SOME INTERTWINING RELATIONS MODULO OPERATOR IDEALS

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 $X \in B(H)$, we say that $X$intertwines$A$and$B$modulo$I$ if $AX - XB \in I$. 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 $J \supseteq I$.