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

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

$Λ^{' *} T Λ^{'} = \oplus_{i = 1}^{n} (\sum_{j = 0}^{m - 1} (\binom{m}{j + 1}) \sum_{α \in 𝔽_{n}^{+}, | α | = j} Λ_{α} T Λ_{α}^{*}),$

where $Λ^{'}$ is the Cauchy dual of $Λ$ and $𝔽_{n}^{+}$ 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 $A_{m} (𝔻)$ , 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 $D_{n}^{m} (ℋ)$ , $n, m \in ℕ^{k}$ , 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

$κ_{m} (z, w) : = \prod_{i = 1}^{k} \frac{1}{{(1 - {\bar{z}}_{i} w_{i})}^{m_{i}}}, z, w \in 𝔻^{k},$

where $m_{i} \geq 1$ . 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

Milestones

Received: 22 January 2019

Revised: 26 January 2020

Accepted: 16 March 2020

Published: 7 September 2021

Authors

Gelu Popescu
Department of Mathematics The University of Texas at San Antonio San Antonio, TX United States

Vol. 14, No. 6, 2021

Gelu Popescu

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors