Let
B
(
H
)
B(\mathcal {H})
denote the algebra of operators on a Hilbert
H
\mathcal {H}
. If
A
j
A_j
and
B
j
∈
B
(
H
)
B_j\in B(\mathcal {H})
are commuting normal operators, and
C
j
C_j
and
D
j
∈
B
(
H
)
D_j\in B(\mathcal {H})
are commuting quasi-nilpotents such that
A
j
C
j
−
C
j
A
j
=
B
j
D
j
−
D
j
B
j
=
0
A_jC_j-C_jA_j=B_jD_j-D_jB_j=0
, then define
M
j
,
N
j
∈
B
(
H
)
M_j, N_j\in B(\mathcal {H})
and
E
,
E
∈
B
(
B
(
H
)
)
{\mathcal E}, E\in B(B(\mathcal {H}))
by
M
j
=
A
j
+
C
j
M_j=A_j+C_j
,
N
j
=
B
j
+
D
j
N_j=B_j+D_j
,
E
(
X
)
=
A
1
X
A
2
+
B
1
X
B
2
{\mathcal E}(X)=A_1XA_2+B_1XB_2
and
E
(
X
)
=
M
1
X
M
2
+
N
1
X
N
2
E(X)=M_1XM_2+N_1XN_2
. It is proved that
E
−
1
(
0
)
⊆
H
0
(
E
)
=
E
−
1
(
0
)
E^{-1}(0)\subseteq H_0({\mathcal E})={\mathcal E}^{-1}(0)
and
X
∈
E
−
1
(
0
)
⟹
|
|
X
|
|
≤
k
dist
(
X
,
E
(
B
(
H
)
)
)
X\in E^{-1}(0)\Longrightarrow ||X||\leq k \textrm {dist}(X, {\mathcal E}(B(\mathcal {H})))
, where
k
≥
1
k\geq 1
is some scalar and
H
0
(
E
)
H_0({\mathcal E})
is the quasi-nilpotent part of the operator
E
{\mathcal E}
.