Zero divisors and topological divisors of zero in certain Banach algebras

In this paper we prove that an element f∈ A(D) is a topological divisor of zero(TDZ) if and only if there exists z₀ ∈ T such that f(z₀)=0. We also give a characterization of TDZ in the Banach algebra L^∞(μ). Further, we prove that the multiplication operator Mₕ is a TDZ in B(Lᵖ(μ))~(1≤ p≤∞) if and only if h is a TDZ in L^∞(μ). Subsequently, we show that a composition operator Cφ is a TDZ in B(L²(μ)) if and only if dμ φ⁻¹dμ is a TDZ in L∞(μ). Lastly, we determine composition operators on the Hardy spaces Hᵖ(D) and ᵖ spaces which are zero-divisors.