10.2140/apde.2021.14.1725 BROWN-HALMOS CHARACTERIZATION OF MULTI-TOEPLITZ OPERATORS ASSOCIATED WITH NONCOMMUTATIVE POLYHYPERBALLS

The noncommutative m-hyperball, m is an element of N, is defined by D-n(m)(H) := {X := (X-1, ..., X-n) is an element of B(H)(n) : (id - Phi(X))(k) (I) >= 0 for 1 <= k <= m}, where Phi(X) : B(H) -> B(H) is the completely positive map given by Phi(X)(Y) := Sigma(n)(i=1) XiY X-i* for Y is an element of B(H). Its right universal model is an n-tuple Lambda = (Lambda(1), ... , Lambda(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 is an element of 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 Lambda'*T Lambda' = circle plus(n)(i=1) (Sigma(m-1)(j=0)((m)(j+1)) Sigma(alpha is an element of Fn+vertical bar alpha vertical bar=j) Lambda T-alpha Lambda(alpha)*), where Lambda' is the Cauchy dual of Lambda 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 Langendorfer 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 is an element of 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 kappa(m)(z, w) := Pi(k)(i=1) 1/(1-(z) over bar (i)w(i))(mi), z, w is an element of D-k, where m(i) >= 1. This includes the Hardy space, the Bergman space, and the weighted Bergman space over the polydisk.

Automated summary

The paper explores noncommutative m-hyperballs and their applications, including multi-Toeplitz operators with harmonic symbols, Cauchy duals, free semigroups, etc. By generalizing the model, it tackles problems related to multi-pluriharmonic functions with operator coefficients.