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Let B(H) denote the algebra of all bounded linear operators on a separable, infinite-dimensional, complex Hilbert space H. Let I be a two-sided ideal in B(H). For operators A, B and X ∈ B(H), we say that Xintertwines A and B modulo I if AX − XB ∈ I. It is easy to see that if X intertwines A and B modulo I, then it intertwines A n and B n modulo I for every integer n >1. However, the converse is not true. In this paper, sufficient conditions on the operators A and B are given so that anydoi:10.1017/s0017089505002910 fatcat:gwv7byn35jcfrfo4la7gultwoq