Few operators on Banach spaces $C_0(L\times L)$

TL;DR

This paper constructs a special non-metrizable space to analyze the structure of operators on the Banach space $C_0(L imes L)$, providing an explicit characterization of all such operators.

Contribution

It introduces a novel non-metrizable space using Ostaszewski's $ullet$-principle and characterizes all operators on the associated Banach space.

Findings

Operators on $C_0(L imes L)$ have a simple, explicit structure.

The space $L$ is non-metrizable, locally compact, and scattered.

Complete classification of operators on the constructed Banach space.

Abstract

Using Ostaszewski's $\clubsuit$-principle, we construct a non-metrizable, locally compact, scattered space $L$ in which the operators on the Banach space $C_0(L \times L)$ exhibit a remarkably simple structure. We provide a detailed analysis and, through a series of decomposition steps, offer an explicit characterization of all operators on $C_0(L \times L)$.