$L$-orthogonality in Daugavet centers and narrow operators

Abstract

We study the presence of $L$-orthogonal elements in connection with Daugavet centers and narrow operators. We prove that, if $\dens(Y)\leq ω_1$ and $G:X⟶Y$ is a Daugavet center, then $G(W)$ contains some $L$-orthogonal for every non-empty $w^{\ast }$-open subset of $B_{X^{\ast \ast }}$. In the context of narrow operators, we show that if $X$ is separable and $T:X⟶Y$ is a narrow operator, then given $y\in B_X$ and any non-empty $w^{\ast }$-open subset $W$ of $B_{X^{\ast \ast }}$ then $W$ contains some $L$-orthogonal $u$ so that $T^{\ast \ast }(u)=T(y)$. In the particular case that $T^{\ast }(Y^{\ast })$ is separable, we extend the previous result to $\dens(X)=ω_1$. 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).

13 pages