A perturbed elementary operator and range-kernel orthogonality

Abstract:

Let $B(\mathcal {H})$ denote the algebra of operators on a Hilbert $\mathcal {H}$. If $A_j$ and $B_j\in B(\mathcal {H})$ are commuting normal operators, and $C_j$ and $D_j\in B(\mathcal {H})$ are commuting quasi-nilpotents such that $A_jC_j-C_jA_j=B_jD_j-D_jB_j=0$, then define $M_j, N_j\in B(\mathcal {H})$ and ${\mathcal E}, E\in B(B(\mathcal {H}))$ by $M_j=A_j+C_j$, $N_j=B_j+D_j$, ${\mathcal E}(X)=A_1XA_2+B_1XB_2$ and $E(X)=M_1XM_2+N_1XN_2$. It is proved that $E^{-1}(0)\subseteq H_0({\mathcal E})={\mathcal E}^{-1}(0)$ and $X\in E^{-1}(0)\Longrightarrow ||X||\leq k \textrm {dist}(X, {\mathcal E}(B(\mathcal {H})))$, where $k\geq 1$ is some scalar and $H_0({\mathcal E})$ is the quasi-nilpotent part of the operator ${\mathcal E}$.