Almost Weighted Composition Operators Between Banach Function Algebras

Abstract

Let A and B be Banach function algebras on compact Hausdorff spaces X and Y, respectively, and $\rho , \tau :I\longrightarrow A$, $S, T :I\longrightarrow B$ be maps on a non-empty set I whose ranges are closed under multiplication and contain exponential functions. In this paper, we first show that if $\Vert S(p)\Vert _{Y}=\Vert \rho (p)\Vert _{X}$, $\Vert T(p)\Vert _Y=\Vert \tau (p)\Vert _X$ and $\Vert S(p)T(q)\Vert _Y=\Vert \rho (p)\tau (q)\Vert _{X}$, for all $p, q \in I$, then there exists a homeomorphism $\varphi$ from Šilov boundary $\partial A$ of A onto $\partial B$ such that for each $x\in \partial A$ and $p\in I$, $\vert S(p)(\varphi (x))\vert =\vert \rho (p)(x)\vert$ and $\vert T(p)(\varphi (x)) \vert =\vert \tau (p)(x)\vert$. Then we prove that, if for some $\varepsilon \ge 0$, $\mathrm {Ran}_\pi (T(p)S(q))$ is contained in an $\varepsilon \Vert \tau (p)\rho (q)\Vert$-neighborhood of $\mathrm {Ran}_\pi (\tau (p)\rho (q))$, for all $p, q \in I$, then, under a certain condition, there exist continuous functions $\alpha , \beta \in B$ such that $\vert \alpha (\varphi (x))T(p)(\varphi (x))-\tau (p)(x)\vert \leqslant 4 \varepsilon \vert \tau (p)(x)\vert$ and $\vert \beta (\varphi (x)) S(p)(\varphi (x))-\rho (p)(x)\vert \leqslant 4 \varepsilon \vert \rho (p)(x)\vert$, for all $p\in I$ and $x\in \partial (A)$. Our results can be applied for Banach algebras of Lipschitz functions.