Let
C
(
H
)
C(H)
denote the lattice of all (closed) subspaces of a complex, separable Hilbert space
H
H
. Let
(
AC)
({\text {AC)}}
be the following condition that a subspace lattice
F
⊆
C
(
H
)
\mathcal {F} \subseteq C(H)
may or may not satisfy: (AC)
\[
F
=
ϕ
(
L
)
for
some
lattice
automorphism
ϕ
of
C
(
H
)
and
some
commutative
subspace
lattice
L
⊆
C
(
H
)
.
\begin {array}{*{20}{c}} {\mathcal {F} = \phi (\mathcal {L})\;{\text {for}}\;{\text {some}}\;{\text {lattice}}\;{\text {automorphism}}\;\phi \;{\text {of}}\;C(H)} \\ {{\text {and}}\;{\text {some}}\;{\text {commutative}}\;{\text {subspace}}\;{\text {lattice}}\;\mathcal {L} \subseteq C(H).} \\ \end {array}
\]
Then
F
\mathcal {F}
satisfies
(
AC
)
({\text {AC}})
if and only if
A
⊆
B
\mathcal {A} \subseteq \mathcal {B}
for some Boolean algebra subspace lattice
B
⊆
C
(
H
)
\mathcal {B} \subseteq C(H)
with the property that, for every
K
,
L
∈
B
K,L \in \mathcal {B}
, the vector sum
K
+
L
K + L
is closed. If
F
\mathcal {F}
is finite, then
F
\mathcal {F}
satisfies
(
AC
)
({\text {AC}})
if and only if
F
\mathcal {F}
is distributive and
K
+
L
K + L
is closed for every
K
,
L
∈
F
K,L \in \mathcal {F}
. In finite dimensions
F
\mathcal {F}
satisfies
(
AC
)
({\text {AC}})
if and only if
F
\mathcal {F}
is distributive. Every
F
\mathcal {F}
satisfying
(
AC
)
({\text {AC}})
is reflexive. For such
F
\mathcal {F}
, given vectors
x
,
y
∈
H
x,y \in H
, the solvability of the equation
T
x
=
y
Tx = y
for
T
∈
Alg
F
T \in \operatorname {Alg}\,\mathcal {F}
is investigated.