Range strongly exposing operators between Banach spaces

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-Nikod\'ym property of $Y$ is sufficient to obtain that RSE operators from $L_1(\mu)$ to $Y$ are dense and that this is also necessary if $\mu$ 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 $L_1(\mu)$ and $L_1 (\nu)$ is dense if and only if at least one of the measures $\mu$ or $\nu$ is purely atomic, in contrast with the classical result by Iwanik (1979) which guarantees the denseness of norm-attaining operators for all measures $\mu$ and $\nu$. 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 $\Gamma$-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.