Let T be a bounded linear operator from a complex Banach space
X
\mathfrak {X}
into itself and let
M
\mathfrak {M}
be a closed invariant subspace of T. Let
T
|
M
T|\mathfrak {M}
denote the restriction of T to
M
\mathfrak {M}
and let
σ
\sigma
denote the spectrum of an operator. The main results say that: (1) If
X
\mathfrak {X}
is the closed linear span of a family
{
M
v
}
\{ {\mathfrak {M}_v}\}
of invariant subspaces, then every component of
σ
(
T
)
\sigma (T)
intersects the closure of the set
∪
v
σ
(
T
|
M
v
)
{ \cup _v}\sigma (T|{\mathfrak {M}_v})
and every point of
σ
(
T
)
∖
∪
v
σ
(
T
|
M
v
)
\sigma (T)\backslash { \cup _v}\sigma (T|{\mathfrak {M}_v})
is an approximate eigenvalue of T. (2) If
X
\mathfrak {X}
is the closed linear span of a finite family
{
M
1
,
…
,
M
n
}
\{ {\mathfrak {M}_1}, \ldots ,{\mathfrak {M}_n}\}
of invariant subspaces, and the spectra
σ
(
T
|
M
j
)
,
j
=
1
,
2
,
…
,
n
\sigma (T|{\mathfrak {M}_j}),j = 1,2, \ldots ,n
, are pairwise disjoint, then
X
\mathfrak {X}
is actually equal to the algebraic direct sum of the
M
j
{\mathfrak {M}_j}
’s, the
M
j
{\mathfrak {M}_j}
’s are hyperinvariant subspaces of T and
σ
(
T
)
=
∪
j
=
1
n
σ
(
T
|
M
j
)
\sigma (T) = \cup _{j = 1}^n\sigma (T|{\mathfrak {M}_j})
. This last result is sharp in a certain specified sense. The results of (1) have a “dual version”
(
1
′
)
(1’)
; (1) and
(
1
′
)
(1’)
are applied to analyze the spectrum of an operator having a chain of invariant subspaces which is “piecewise well-ordered by inclusion", extending in several ways recent results of J. D. Stafney on the spectra of lower triangular matrices.