Operators on C_{0}(L,X) whose range does not contain c_{0}

TL;DR

This paper investigates operators on spaces of continuous functions and c0-sums, establishing conditions under which certain operators satisfy the Daugavet equation and are weakly compact, revealing structural properties of these Banach spaces.

Contribution

It proves that operators not containing c0 in their range satisfy the Daugavet equality and that operators from c0-sums into spaces without c0 are weakly compact, extending operator theory in Banach spaces.

Findings

Operators on C_0(L,X) satisfy the Daugavet equation if their range does not contain c_0.

Operators from c_0-sums into spaces without c_0 are weakly compact.

Structural properties of Banach spaces related to c_0 and reflexivity are characterized.

Abstract

This paper contains the following results: a) Suppose that X is a non-trivial Banach space and L is a non-empty locally compact Hausdorff space without any isolated points. Then each linear operator T: C_{0}(L,X)\to C_{0}(L,X), whose range does not contain C_{00} isomorphically, satisfies the Daugavet equality ||I+T||=1+||T||. b) Let \Gamma be a non-empty set and X, Y be Banach spaces such that X is reflexive and Y does not contain c_{0} isomorphically. Then any continuous linear operator T: c_{0}(\Gamma,X)\to Y is weakly compact.