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^ (). Further, we prove that the multiplication operator Mₕ is a TDZ in B (Lᵖ () ) ~ (1 p) if and only if h is a TDZ in L^ (). Subsequently, we show that a composition operator C_ is a TDZ in B (L² () ) if and only if d ^-1d is a TDZ in L^ (). Lastly, we determine composition operators on the Hardy spaces Hᵖ (D) and ᵖ spaces which are zero-divisors.