Spaceability in sets of operators on $C(K)$

TL;DR

This paper investigates the structure of operators on certain $C(K)$ spaces, showing spaceability of non-weak multiplier operators in some cases and characterizing operators in others, revealing new structural insights.

Contribution

It demonstrates the spaceability of non-weak multiplier operators on $C(K)$ spaces lacking few operators and characterizes all operators on a specific $C(K)$ space as a sum involving functions and strictly singular operators.

Findings

Non-weak multiplier operators form a spaceable set in certain $C(K)$ spaces.

Existence of a $C(K)$ space where all operators are of a specific form involving functions and strictly singular operators.

Contrast with general Banach spaces lacking few operators.

Abstract

We show that when $C(K)$ does not have few operator -- in the sense of Koszmider [P. Koszmider, Banach spaces of continuous functions with few operators. Math. Ann. 300 (2004), no. 1, 151 - 183.] -- the sets of operators which are not weak multipliers is spaceable. This shows a contrast with what happens in general Banach spaces that do not have few operators. In addition, we show that there exist a $C(K)$ space such that each operator on it is of the form $gI+hJ+S$, where $g,h\in C(K)$ and $S$ is strictly singular, in connection to a result by Ferenczi [V. Ferenczi,Uniqueness of complex structure and real hereditarily indecomposable Banach spaces. Adv. Math. 213 (2007), no. 1, 462 - 488.].