Free holomorphic functions and interpolation

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• 作者：Gelu Popescu
• 关键词：Mathematics Subject Classification (2000) Primary 47A57 ; 47A56 ; 47A13 ; Secondary 46L52 ; 46T25
• 刊名：Mathematische Annalen
• 出版年：2008
• 出版时间：September, 2008
• 年：2008
• 卷：342
• 期：1
• 页码：1-30
• 全文大小：343.5 KB

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In this paper we obtain a noncommutative multivariable analogue of the classical Nevanlinna–Pick interpolation problem for analytic functions with positive real parts on the open unit disc. Given a function f : L ? \mathbb Cf : \Lambda \to \mathbb {C} , where L\Lambda is an arbitrary subset of the open unit ball \mathbbB_n:={z ? \mathbb Cn: ||z|| < 1}\mathbb{B}_n:=\{z\in \mathbb {C}^n: \|z\| , we find necessary and sufficient conditions for the existence of a free holomorphic function g with complex coefficients on the noncommutative open unit ball [B(H)n]₁[B({\mathcal H})^n]_1 such that Re g 3 0 and g(z)=f(z), z ? L,{\rm Re} \ g \geq 0 \quad {\rm and} \quad g(z)=f(z),\quad z\in \Lambda,