Weak-star quasi norm attaining operators

TL;DR

This paper introduces the concept of weak-star quasi norm attaining operators between Banach spaces, proves their density in the space of bounded operators, and explores their properties and differences from other norm attaining operators.

Contribution

It defines weak-star quasi norm attaining operators, proves their density in operator spaces, and analyzes their properties compared to other norm attaining operators.

Findings

Weak-star quasi norm attaining operators are dense in the space of bounded linear operators.

These operators can be approximated with additional desirable properties.

They exhibit both similarities and differences with other types of norm attaining operators.

Abstract

For Banach spaces $X$ and $Y$, a bounded linear operator $T\colon X \longrightarrow Y^*$ is said to weak-star quasi attain its norm if the $\sigma(Y^*,Y)$-closure of the image by $T$ of the unit ball of $X$ intersects the sphere of radius $\|T\|$ centred at the origin in $Y^*$. This notion is inspired by the quasi-norm attainment of operators introduced and studied in \cite{CCJM}. As a main result, we prove that the set of weak-star quasi norm attaining operators is dense in the space of bounded linear operators regardless of the choice of the Banach spaces, furthermore, that the approximating operator can be chosen with additional properties. This allows us to distinguish the properties of weak-star quasi norm attaining operators from those of quasi norm attaining operators. It is also shown that, under certain conditions, weak-star quasi norm attaining operators share numbers of equivalent properties with other types of norm attaining operators, but that there are also a number of situations in which they behave differently from the others.