Free holomorphic functions and interpolation

Free Holomorphic Functions and Interpolation

In this paper we obtain a noncommutative multivariable analogue of the classical NevanlinnaPick interpolation problem for analytic functions with positive real parts on the open unit disc. Given a function f : Λ → C, where Λ is an arbitrary subset of the open unit ball Bn := {z ∈ Cn : ‖z‖ < 1}, we find necessary and sufficient conditions for the existence of a free holomorphic function g with c...

Pick Interpolation for free holomorphic functions

Monotonicity and holomorphic functions

This is a survey on some recent works, mainly by the author on the relation between holomorphic functions on Kähler manifolds, monotonicity and the geometry of complex manifolds. We also use this opportunity to give details of a sketched step in the proof of a previously established result. Math Subject Classifications. Primary: 58G11, 53C44.

Free Holomorphic Functions on the Unit Ball of B(h)

1. Free holomorphic functions and Hausdorff derivations 2. Cauchy, Liouville, and Schwartz type results for free holomorphic functions 3. Algebras of free holomorphic functions 4. Free analytic functional calculus and noncommutative Cauchy transforms 5. Weierstrass and Montel theorems for free holomorphic functions 6. Free pluriharmonic functions and noncommutative Poisson transforms 7. Hardy s...

Free Holomorphic Functions on the Unit Ball Of

In this paper we continue the study of free holomorphic functions on the noncommutative ball [B(H)]1 := n (X1, . . . , Xn) ∈ B(H) n : ‖X1X ∗ 1 + · · ·+ XnX ∗ n‖ 1/2 < 1 o , where B(H) is the algebra of all bounded linear operators on a Hilbert space H, and n = 1, 2, . . . or n = ∞. Several classical results from complex analysis have free analogues in our noncommutative setting. We prove a maxi...