Weighted composition operators as Daugavet centers

TL;DR

This paper characterizes weighted composition operators on spaces of continuous functions and disk algebra that satisfy a specific norm identity, identifying those that act as Daugavet centers.

Contribution

It provides a complete characterization of weighted composition operators that satisfy the Daugavet equation on $C(S)$ and $A(D)$, extending understanding of operator norm identities.

Findings

Identifies weighted composition operators satisfying the Daugavet equation on $C(S)$.

Characterizes such operators on the disk algebra $A(D)$.

Provides conditions under which these operators act as Daugavet centers.

Abstract

We investigate the norm identity $\|uC_\phi + T\| = \|u\|_\infty + \|T\|$ for classes of operators on $C(S)$, where $S$ is a compact Hausdorff space without isolated point, and characterize those weighted composition operators which satisfy this equation for every weakly compact operator $T : C(S)\to C(S)$. We also give a characterization of such weighted composition operator acting on the disk algebra $A(D).$