On commutants and operator equations

Abstract

Let B(H) denote the algebra of bounded linear operators on a Hilbert Space H into itself. Given A,B ∈ B(H) define C(A,B) and R(A,B) : B(H) −→ B(H) by C(A,B)X = AX − XB and R(A,B)X = AXB − X. Our task in this note is to show that if A is one-one and B has dense range then C(A2,B2)X = 0 and C(A3,B3)X = 0 imply C(A,B)X = 0 for some X ∈ B(H). Similarly, if R(A2,B2)X = 0 and R(A3,B3)X = 0 then R(A,B)X = 0 for some X ∈ B(H).