Daugavet centers and direct sums of Banach spaces

TL;DR

This paper characterizes the Banach space sums for which Daugavet centers exist, providing a complete description of the conditions on the sum space F and illustrating with examples.

Contribution

It offers a full classification of the two-dimensional Banach spaces F that admit Daugavet centers in sums of Banach spaces.

Findings

Identifies classes of F allowing Daugavet centers from sums

Identifies classes of F allowing Daugavet centers into sums

Provides examples of Daugavet centers in specific sums

Abstract

A linear continuous nonzero operator G:X->Y is a Daugavet center if every rank-1 operator T:X->Y satisfies ||G+T||=||G||+||T||. We study the case when either X or Y is a sum $X_1 \oplus_F X_2$ of two Banach spaces $X_1$ and $X_2$ by some two-dimensional Banach space F. We completely describe the class of those F such that for some spaces $X_1$ and $X_2$ there exists a Daugavet center acting from $X_1\oplus_F X_2$, and the class of those F such that for some pair of spaces $X_1$ and $X_2$ there is a Daugavet center acting into $X_1\oplus_F X_2$. We also present several examples of such Daugavet centers.