Let T be a bounded linear operator from a complex Banach space
X
\mathfrak {X}
into itself. Let
A
T
{\mathcal {A}_T}
and
A
T
a
\mathcal {A}_T^a
denote the weak closure of the polynomials and the rational functions (with poles outside the spectrum
σ
(
T
)
\sigma (T)
of T) in T, respectively. The lattice
Lat
A
T
a
{\operatorname {Lat}}\;\mathcal {A}_T^a
of (closed) invariant subspaces of
A
T
a
\mathcal {A}_T^a
is a very particular subset of the invariant subspace lattice
Lat
A
T
=
Lat
T
{\operatorname {Lat}}\;{\mathcal {A}_T} = {\operatorname {Lat}}\;T
of T. It is shown that: (1) If the resolvent set of T has finitely many components, then
Lat
A
T
a
{\operatorname {Lat}}\;\mathcal {A}_T^a
is a clopen (i.e., closed and open) sublattice of
Lat
T
{\operatorname {Lat}}\;T
, with respect to the “gap topology” between subspaces. (2) If
M
1
,
M
2
∈
Lat
T
,
M
1
∩
M
2
∈
Lat
A
T
a
{\mathfrak {M}_1},{\mathfrak {M}_2} \in {\operatorname {Lat}}\;T,{\mathfrak {M}_1} \cap {\mathfrak {M}_2} \in {\operatorname {Lat}}\;\mathcal {A}_T^a
and
M
1
+
M
2
{\mathfrak {M}_1} + {\mathfrak {M}_2}
is closed in
X
\mathfrak {X}
and belongs to
Lat
A
T
a
{\operatorname {Lat}}\;\mathcal {A}_T^a
, then
M
1
{\mathfrak {M}_1}
and
M
2
{\mathfrak {M}_2}
also belong to
Lat
A
T
a
{\operatorname {Lat}}\;\mathcal {A}_T^a
. (3) If
M
∈
Lat
T
,
R
\mathfrak {M} \in {\operatorname {Lat}}\;T,R
is the restriction of T to
M
\mathfrak {M}
and
T
¯
\bar T
is the operator induced by T on the quotient space
X
/
M
\mathfrak {X}/\mathfrak {M}
, then
σ
(
T
)
⊂
σ
(
R
)
∪
σ
(
T
¯
)
\sigma (T) \subset \sigma (R) \cup \sigma (\bar T)
. Moreover,
σ
(
T
)
=
σ
(
R
)
∪
σ
(
T
¯
)
\sigma (T) = \sigma (R) \cup \sigma (\bar T)
if and only if
M
∈
Lat
A
T
a
\mathfrak {M} \in {\operatorname {Lat}}\;\mathcal {A}_T^a
. The results also include an analysis of the semi-Fredholm index of R and
T
¯
\bar T
at a point
λ
∈
σ
(
R
)
∪
σ
(
T
¯
)
∖
σ
(
T
)
\lambda \in \sigma (R) \cup \sigma (\bar T)\backslash \sigma (T)
and extensions of the results to algebras between
A
T
{\mathcal {A}_T}
and
A
T
a
\mathcal {A}_T^a
.