Vol. 14, No. 6, 2021

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Brown–Halmos characterization of multi-Toeplitz operators associated with noncommutative polyhyperballs

Gelu Popescu

Vol. 14 (2021), No. 6, 1725–1760

The noncommutative m-hyperball, m , is defined by

𝒟nm() := {X := (X 1,,Xn) B()n : (idΦ X)k(I) 0  for 1 k m},

where ΦX : B() B() is the completely positive map given by ΦX(Y ) := i=1nXiY Xi for Y B(). Its right universal model is an n-tuple Λ = (Λ1,,Λn) of weighted right creation operators acting on the full Fock space F2(Hn) with n generators. We prove that an operator T B(F2(Hn)) is a multi-Toeplitz operator with free pluriharmonic symbol on 𝒟nm() if and only if it satisfies the Brown–Halmos-type equation

ΛTΛ = i=1n( j=0m1 m j+1 α𝔽n+,|α|=jΛαTΛα),

where Λ is the Cauchy dual of Λ and 𝔽n+ is the free unital semigroup with n generators. This is a noncommutative multivariable analogue of Louhichi and Olofsson characterization of Toeplitz operators with harmonic symbols on the weighted Bergman space Am(𝔻), as well as Eschmeier and Langendörfer extension to the unit ball of n.

All our results are proved in the more general setting of noncommutative polyhyperballs Dnm(), n,m k , and are used to characterize the bounded free k-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

κm(z,w) := i=1k 1 (1z̄iwi)mi,z,w 𝔻k,

where mi 1. This includes the Hardy space, the Bergman space, and the weighted Bergman space over the polydisk.

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multivariable operator theory, multi-Toeplitz operator, noncommutative domain, Fock space, Bergman space, pluriharmonic function
Mathematical Subject Classification 2010
Primary: 47B35, 47A62
Secondary: 47A56, 47B37
Received: 22 January 2019
Revised: 26 January 2020
Accepted: 16 March 2020
Published: 7 September 2021
Gelu Popescu
Department of Mathematics
The University of Texas at San Antonio
San Antonio, TX
United States