Stability and instability of weighted composition operators

Abstract

Let $ε>0$. A continuous linear operator $T:C(X) \ra C(Y)$ is said to be {\em $ε$-disjointness preserving} if $\vc (Tf)(Tg)\vd_{\infty} \le ε$, 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 $ε$-disjointness preserving operator? 2.- How far can the set of weighted composition operators be from a given $ε$-disjointness preserving operator? We address these two questions distinguishing among three cases: $X$ infinite, $X$ finite, and $Y$ a singleton ($ε$-disjointness preserving functionals). We provide sharp stability and instability bounds for the three cases.