On the range and the kernel of the operator $X\mapsto AXB-X$

Let $L(H)$ denote the algebra of (bounded linear) operators on the separable complex Hilbert space $H$, and let $(\mathfrak I;\|\,.\,\|_{\mathfrak I})$ denote a norm ideal in $L(H)$. For $A,B\in L(H)$, let the derivation $\delta _{A,B}\colon L(H)\to L(H)$ be defined by $\delta _{A,B}(X)=AX-XB$, and let $\Delta _{A,B}:L(H)\to L(H)$ be defined by $\Delta _{A,B}(X)=AXB-X$. The main result of this paper is to show that if $A$, $B$ are contractions, then for every operator $T\in\mathfrak J$ such that $ATB=T$, then $\|AXB-X+T\|_{\mathfrak J}\ge \|T\|_{\mathfrak J}$ for all $X\in\mathfrak J$.