Every pair
{
M
,
N
}
\{M,N\}
of non-trivial topologically complementary subspaces of a Hilbert space is unitarily equivalent to a pair of the form
{
G
(
−
A
)
⊕
K
,
G
(
A
)
⊕
(
0
)
}
\left \{G(-A)\oplus K,G(A)\oplus (0)\right \}
on a Hilbert space
H
⊕
H
⊕
K
H\oplus H\oplus K
. Here
K
K
is possibly
(
0
)
(0)
,
A
∈
B
(
H
)
A\in \mathcal {B}(H)
is a positive injective contraction and
G
(
±
A
)
G(\pm A)
denotes the graph of
±
A
\pm A
. For such a pair
{
M
,
N
}
\{M,N\}
the following are equivalent: (i)
{
M
,
N
}
\{M,N\}
is similar to a pair in generic position; (ii)
M
M
and
N
N
have a common algebraic complement; (iii)
{
M
,
N
}
\{M,N\}
is similar to
{
G
(
X
)
,
G
(
Y
)
}
\left \{G(X),G(Y)\right \}
for some operators
X
,
Y
X,Y
on a Hilbert space. These conditions need not be satisfied. A second example is given (the first due to T. Kato), involving only compact operators, of a double triangle subspace lattice which is not similar to any operator double triangle.