Stability and instability of weighted composition operators

TL;DR

This paper investigates the stability and instability bounds of weighted composition operators in the context of $ ext{ extbackslash epsilon}$-disjointness preserving operators on spaces of continuous functions, considering different domain and codomain cases.

Contribution

It provides sharp bounds on how close $ ext{ extbackslash epsilon}$-disjointness preserving operators are to weighted composition operators, addressing stability and instability in various cases.

Findings

Sharp stability bounds for infinite $X$

Instability bounds for finite $X$

Results for $ ext{ extbackslash epsilon}$-disjointness preserving functionals

Abstract

Let $\epsilon >0$. A continuous linear operator $T:C(X) \ra C(Y)$ is said to be {\em $\epsilon$-disjointness preserving} if $\vc (Tf)(Tg)\vd_{\infty} \le \epsilon$, whenever $f,g\in C(X)$ satisfy $\vc f\vd_{\infty} =\vc g\vd_{\infty} =1$ and $fg\equiv 0$. In this paper we address basically two main questions: 1.- How close there must be a weighted composition operator to a given $\epsilon$-disjointness preserving operator? 2.- How far can the set of weighted composition operators be from a given $\epsilon$-disjointness preserving operator? We address these two questions distinguishing among three cases: $X$ infinite, $X$ finite, and $Y$ a singleton ($\epsilon$-disjointness preserving functionals). We provide sharp stability and instability bounds for the three cases.