The
noncommutative -hyperball,
, is
defined by
where
is the completely
positive map given by
for
. Its right universal
model is an
-tuple
of weighted right creation operators acting on the full Fock space
with
generators. We prove
that an operator
is a multi-Toeplitz operator with free pluriharmonic symbol on
if and
only if it satisfies the Brown–Halmos-type equation
where
is the
Cauchy dual of
and
is the free unital
semigroup with
generators. This is a noncommutative multivariable analogue of Louhichi and Olofsson
characterization of Toeplitz operators with harmonic symbols on the weighted Bergman
space
,
as well as Eschmeier and Langendörfer extension to the unit ball of
.
All our results are proved in the more general setting of
noncommutative polyhyperballs
,
,
and are used to characterize the bounded free
-pluriharmonic
functions with operator coefficients on polyhyperballs and to solve the associated
Dirichlet extension problem. In particular, the results hold for the reproducing kernel
Hilbert space with kernel
where
.
This includes the Hardy space, the Bergman space, and the weighted Bergman space
over the polydisk.
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