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Weak-star quasi norm attaining operators

Geunsu Choi, Mingu Jung, Sun Kwang Kim, Miguel Martin

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 $σ(Y^*,Y)$-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.

Weak-star quasi norm attaining operators

Abstract

For Banach spaces and , a bounded linear operator is said to weak-star quasi attain its norm if the -closure of the image by of the unit ball of intersects the sphere of radius centred at the origin in . 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.
Paper Structure (4 sections, 14 theorems, 36 equations)

This paper contains 4 sections, 14 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Denseness results
  3. Some remarks on weak-star quasi norm attaining operators
  4. Relations between the sets defined by different notions of the (quasi) norm attainment

Key Result

Theorem 2.1

For arbitary Banach spaces $X$ and $Y$, the set $w^*\!\operatorname{QNA} (X, Y^*)$ is dense in $\mathcal{L}(X, Y^*)$. Indeed, given $T\in \mathcal{L} (X, Y^*)$ and $\varepsilon>0$, there is a rank-one perturbation $S\in w^*\!\operatorname{QNA} (X, Y^*)$ of $T$ such that $\|T-S\|<\varepsilon$.

Theorems & Definitions (44)

  • Definition 1.1
  • Theorem 2.1
  • Lemma 2.2: PZ
  • proof : Proof of Theorem \ref{['wQNA_dense']}
  • Proposition 2.3
  • proof
  • proof : Alternative proof of the denseness of $w^*\!\operatorname{QNA} (X,Y^*)$
  • Theorem 2.4
  • proof : Proof of Theorem \ref{['theorem:wQNA-dense-2']}
  • Example 2.5
  • ...and 34 more