2020/01/30 by Popescu, Gelu
#FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2002.00462
In this paper we introduce and study the class of weighted multi-Toeplitz operators associated with noncommutative polydomains \bf Dfm, \bf m:=(m1,…, mk)∈ \bf Nk, generated by k-tuples \bf f:=(f1,…, fk) of positive regular free holomorphic functions in a neighborhood of the origin. These operators are acting on the tensor product F2(Hn1)⊗ ⋯ ⊗ F2(Hnk) of full Fock spaces with ni 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 Dfm 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 Dfm.