Measures of weak non-compactness in spaces of nuclear operators

TL;DR

This paper demonstrates that in spaces of nuclear operators between certain sequence spaces, two common measures of weak non-compactness are equivalent and provides explicit formulas, extending results to preduals of atomic von Neumann algebras.

Contribution

It establishes the equivalence of two measures of weak non-compactness in nuclear operator spaces and derives explicit formulas, also applying to preduals of atomic von Neumann algebras.

Findings

The two measures of weak non-compactness coincide in these spaces.

Explicit formulas for the measures are provided.

Results extend to preduals of atomic von Neumann algebras.

Abstract

We show that in the space of nuclear operators from $\ell^q(\Lambda)$ to $\ell^p(J)$ the two natural ways of measuring weak non-compactness coincide. We also provide explicit formulas for these measures. As a consequence the same is proved for preduals of atomic von Neumann algebras.