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.