We study the invariant subspace structure of the operator of multiplication by
z
z
,
M
z
{M_z}
, on a class of Banach spaces of analytic functions. For operators on Hilbert spaces our class coincides with the adjoints of the operators in the Cowen-Douglas class
B
1
(
Ω
¯
)
{\mathcal {B}_1}(\overline \Omega )
. We say that an invariant subspace
M
\mathcal {M}
satisfies
cod
M
=
1
\operatorname {cod} \mathcal {M} = 1
if
z
M
z\mathcal {M}
has codimension one in
M
\mathcal {M}
. We give various conditions on invariant subspaces which imply that
cod
M
=
1
\operatorname {cod} \mathcal {M} = 1
. In particular, we give a necessary and sufficient condition on two invariant subspaces
M
\mathcal {M}
,
N
\mathcal {N}
with
cod
M
=
cod
N
=
1
\operatorname {cod} \mathcal {M} = \operatorname {cod} \mathcal {N} = 1
so that their span again satisfies
cod
(
M
∨
N
)
=
1
\operatorname {cod} (\mathcal {M} \vee \mathcal {N}) = 1
. This result will be used to show that any invariant subspace of the Bergman space
L
a
p
,
p
⩾
1
L_a^p,\,p \geqslant 1
, which is generated by functions in
L
a
2
p
L_a^{2p}
, must satisfy
cod
M
=
1
\operatorname {cod} \mathcal {M} = 1
. For an invariant subspace
M
\mathcal {M}
we then consider the operator
S
=
M
z
∗
|
M
⊥
S = M_z^{\ast }|{\mathcal {M}^ \bot }
. Under some extra assumption on the domain of holomorphy we show that the spectrum of
S
S
coincides with the approximate point spectrum iff
cod
M
=
1
\operatorname {cod} \mathcal {M} = 1
. Finally, in the last section we obtain a structure theorem for invariant subspaces with
cod
M
=
1
\operatorname {cod} \mathcal {M} = 1
. This theorem applies to Dirichlet-type spaces.