Range strongly exposing operators between Banach spaces

TL;DR

The paper introduces range-strongly exposing (RSE) operators as a natural middle ground between norm-attaining and quasi-norm-attaining notions, establishing that $\mathop{\mathrm{RSE}}(X,Y)=\mathop{\mathrm{NA}}(X,Y)\cap\mathop{\mathrm{UQNA}}(X,Y)$. It proves several density and approximation results: there is no universal infinite-dimensional target for RSE (universal domain fails); strong RN operators from $L_1(\mu)$ can be approximated by RSE, with RNP playing a key role in density characterizations; weakly compact operators from $C(K)$ can be approximated by RSE, extending Schachermayer; finite-rank and compact operators admit RSE-approximations under suitable structural hypotheses and the framework extends Johnson–Wolfe-type results. The paper further develops density results when the range has quasi-ACK structure and explores when adjoint-related RSE approximations are possible, including cases with closed range and Asplund property. Overall, the work broadens norm-attainment theory by anchoring new RSE-type results to classical settings (L1, C(K), compact operators) and connecting to RNP, ACK structures, and adjoint behavior, highlighting both universal limitations and new dense approximation phenomena.

Abstract

We introduce a new class of bounded linear operators, called range strongly exposing (RSE) operators, which form a natural intermediate class: weaker than Bourgain's absolutely strongly exposing operators, yet stronger than both uniquely quasi norm-attaining and classical norm-attaining operators. Several foundational results on norm-attaining operators are extended to the RSE setting. Among our main contributions, we establish that for every infinite-dimensional Banach space $Y$, there exists a Banach space $X$ such that the RSE operators from $X$ to $Y$ are not dense - an RSE analogue of a result by Acosta (1999) which applies only when $Y$ is strictly convex. We also show that the Radon-Nikodým property of $Y$ is sufficient to obtain that RSE operators from $L_1(μ)$ to $Y$ are dense and that this is also necessary if $μ$ is not purely atomic. This extends and sharpens classical results by Uhl (1976). As a consequence, we prove that the set of RSE operators between $L_1(μ)$ and $L_1 (ν)$ is dense if and only if at least one of the measures $μ$ or $ν$ is purely atomic, in contrast with the classical result by Iwanik (1979) which guarantees the denseness of norm-attaining operators for all measures $μ$ and $ν$. We also prove that weakly compact operators from any $C(K)$ space can always be approximated by (weakly compact) RSE operators, thereby strengthening a result of Schachermayer (1983). Additionally, we present several improvements of more recent results concerning finite-rank operators and $Γ$-flat operators which give, in particular, RSE versions of classical results on compact operators by Johnson-Wolfe (1979). Finally, we discuss RSE counterparts of results by Zizler and Lindenstrauss on the denseness of operators whose adjoints attain their norm.

Figures (2)

• Figure 1: Relation between the different kinds of norm attainment
• Figure 2: Canonical factorization of $T\in\mathcal{L}(X,Y)$

Theorems & Definitions (82)

• Definition 1.1
• Remark 2.1
• Example 2.2
• Proposition 2.3
• proof
• Remark 2.4
• proof
• Proposition 2.5
• proof
• Lemma 2.6