A class of Zero Divisors and Topological Divisors of Zero in some Banach algebras

Abstract

In this paper, we establish necessary and sufficient conditions that must be met for weighted composition operators to act as zero divisors in $\mathcal{B}(\ell^p).$ We also give a necessary condition and a sufficient condition for a composition operators to act as zero divisors in $\mathcal{B}(L^p(\mu)).$ Subsequently, we characterize TDZ in $C(X)$. Afterward, we establish that a multiplication operator $M_h$ in $\mathcal{B}(C(X))$ becomes a TDZ if and only if $h$ is a TDZ in $C(X).$ Further, motivated by the definition of TDZ, we introduce notions of polynomially TDZ and strongly TDZ and prove that every element in $C(X)$ and in $L^\infty(\mu)$ is a polynomially TDZ. We then prove that a multiplication operator $M_h$ in $\mathcal{B}(C(X))$ as well as in $\mathcal{B}(L^p(\mu))$ is a polynomially TDZ. Lastly, we show that each $T\in \mathcal{B}(H)$, where $H$ is a separable Hilbert space, is a strongly TDZ.