences and the existence of universal operators for various classes of operators between Banach spaces. We also obtain an operator-theoretic characterisation of separable Banach spaces with non-separable dual.