Some results on disjointness preserving Fredholm operators between certain Banach function algebras

For two algebras $\A$ and $\B$, a linear map $T : \A \lo \B$ is disjointness preserving if $x \cdot y = ۰$ implies $Tx \cdot Ty = ۰$ for all $x, y \in \A$ and is said Fredholm if dim(ker($T$)) i.e. the  nullity of $T$ and codim($T(E)$) i.e. the corank of $T$ are finite. We develop some results of  Fredholm linear disjointness preserving operators from $C_۰(X)$ into $C_۰(Y)$ for locally compact  Hausdorff spaces $X$ and $Y $in \cite{JW۲۸}, into regular Banach function algebras. In particular,  we consider weighted composition Fredholm operators as a typical example of disjointness preserving  Fredholm operators on certain regular Banach function algebras.

Keywords:

Disjointness preserving , Weighted composition , Fredholm