Free holomorphic functions on the unit ball of B(H)n

By Gelu Popescu. 2006

161 citations recorded by the source corpus.

Abstract In this paper we continue the study of free holomorphic functions on the noncommutative ball [ B ( H ) n ] 1 : = { ( X 1 , … , X n ) ∈ B ( H ) n : ‖ X 1 X 1 ∗ + ⋯ + X n X n ∗ ‖ 1 / 2 1 } , 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 maximum principle, a Naimark type representation theorem, and a Vitali convergence theorem, for free holomorphic functions with operator-valued coefficients. We introduce the class of free holomorphic functions with the radial infimum property and study it in connection with factorizations and noncommutative generalizations of some classical inequalities obtained by Schwarz and Harnack. The Borel–Caratheodory theorem is extended to our noncommutative setting. Using a noncommutative generalization of Schwarz's lemma and basic facts concerning the free holomorphic automorphisms of the noncommutative ball [ B ( H ) n ] 1 , we obtain an analogue of Julia's lemma for free holomorphic functions F : [ B ( H ) n ] 1 → [ B ( H ) m ] 1 . We also obtain Pick–Julia theorems for free holomorphic functions with operator-valued coefficients. We provide a noncommutative generalization of a classical inequality due to Lindelof, which turns out to be sharper then the noncommutative von Neumann inequality. Finally, we introduce a pseudohyperbolic metric on [ B ( H ) n ] 1 which is invariant under the action of the free holomorphic automorphism group of [ B ( H ) n ] 1 and turns out to be a noncommutative extension of the pseudohyperbolic distance on B n , the open unit ball of C n . In this setting, we obtain a Schwarz–Pick type lemma. We also provide commutative versions of these results for operator-valued multipliers of the Drury–Arveson space.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/add475961d4de696499d43986288c68e37a9f08a

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Citations in the corpus

Free holomorphic functions and interpolation (2007, background): “Due to [39], [41], there is a unique g ∈ [H(B(H)1 )]≤1 having the boundary function G, i.”
Foundations of Free Noncommutative Function Theory (2012, methodology): “[89] G. Popescu.”
The implicit function theorem and free algebraic sets (2014, unknown): “…[4] as a generalization of the Weyl calculus; by Taylor [20] in the context of the functional calculus for non-commuting operators; Popescu [14–17] in the context of extending classical function theory to d -tuples of bounded operators; Ball, Groenewald and Malakorn [5] in the context of…”
Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball (2017, methodology): “Popescu’s observation in [67] after equation (3.”
Operator Theory on Noncommutative Domains (2007, unknown): “In [87], we developed a theory of holomorphic functions in several noncommuting (free) variables and provide a framework for the study of arbitrary n-tuples of operators.”
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 ”
Free holomorphic automorphisms of the unit ball of B(ℋ) n (2008, background): “In [27], [31], and [32] we developed a theory of free holomorphic (resp.”
Pick Interpolation for free holomorphic functions (2013, background): “An nc-function is a graded function φ defined on a set D ⊆ M[d] such that i) If x, y, x⊕ y ∈ D, then φ(x⊕ y) = φ(x)⊕ φ(y). ii) If s ∈ Mn is invertible and x, s −1xs ∈ D ∩ Mdn then φ(s −1xs) = s−1φ(x)s. ∗Partially supported by National Science Foundation Grant DMS 1068830 †Partially supported by Nati”