$L$-orthogonality in Daugavet centers and narrow operators

TL;DR

This paper investigates the presence of $L$-orthogonal elements in relation to Daugavet centers and narrow operators, establishing conditions under which such elements exist and providing counterexamples in larger density settings.

Contribution

It introduces new results linking $L$-orthogonality with Daugavet centers and narrow operators, extending known properties to larger density characters and providing counterexamples.

Findings

$L$-orthogonal elements exist in images of Daugavet centers under certain conditions.

Narrow operators on separable spaces have $L$-orthogonal elements in specific $w^*$-open subsets.

Counterexamples show these properties fail in larger density characters like $oldsymbol{ ho=\omega_2}$.

Abstract

We study the presence of $L$-orthogonal elements in connection with Daugavet centers and narrow operators. We prove that, if $\dens(Y)\leq \omega_1$ and $G:X\longrightarrow Y$ is a Daugavet center, then $G(W)$ contains some $L$-orthogonal for every non-empty $w^*$-open subset of $B_{X^{**}}$. In the context of narrow operators, we show that if $X$ is separable and $T:X\longrightarrow Y$ is a narrow operator, then given $y\in B_X$ and any non-empty $w^*$-open subset $W$ of $B_{X^{**}}$ then $W$ contains some $L$-orthogonal $u$ so that $T^{**}(u)=T(y)$. In the particular case that $T^*(Y^*)$ is separable, we extend the previous result to $\dens(X)=\omega_1$. Finally, we prove that none of the previous results holds in larger density characters (in particular, a counterexample is shown for $\omega_2$ under continuum hypothesis).