An element a in a Banach algebraA has ps-Drazin inverse if there exists p2 = p ∈ comm2(a) such that (a − p)k ∈ J(A) for some k ∈ N. LetA be a Banach algebra, and let a, b ∈ A have ps-Drazin inverses. If a2b = aba and b2a = bab, we prove that 1. ab ∈ A has ps-Drazin inverse. 2. a + b ∈ A has ps-Drazin inverse if and only if 1 + adb ∈ A has ps-Drazin inverse. As applications, we present various conditions under which a 2 × 2 matrix over a Banach algebra has ps-Drazin inverse