For a linear transformation
A
A
on a Banach space, let
L
(
A
)
\mathcal {L}(A)
be the lattice of (not necessarily closed) invariant subspaces of
A
A
. For
A
A
bounded it is shown that if
L
(
A
⊕
A
)
⊂
L
(
T
⊕
T
)
\mathcal {L}(A \oplus A) \subset \mathcal {L}(T \oplus T)
, or if
L
(
A
)
⊂
L
(
T
)
\mathcal {L}(A) \subset \mathcal {L}(T)
and
T
T
commutes with
A
A
, then
T
T
is a polynomial in
A
A
. In the case of a Hilbert space, if
L
(
A
)
⊂
L
(
A
∗
)
\mathcal {L}(A) \subset \mathcal {L}({A^ \ast })
then
A
∗
{A^ \ast }
is a polynomial in
A
A
.