Zero divisors and topological divisors of zero in certain Banach algebras

Abstract

In this paper we prove that an element $f\in \mathcal{A}(\mathbb{D})$ is a topological divisor of zero(TDZ) if and only if there exists $z_0 \in \mathbb{T}$ such that $f(z_0)=0.$ We also give a characterization of TDZ in the Banach algebra $L^\infty(\mu).$ Further, we prove that the multiplication operator $M_h$ is a TDZ in $\mathcal{B}(L^p(\mu))~(1\leq p\leq\infty)$ if and only if $h$ is a TDZ in $L^\infty(\mu).$ Subsequently, we show that a composition operator $C_{\phi}$ is a TDZ in $\mathcal{B}(L^2(\mu))$ if and only if $\frac{d\mu \phi^{-1}}{d\mu}$ is a TDZ in $L^{\infty}(\mu).$ Lastly, we determine composition operators on the Hardy spaces $\mathbb{H}^p(\mathbb{D})$ and $\ell^p$ spaces which are zero-divisors.