Global Holomorphic Functions in Several Non-Commuting Variables II

Agler, Jim; McCarthy, John E.

doi:10.4153/cmb-2017-044-4

Cited by 28 publications

(82 citation statements)

References 34 publications

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“…The case where τ is the free topology will be of special interest here, and in this case we refer to τ -holomorphic functions as free (or freely) holomorphic functions. Such functions are particularly well behaved on account of the following theorem [1] (it is also proved in [3,6]…”

Section: Holomorphy With Respect To Admissible Topologiesmentioning

confidence: 91%

“…To prove Claim 1, first notice that by Proposition , it suffices to show that

G

is a free domain. In the light of equation and the fact that

S

is finite it will follow that

G

is a free domain if we can show that for each

w \in C

and each

C > 0

{x \in {double-struckM}^{1} | w \notin σ false(x false) and ∥ {(w - x)}^{- 1} ∥ < C} is a free domain .

But by [, Theorem 10.1] for each fixed

w \in C

g_{w} false(x false) = {(w - x)}^{- 1}

is a free holomorphic function on the free domain

{x \in {double-struckM}^{1} | w \notin σ false(x false)}

. As Proposition guarantees that

g_{w}

is freely continuous, it follows that

false{x \in {double-struckM}^{1} {| w \notin σ false(x false) and ∥ false(w - x false)}^{- 1} false∥ < C false} = g_{w}^{- 1} false(false{y \in {double-struckM}^{1} false| false∥ y false∥ < C false} false)

is a free domain.…”

Section: Free Holomorphic Functions In One Variablementioning

confidence: 99%

“…Replacing

δ (z)

t (δ (z) - δ (0))

, where

t ⩾ 1 / (1 - ∥ δ (0) ∥)

, we can assume that

δ (0) = 0

. By ,

F (z)

can be represented by a convergent series on

B_{δ}

whose terms are non‐commutative polynomials in the entries

δ_{i j} false(z false)

. If

F \circ π = f

, then

F false(u, v^{2}, u v u false) = 16 v u^{2} v .

Expand

F

in a power series in

double-struckD

, and group terms by homogeneity.…”

Section: Symmetric Free Holomorphic Functionsmentioning

confidence: 99%

“…Then U ∩ V is an nc-set and is a τ -neighborhood of x in D on which ϕ is a bounded nc-function. [1,Theorem 4.6] asserts that under these hypotheses ϕ is analytic on U ∩ V ∩ M d n . Thus, ϕ is analytic on some neighborhood of an arbitrary point of D ∩ M d n , and therefore ϕ is analytic on D ∩ M d n .…”

Section: Holomorphy With Respect To Admissible Topologiesmentioning

confidence: 99%

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Non‐commutative manifolds, the free square root and symmetric functions in two non‐commuting variables

Agler

McCarthy

Young

2018

Trans. London Math. Soc.

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The richly developed theory of complex manifolds plays important roles in our understanding of holomorphic functions in several complex variables. It is natural to consider manifolds that will play similar roles in the theory of holomorphic functions in several non‐commuting variables. In this paper we introduce the class of nc‐manifolds, the mathematical objects that at each point possess a neighborhood that has the structure of an nc‐domain in the d‐dimensional nc‐universe Md. We illustrate the use of such manifolds in free analysis through the construction of the non‐commutative Riemann surface for the matricial square root function. A second illustration is the construction of a non‐commutative analog of the elementary symmetric functions in two variables. For any symmetric domain in M2 we construct a two‐dimensional non‐commutative manifold such that the symmetric holomorphic functions on the domain are in bijective correspondence with the holomorphic functions on the manifold. We also derive a version of the classical Newton–Girard formulae for power sums of two non‐commuting variables.

