Let
V
=
B
(
H
)
\mathbf V = B(H)
or
S
(
H
)
S(H)
, where
B
(
H
)
B(H)
is the algebra of a bounded linear operator acting on the Hilbert space
H
H
, and
S
(
H
)
S(H)
is the set of self-adjoint operators in
B
(
H
)
B(H)
. Denote the numerical range of
A
∈
B
(
H
)
A \in B(H)
by
W
(
A
)
=
{
(
A
x
,
x
)
:
x
∈
H
,
(
x
,
x
)
=
1
}
W(A) = \{\, (Ax,x): x \in H, (x,x) = 1\,\}
. It is shown that a surjective map
ϕ
:
V
→
V
\phi : \mathbf V \rightarrow \mathbf V
satisfies
\[
W
(
A
B
+
B
A
)
=
W
(
ϕ
(
A
)
ϕ
(
B
)
+
ϕ
(
B
)
ϕ
(
A
)
)
for all
A
,
B
∈
V
W(AB+BA) = W(\phi (A)\phi (B)+\phi (B)\phi (A)) \qquad \text {for all $A$, $B \in \mathbf {V}$}
\]
if and only if there is a unitary operator
U
∈
B
(
H
)
U \in B(H)
such that
ϕ
\phi
has the form
\[
X
↦
±
U
∗
X
U
o
r
X
↦
±
U
∗
X
t
U
,
X \mapsto \pm U^*XU \quad \mathrm {or} \quad X \mapsto \pm U^*X^tU,
\]
where
X
t
X^t
is the transpose of
X
X
with respect to a fixed orthonormal basis. In other words, the map
ϕ
\phi
or
−
ϕ
-\phi
is a
C
∗
C^*
-isomorphism on
B
(
H
)
B(H)
and a Jordan isomorphism on
S
(
H
)
S(H)
. Moreover, if
H
H
has finite dimension, then the surjective assumption on
ϕ
\phi
can be removed.