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Spectral conditions for almost composition operators between algebras of functions


Author: T. Tonev
Journal: Proc. Amer. Math. Soc. 142 (2014), 2721-2732
MSC (2010): Primary 46J10
Published electronically: April 30, 2014
MathSciNet review: 3209327
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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 $ \Vert Tf\,Tg\Vert=\Vert fg\Vert$ and there is an $ \varepsilon ,\,0\leq \varepsilon <2/3$, so that the peripheral spectrum $ \sigma _\pi (Tf\,Tg)$ is contained in an $ \varepsilon \,\Vert fg\Vert$-neighborhood of $ \sigma _\pi (fg)$ for all $ f\in A$ and all $ g\in A$ with $ \Vert g\Vert=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 $ \vert(Tf)(y)-\alpha (y)\,f(\psi (y))\vert\leq 2\varepsilon \,\vert f(\psi (y))\vert$ for each $ f\in A$ and every $ y\in \delta B$. Therefore, $ T$ is an almost weighted composition operator on $ \delta B$. If $ \Vert Tf\,Tg\Vert=\Vert fg\Vert$, 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 \,\Vert fg\Vert$, 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 $ \Vert g\Vert=1$. Then $ \vert(Tf)(y)-f(\psi (y))\vert\leq (\varepsilon +\eta )\,\vert f(\psi (y))\vert$ 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 $ \Vert g\Vert=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.


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Additional Information

T. Tonev
Affiliation: Department of Mathematics, University of Montana, Missoula, Montana 59812
Email: tonevtv@mso.umt.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12005-3
Keywords: Function algebra, uniform algebra, peripheral spectrum, composition operator, almost composition operator, almost weighted composition operator, algebra isomorphism, Choquet boundary
Received by editor(s): April 30, 2012
Received by editor(s) in revised form: August 31, 2012
Published electronically: April 30, 2014
Additional Notes: This research was partially supported by grant No. 209762 from the Simons Foundation
Communicated by: Richard Rochberg
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.