Free holomorphic functions and interpolation

By Gelu Popescu. 2007

61 citations recorded by the source corpus.

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 $$f : \Lambda \to \mathbb {C}$$ , where $$\Lambda$$ is an arbitrary subset of the open unit ball $$\mathbb{B}_n:=\{z\in \mathbb {C}^n: \|z\| < 1\}$$ , we find necessary and sufficient conditions for the existence of a free holomorphic function g with complex coefficients on the noncommutative open unit ball $$[B({\mathcal H})^n]_1$$ such that$${\rm Re} \ g \geq 0 \quad {\rm and} \quad g(z)=f(z),\quad z\in \Lambda,$$where $$B({\mathcal H})$$ is the algebra of all bounded linear operators on a Hilbert space $${\mathcal H}$$ . The proof employs several results from noncommutative multivariable operator theory and a noncommutative Cayley transform (introduced and studied in the present paper) acting from the set of all free holomorphic functions with positive real parts to the set of all bounded free holomorphic functions. All the results of this paper are obtained in the more general setting of free holomorphic functions with operator-valued coefficients. As consequences, we deduce some results concerning operator-valued analytic interpolation on the unit ball $${\mathbb B}_n$$.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/98d100bf678e86ba7fbfdf90b6bdbc13fcbb1b54

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In Applied Mathematics

Citations in the corpus

Free holomorphic automorphisms of the unit ball of B(ℋ) n (2008, background): “We mention that the results from [27], [31], and [32] hold true in the context of free holomorphic (resp. pluriharmonic) functions with operator-valued coefficients.”
Free holomorphic functions on the unit ball of B(H)n (2006, methodology): “According to [39], the noncommutative Cayley transform defined by C[F ] := [F − 1][1 + F ]−1 is a bijection between H+ ball(B(E )) and the unit ball of [H∞ ball(B(E ))]inv.”
Pick Interpolation for free holomorphic functions (2013, background): “functions have been studied for a variety of reasons: by Taylor [25], in thecontext ofthefunctionalcalculus fornon-commutingoperators; Voiculescu [26, 27], in the context of free probability; Popescu [19, 20, 21, 22], in the context of extending classical function theory to d-tuples of bounded opera”
The implicit function theorem and free algebraic sets (2014, unknown): “Nc-functions have been studied for a variety of reasons: by Anderson [4] as a generalization of the Weyl calculus; by Taylor [20], in the context of the functional calculus for non-commuting operators; Popescu [14, 15, 16, 17], in the context of extending classical function theory to d-tuples of bou”
Global Holomorphic Functions in Several Noncommuting Variables (2015, unknown): “This has been done in a variety of ways: by Taylor [23], in the context of the functional calculus for non-commuting operators; Voiculescu [24, 25], in the context of free probability; Popescu [18, 19, 20, 21], in the context of extending classical function theory to d-tuples Partially supported by ”