Free pluriharmonic majorants and commutant lifting

In this paper we initiate the study of sub-pluriharmonic curves and free pluriharmonic majorants on the noncommutative open ball[B(H)n]1:={(X1,…,Xn)∈B(H)n:‖X1X1∗+⋯+XnXn∗‖1/2<1}, where B(H)B(H) is the algebra of all bounded linear operators on a Hilbert space HH. Several classical results from complex analysis have analogues in this noncommutative multivariable setting.We present basic properties for sub-pluriharmonic curves, characterize the class of sub-pluriharmonic curves that admit free pluriharmonic majorants and find, in this case, the least free pluriharmonic majorants. We show that, for any free holomorphic function Θ   on [Bn(H)]1[B(H)n]1, the mapφ:[0,1)→C∗(R1,…,Rn),φ(r):=Θ(rR1,…,rRn)∗Θ(rR1,…,rRn), is a sub-pluriharmonic curve in the Cuntz–Toeplitz algebra generated by the right creation operators R1,…,RnR1,…,Rn on the full Fock space with n generators. We prove that Θ   is in the noncommutative Hardy space Hball2 if and only if φ has a free pluriharmonic majorant. In this case, we find Herglotz–Riesz and Poisson type representations for the least pluriharmonic majorant of φ  . Moreover, we obtain a characterization of the unit ball of Hball2 and provide a parametrization and concrete representations for all free pluriharmonic majorants of φ, when Θ   is in the unit ball of Hball2.In the second part of this paper, we introduce a generalized noncommutative commutant lifting (GNCL) problem which extends, to our noncommutative multivariable setting, several lifting problems including the classical Sz.-Nagy–Foiaş commutant lifting problem and the extensions obtained by Treil–Volberg, Foiaş–Frazho–Kaashoek, and Biswas–Foiaş–Frazho, as well as their multivariable noncommutative versions. We solve the GNCL problem and, using the results regarding sub-pluriharmonic curves and free pluriharmonic majorants on noncommutative balls, we provide a complete description of all solutions. In particular, we obtain a concrete Schur type description of all solutions in the noncommutative commutant lifting theorem.