On domains of noncommutative rational functions

Volčič, Jurij

doi:10.1016/j.laa.2016.11.031

Cited by 24 publications

(39 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proposition 6.4 below is a variant of the main result of [HM14]. Taking advantage of recent advances in our understanding of the singularities of free rational functions (e.g., [Vol17]), the proof given here is rather short, compared to that of the similar result in [HM14]. Proposition 6.…”

Section: Convex Sets Defined By Rational Functionsmentioning

confidence: 84%

“…Free rational functions regular at 0 are determined by their evaluations near 0; that is if f (X) = g(X) in some neighborhood of 0 in dom(f ) ∩ dom(g), then f = g. In what follows, we often omit regular at 0 when it is understood from context. We refer the reader to [Vol17,KVV09] for a fuller discussion of the domain of a free rational function.…”

Section: Free Rational Functions and Mappingsmentioning

confidence: 99%

“…[Vol17], and also[KVV09], for full details.) By Lemma 2.3, D contains D J , (as X ∈ D J implies I + Λ J (X) is invertible).…”

mentioning

confidence: 99%

“…From[KVV09,Vol17] the free rational function s has an observable and controllable realization. By[HMV06], since s is symmetric, this realization can be symmetrized.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Bianalytic free maps between spectrahedra and spectraballs

Helton

Klep

McCullough

et al. 2020

Journal of Functional Analysis

Self Cite

View full text Add to dashboard Cite

Linear matrix inequalities (LMIs) are ubiquitous in real algebraic geometry, semidefinite programming, control theory and signal processing. LMIs with (dimension free) matrix unknowns are central to the theories of completely positive maps and operator algebras, operator systems and spaces, and serve as the paradigm for matrix convex sets. The matricial feasibility set of an LMI is called a free spectrahedron.In this article, the bianalytic maps between a very general class of ball-like free spectrahedra (examples of which include row or column contractions, and tuples of contractions) and arbitrary free spectrahedra are characterized and seen to have an elegant algebraic form. They are all highly structured rational maps. In the case that both the domain and codomain are ball-like, these bianalytic maps are explicitly determined and the article gives necessary and sufficient conditions for the existence of such a map with a specified value and derivative at a point. In particular, this result leads to a classification of automorphism groups of ball-like free spectrahedra. The proofs depend on a novel free Nullstellensatz, established only after new tools in free analysis are developed and applied to obtain fine detail, geometric in nature locally and algebraic in nature globally, about the boundary of ball-like free spectrahedra.

show abstract

Section: Convex Sets Defined By Rational Functionsmentioning

confidence: 84%

Section: Free Rational Functions and Mappingsmentioning

confidence: 99%

See 2 more Smart Citations

Bianalytic free maps between spectrahedra and spectraballs

Helton

Klep

McCullough

et al. 2020

Journal of Functional Analysis

Self Cite

View full text Add to dashboard Cite

show abstract

“…Such expressions are called equivalent, and a non‐commutative rational function is formally an equivalence class of non‐degenerate non‐commutative rational expressions. See for example for a discussion. For the expressions

P_{- n}

it is easy to see that they are non‐degenerate; from equation we see that the functions can be evaluated as long as all four of the expressions

α - β γ^{- 1} β, β - γ β^{- 1} α, β - α β^{- 1} γ, γ - β α^{- 1} β

are invertible.…”

Section: Nc Newton‐girard Formulaementioning

confidence: 99%

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

Agler

McCarthy

Young

2018

Trans. London Math. Soc.

View full text Add to dashboard Cite

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.

show abstract

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

View full text Add to dashboard Cite

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.

show abstract

On domains of noncommutative rational functions

Cited by 24 publications

References 20 publications

Bianalytic free maps between spectrahedra and spectraballs

Bianalytic free maps between spectrahedra and spectraballs

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

Free Bianalytic Maps between Spectrahedra and Spectraballs in a Generic Setting

Contact Info

Product

Resources

About