Some Intertwining Relations Modulo Operator Ideals

Abstract

Abstract. 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 any operator X which intertwines certain powers of A and B modulo I also intertwines A and B modulo J for some two-sided ideal J ⊇ I.2000 Mathematics Subject Classification. 47A53, 47B10, 47B15, 47B20, 47B47.