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 $\mathcal{B}(L^p(\mu ))(1\le p\le \infty )$ if and only if $h$ is a TDZ in $L^{\infty }(\mu ).$ Subsequently, we show that a composition operator $C_{\varphi }$ is a TDZ in $\mathcal{B}(L^2(\mu ))$ if and only if $\frac{d\mu {\varphi }^{-1}}{d\mu }$ 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.