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Multi-Toeplitz operators associated with regular polydomains

In this paper we introduce and study the class of weighted multi-Toeplitz operators associated with noncommutative polydomains ${\bf D_f^m}$, ${\bf m}:=(m_1,\ldots, m_k)\in {\bf N}^k$, generated by $k$-tuples ${\bf f}:=(f_1,\ldots, f_k)$ of positive regular free holomorphic functions in a neighborhood of the origin. These operators are acting on the tensor product $F^2(H_{n_1})\otimes \cdots \otimes F^2(H_{n_k})$ of full Fock spaces with $n_i$ generators or, equivalently, they can be viewed as multi-Toeplitz operators acting on tensor products of weighted full Fock spaces. For a large class of polydomains, we show that there are no non-zero compact multi-Toeplitz operators. We characterize the weighted multi-Toeplitz operators in terms of bounded free $k$-pluriharmonic functions on the radial part of ${\bf D_f^m}$ and use the result to obtain an analogue of the Dirichlet extension problem for free $k$-pluriharmonic functions. We show that the weighted multi-Toeplitz operators have noncommutative Fourier representations which can be viewed as noncommutative symbols and can be used to recover the associated operators. We also prove that the weighted multi-Toeplitz operators satisfy a Brown-Halmos type equation associated with the polydomain ${\bf D_f^m}$.