Multi-Toeplitz operators and free pluriharmonic functions

Abstract

We initiate the study of weighted multi-Toeplitz operators associated with noncommutative regular domains Dqm(H)⊂B(H)n, m,n≥1, where B(H) is the algebra of all bounded linear operators on a Hilbert space H. These operators are acting on the full Fock space with n generators and have as symbols free pluriharmonic functions on the interior of the domain Dqm(H). We prove that the set of all weighted multi-Toeplitz operators coincides withA(Dqm)⁎+A(Dqm)‾WOT, where the domain algebra A(Dqm) is the norm-closed unital non-selfadjoint algebra generated by the universal model (W1,…,Wn) of the noncommutative domain Dqm(H). These results are used to study the class of free pluriharmonic functions on Dqm(H)∘. Several classical results from complex analysis concerning harmonic functions have analogues in our noncommutative setting. In particular, we show that the bounded free pluriharmonic functions are precisely those which are noncommutative Berezin transforms of weighted multi-Toeplitz operators, and solve the Dirichlet extension problem in this setting. Using noncommutative Cauchy transforms, we provide a free analytic functional calculus for n-tuples of operators, which extends to free pluriharmonic functions. Our study of weighted multi-Toeplitz operators on Fock spaces is a blend of multi-variable operator theory, noncommutative function theory, operator spaces, and harmonic analysis.