Analysis & PDE

  • Anal. PDE
  • Volume 9, Number 5 (2016), 1185-1234.

Free pluriharmonic functions on noncommutative polyballs

Gelu Popescu

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Abstract

We study free k-pluriharmonic functions on the noncommutative regular polyball Bn, n = (n1,,nk) k , which is an analogue of the scalar polyball (n1)1 × × (nk)1. The regular polyball has a universal model S := {Si,j} consisting of left creation operators acting on the tensor product F2(Hn1) F2(Hnk) of full Fock spaces. We introduce the class Tn of k-multi-Toeplitz operators on this tensor product and prove that T n = span{AnAn} - SOT, where An is the noncommutative polyball algebra generated by S and the identity. We show that the bounded free k-pluriharmonic functions on Bn are precisely the noncommutative Berezin transforms of k-multi-Toeplitz operators. The Dirichlet extension problem on regular polyballs is also solved. It is proved that a free k-pluriharmonic function has continuous extension to the closed polyball Bn if and only if it is the noncommutative Berezin transform of a k-multi-Toeplitz operator in span{AnAn} -.

We provide a Naimark-type dilation theorem for direct products Fn1+ × × Fnk+ of unital free semigroups, and use it to obtain a structure theorem which characterizes the positive free k-pluriharmonic functions on the regular polyball with operator-valued coefficients. We define the noncommutative Berezin (resp. Poisson) transform of a completely bounded linear map on C(S), the C-algebra generated by Si,j, and give necessary and sufficient conditions for a function to be the Poisson transform of a completely bounded (resp. completely positive) map. In the last section of the paper, we obtain Herglotz–Riesz representation theorems for free holomorphic functions on regular polyballs with positive real parts, extending the classical result as well as the Korányi–Pukánszky version in scalar polydisks.

Article information

Source
Anal. PDE, Volume 9, Number 5 (2016), 1185-1234.

Dates
Received: 3 December 2015
Revised: 29 February 2016
Accepted: 12 April 2016
First available in Project Euclid: 12 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513097071

Digital Object Identifier
doi:10.2140/apde.2016.9.1185

Mathematical Reviews number (MathSciNet)
MR3531370

Zentralblatt MATH identifier
1353.47007

Subjects
Primary: 47A13: Several-variable operator theory (spectral, Fredholm, etc.) 47A56: Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones)
Secondary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15] 46L52: Noncommutative function spaces

Keywords
noncommutative polyball Berezin transform Poisson transform Fock space multi-Toeplitz operator Naimark dilation completely bounded map pluriharmonic function free holomorphic function Herglotz–Riesz representation

Citation

Popescu, Gelu. Free pluriharmonic functions on noncommutative polyballs. Anal. PDE 9 (2016), no. 5, 1185--1234. doi:10.2140/apde.2016.9.1185. https://projecteuclid.org/euclid.apde/1513097071


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