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Section: Holomorphy With Respect To Admissible Topologiesmentioning

confidence: 91%

“…To prove Claim 1, first notice that by Proposition , it suffices to show that

G

is a free domain. In the light of equation and the fact that

S

is finite it will follow that

G

is a free domain if we can show that for each

w \in C

and each

C > 0

{x \in {double-struckM}^{1} | w \notin σ false(x false) and ∥ {(w - x)}^{- 1} ∥ < C} is a free domain .

But by [, Theorem 10.1] for each fixed

w \in C

g_{w} false(x false) = {(w - x)}^{- 1}

is a free holomorphic function on the free domain

{x \in {double-struckM}^{1} | w \notin σ false(x false)}

. As Proposition guarantees that

g_{w}

is freely continuous, it follows that

false{x \in {double-struckM}^{1} {| w \notin σ false(x false) and ∥ false(w - x false)}^{- 1} false∥ < C false} = g_{w}^{- 1} false(false{y \in {double-struckM}^{1} false| false∥ y false∥ < C false} false)

is a free domain.…”

Section: Free Holomorphic Functions In One Variablementioning

confidence: 99%

“…Replacing

δ (z)

t (δ (z) - δ (0))

, where

t ⩾ 1 / (1 - ∥ δ (0) ∥)

, we can assume that

δ (0) = 0

. By ,

F (z)

can be represented by a convergent series on

B_{δ}

whose terms are non‐commutative polynomials in the entries

δ_{i j} false(z false)

. If

F \circ π = f

, then

F false(u, v^{2}, u v u false) = 16 v u^{2} v .

Expand

F

in a power series in

double-struckD

, and group terms by homogeneity.…”

Section: Symmetric Free Holomorphic Functionsmentioning

confidence: 99%

Section: Holomorphy With Respect To Admissible Topologiesmentioning

confidence: 99%

See 2 more Smart Citations

Non‐commutative manifolds, the free square root and symmetric functions in two non‐commuting variables

Agler

McCarthy

Young

2018

Trans. London Math. Soc.

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“…The free topology has good polynomial approximation properties -not only is every free holomorphic function pointwise approximable by free polynomials, there is an Oka-Weil theorem which says that on sets of the form G ı , bounded free holomorphic functions are uniformly approximable on compact sets by free polynomials [2]. In contrast, neither the fine nor fat topology admit even pointwise polynomial approximation.…”

Section: -Holomorphic Functionsmentioning

confidence: 99%

Aspects of non-commutative function theory

Agler

McCarthy

2016

Concrete Operators

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Free Bianalytic Maps between Spectrahedra and Spectraballs in a Generic Setting

Augat

Helton

Klep

et al. 2019

Interpolation and Realization Theory With Applications to Control Theory

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Given a tuple E = (E 1 , . . . , E g ) of d×d matrices, the collection B E of those tuples of matrices X = (X 1 , . . . , X g ) (of the same size) such that E j ⊗ X j ≤ 1 is a spectraball. Likewise, given a tuple B = (B 1 , . . . , B g ) of e × e matrices the collection D B of tuples of matrices X = (X 1 , . . . , X g ) (of the same size) such thatj 0 is a free spectrahedron. Assuming E and B are irreducible, plus an additional mild hypothesis, there is a free bianalytic map p : B E → D B normalized by p(0) = 0 and p ′ (0) = I if and only if B E = B B and B spans an algebra. Moreover p is unique, rational and has an elegant algebraic representation.

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Global Holomorphic Functions in Several Non-Commuting Variables II

Abstract: We give a new proof that bounded non-commutative functions on polynomial polyhedra can be represented by a realization formula, a generalization of the transfer function realization formula for bounded analytic functions on the unit disk.

Cited by 28 publications

References 34 publications

Non‐commutative manifolds, the free square root and symmetric functions in two non‐commuting variables

Non‐commutative manifolds, the free square root and symmetric functions in two non‐commuting variables

Aspects of non-commutative function theory

Free Bianalytic Maps between Spectrahedra and Spectraballs in a Generic Setting

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