LetB(H)B(\mathcal {H})denote the algebra of operators on a HilbertH\mathcal {H}. IfAjA_jandBj∈B(H)B_j\in B(\mathcal {H})are commuting normal operators, andCjC_jandDj∈B(H)D_j\in B(\mathcal {H})are commuting quasi-nilpotents such thatAjCj−CjAj=BjDj−DjBj=0A_jC_j-C_jA_j=B_jD_j-D_jB_j=0, then defineMj,Nj∈B(H)M_j, N_j\in B(\mathcal {H})andE,E∈B(B(H)){\mathcal E}, E\in B(B(\mathcal {H}))byMj=Aj+CjM_j=A_j+C_j,Nj=Bj+DjN_j=B_j+D_j,E(X)=A1XA2+B1XB2{\mathcal E}(X)=A_1XA_2+B_1XB_2andE(X)=M1XM2+N1XN2E(X)=M_1XM_2+N_1XN_2. It is proved thatE−1(0)⊆H0(E)=E−1(0)E^{-1}(0)\subseteq H_0({\mathcal E})={\mathcal E}^{-1}(0)andX∈E−1(0)⟹||X||≤kdist(X,E(B(H)))X\in E^{-1}(0)\Longrightarrow ||X||\leq k \textrm {dist}(X, {\mathcal E}(B(\mathcal {H}))), wherek≥1k\geq 1is some scalar andH0(E)H_0({\mathcal E})is the quasi-nilpotent part of the operatorE{\mathcal E}.