#### 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
##### Abstract

The noncommutative $m$-hyperball, $m\in ℕ$, is defined by

where ${\Phi }_{X}:B\left(\mathsc{ℋ}\right)\to B\left(\mathsc{ℋ}\right)$ is the completely positive map given by ${\Phi }_{X}\left(Y\right):={\sum }_{i=1}^{n}{X}_{i}Y{X}_{i}^{\ast }$ for $Y\in B\left(\mathsc{ℋ}\right)$. Its right universal model is an $n$-tuple $\Lambda =\left({\Lambda }_{1},\dots ,{\Lambda }_{n}\right)$ of weighted right creation operators acting on the full Fock space ${F}^{2}\left({H}_{n}\right)$ with $n$ generators. We prove that an operator $T\in B\left({F}^{2}\left({H}_{n}\right)\right)$ is a multi-Toeplitz operator with free pluriharmonic symbol on ${\mathsc{𝒟}}_{n}^{m}\left(\mathsc{ℋ}\right)$ if and only if it satisfies the Brown–Halmos-type equation

${\Lambda }^{\prime \ast }T{\Lambda }^{\prime }={\oplus }_{i=1}^{n}\left({\sum }_{j=0}^{m-1}\left(\genfrac{}{}{0.0pt}{}{m}{j+1}\right){\sum }_{\alpha \in {\mathbb{𝔽}}_{n}^{+},|\alpha |=j}\phantom{\rule{-0.17em}{0ex}}\phantom{\rule{-0.17em}{0ex}}{\Lambda }_{\alpha }T{\Lambda }_{\alpha }^{\ast }\right),$

where ${\Lambda }^{\prime }$ is the Cauchy dual of $\Lambda$ and ${\mathbb{𝔽}}_{n}^{+}$ 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 ${A}_{m}\left(\mathbb{𝔻}\right)$, as well as Eschmeier and Langendörfer extension to the unit ball of ${ℂ}^{n}$.

All our results are proved in the more general setting of noncommutative polyhyperballs ${D}_{n}^{m}\left(\mathsc{ℋ}\right)$, $n,m\in {ℕ}^{k}$, 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

${\kappa }_{m}\left(z,w\right):={\prod }_{i=1}^{k}\frac{1}{{\left(1-{\stackrel{̄}{z}}_{i}{w}_{i}\right)}^{{m}_{i}}},\phantom{\rule{1em}{0ex}}z,w\in {\mathbb{𝔻}}^{k},$

where ${m}_{i}\ge 1$. This includes the Hardy space, the Bergman space, and the weighted Bergman space over the polydisk.

##### Keywords
multivariable operator theory, multi-Toeplitz operator, noncommutative domain, Fock space, Bergman space, pluriharmonic function
##### Mathematical Subject Classification 2010
Primary: 47B35, 47A62
Secondary: 47A56, 47B37