A commuting tuple
T
=
(
T
1
,
…
,
T
n
)
∈
B
(
H
)
n
T =(T_1, \ldots , T_n) \in B(H)^n
of bounded Hilbert-space operators is called a spherical isometry if
∑
i
=
1
n
T
i
∗
T
i
=
1
H
\sum _{i=1}^n T_i^*T_i = 1_H
. B. Prunaru initiated the study of
T
T
-Toeplitz operators, which he defined to be the solutions
X
∈
B
(
H
)
X \in B(H)
of the fixed-point equation
∑
i
=
1
n
T
i
∗
X
T
i
=
X
\sum _{i=1}^n T_i^*XT_i = X
. Using results of Aleksandrov on abstract inner functions, we show that
X
∈
B
(
H
)
X \in B(H)
is a
T
T
-Toeplitz operator precisely when
X
X
satisfies
J
∗
X
J
=
X
J^*XJ=X
for every isometry
J
J
in the unital dual algebra
A
T
⊂
B
(
H
)
\mathcal {A}_T \subset B(H)
generated by
T
T
. As a consequence we deduce that a spherical isometry
T
T
has empty point spectrum if and only if the only compact
T
T
-Toeplitz operator is the zero operator. Moreover, we show that if
σ
p
(
T
)
=
∅
\sigma _p(T) = \emptyset
, then an operator which commutes modulo the finite-rank operators with
A
T
\mathcal {A}_T
is a finite-rank perturbation of a
T
T
-Toeplitz operator.