Brown-Halmos characterization of multi-Toeplitz operators associated with noncommutative poly-hyperballs

Abstract

We obtain a noncommutative multivariable analogue of Louhichi and Olofsson characterization of Toeplitz operators with harmonic symbols on the weighted Bergman space $A_m(\mathbf{D})$, as well as Eschmeier and Langendorfer extension to the unit ball of ${\mathbf{C}}^n$. All our results are proved in the more general setting of noncommutative poly-hyperballs ${\mathbf{D}}_{\mathbf{n}}^{\mathbf{m}}(H)$, $\mathbf{n},\mathbf{m}\in {\mathbf{N}}^k$, and are used to characterize the bounded free $k$-pluriharmonic functions with operator coefficients on poly-hyperballs and to solve the associated Dirichlet extension problem. In particular, the results hold for the reproducing kernel Hilbert space with kernel $${\kappa }_{\mathbf{m}}(z,w):=\prod _{i=1}^k\frac{1}{(1-{\bar{z}}_i{w}_i{)}^{m_i}},z,w\in {\mathbf{D}}^k,$$ where $m_i\ge 1$. This includes the Hardy space, the Bergman space, and the weighted Bergman space over the polydisk.

31 pages