Let
X
,
Y
X, Y
be given Banach spaces. For
A
∈
L
(
X
)
,
B
∈
L
(
Y
)
A\in {\mathcal L}(X),\,B\in {\mathcal L}(Y)
and
C
∈
L
(
Y
,
X
)
C\in {\mathcal L}(Y,X)
, let
M
C
M_C
be the operator defined on
X
⊕
Y
X\oplus Y
by
M
C
=
[
A
a
m
p
;
C
0
a
m
p
;
B
]
M_C = [\begin {smallmatrix} A & C 0 & B \end {smallmatrix}]
. We give sufficient conditions on
C
C
to get
Σ
(
M
C
)
=
Σ
(
M
0
)
,
\Sigma (M_C) = \Sigma (M_0),
where
Σ
\Sigma
runs over a large class of spectra. We also discuss the case of some spectra for which the latter equality fails.