A generalized ACK structure and the denseness of norm attaining operators

TL;DR

This paper introduces a generalized ACK structure called quasi-ACK in Banach spaces, extending properties related to norm attaining operators, and explores its stability, examples, and implications for vector-valued holomorphic functions.

Contribution

It defines the quasi-ACK structure, extends universal properties of norm attaining operators, and provides new examples and stability results for property B$^k$ in Banach spaces.

Findings

Quasi-ACK structure extends known properties of norm attaining operators.

Property B$^k$ is stable under certain tensor products.

New examples of spaces with property B$^k$ are provided.

Abstract

Inspired by the recent work of Cascales et al., we introduce a generalized concept of ACK structure on Banach spaces. Using this property, which we call by the quasi-ACK structure, we are able to extend known universal properties on range spaces concerning the density of norm attaining operators. We provide sufficient conditions for quasi-ACK structure of spaces and results on the stability of quasi-ACK structure. As a consequence, we present new examples satisfying the (Lindenstrauss) property B$^k$, which have not been known previously. We also prove that property B$^k$ is stable under injective tensor products in certain cases. Moreover, ACK structure of some Banach spaces of vector-valued holomorphic functions is also discussed, leading to new examples of universal BPB range spaces for certain operator ideals.