Spectral conditions for almost composition operators between algebras of functions

Abstract:

In this article we develop a unifying theory of many years of work by a number of researchers. Especially, we establish general sufficient conditions for maps between algebras of bounded continuous functions on locally compact Hausdorff spaces to be almost composition or almost weighted composition operators, which extend the main results of many previous works on this subject. The following are some typical results. Let $T\colon A \to B$ be a surjective map between two function algebras on locally compact Hausdorff spaces $X$ and $Y$ with Choquet boundaries $\delta A\subset X$ and $\delta B\subset Y$. If $\|Tf Tg\|=\|fg\|$ and there is an $\varepsilon , 0\leq \varepsilon <2/3$, so that the peripheral spectrum $\sigma _\pi (Tf Tg)$ is contained in an $\varepsilon \|fg\|$-neighborhood of $\sigma _\pi (fg)$ for all $f\in A$ and all $g\in A$ with $\|g\|=1$, then there is a continuous function $\alpha \colon \delta B\to \{\pm 1\}$ and a homeomorphism $\psi \colon \delta B\to \delta A$ so that $|(Tf)(y)-\alpha (y) f(\psi (y))|\leq 2\varepsilon |f(\psi (y))|$ for each $f\in A$ and every $y\in \delta B$. Therefore, $T$ is an almost weighted composition operator on $\delta B$. If $\|Tf Tg\|=\|fg\|$, there are $\varepsilon , 0\leq \varepsilon <1$, and $\eta ,\ 0\leq \eta <1$, so that $d(\sigma _\pi (Tf Tg),\sigma _\pi (fg))\leq \varepsilon \|fg\|$, while $\sigma _\pi (Tf)$ is contained in an $\eta$-neighborhood of $\sigma _\pi (f)$ for all $f\in A$ and all $g\in A$ with $\|g\|=1$. Then $|(Tf)(y)-f(\psi (y))|\leq (\varepsilon +\eta ) |f(\psi (y))|$ for each $y\in \delta B$ and every $f\in A$; i.e. $T$ is an almost composition operator on $\delta B$. If $\sigma _\pi (Tf Tg)\subset \sigma _\pi (fg)$ $($or $\sigma _\pi (fg)\subset \sigma _\pi (Tf Tg))$, and $d(\sigma _\pi (Tg),\sigma _\pi (g))\leq \eta$ for some $\eta , 0\leq \eta <1$, for all $f\in A$ and all $g\in A$ with $\|g\|=1$, then $(Tf)(y)=f(\psi (y))$ for each $f\in A$ and every $y\in \delta B$. Consequently, $T$ is a composition operator on $\delta B$ and, therefore, an algebra isomorphism.