Noncommutative Partial Convexity Via $$\Gamma $$-Convexity

Jury, Michael T.; Klep, Igor; Mancuso, Mark E.; McCullough, Scott; Pascoe, J. E.

doi:10.1007/s12220-020-00387-1

Cited by 5 publications

(6 citation statements)

References 22 publications

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“…Such a pair ..X; Y /; V / is called an xy-pair. Sublevel sets of xy-convex functions are delineated by (perhaps infinitely many) BMIs, as proved in [32].…”

Section: Resultsmentioning

confidence: 95%

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Noncommutative partially convex rational functions

et al. 2021

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Motivated by classical notions of bilinear matrix inequalities (BMIs) and partial convexity, this article investigates partial convexity for noncommutative functions. It is shown that noncommutative rational functions that are partially convex admit novel butterfly-type realizations that necessitate square roots. A strengthening of partial convexity arising in connection with BMIs -xy-convexity -is also considered. A characterization of xy-convex polynomials is given.

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“…Such a pair ..X; Y /; V / is called an xy-pair. Sublevel sets of xy-convex functions are delineated by (perhaps infinitely many) BMIs, as proved in [32].…”

Section: Resultsmentioning

confidence: 95%

“…The notions of partial convexity and xy-convexity are two instantiations of -convexity [32]. Let D Â S h S g be a given free open set that is also closed with respect to restrictions to reducing subspaces; that is, if .A; X / 2 D and V is an isometry whose range reduces each A j and X k , then…”

Section: Resultsmentioning

confidence: 99%

Noncommutative partially convex rational functions

et al. 2021

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“…An aim of operatorial free function theory is to understand functional calculus for several noncommuting operators and its applications. (See e. g. [4,5,32,15,29,26].) We will see that invariant structure preserving functions are in the pointwise weak closure of the polynomials and therefore free.…”

Section: Case Study: Weak Algebraicity In Noncommutative Function Theorymentioning

confidence: 90%

Invariant structure preserving functions and an Oka-Weil Kaplansky density type theorem

Pascoe

2021

Preprint

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We develop the theory of invariant structure preserving and free functions on a general structured topological space. We show that an invariant structure preserving function is pointwise approximiable by the appropriate analog of polynomials in the strong topology and therefore a free function. Moreover, if a domain of operators on a Hilbert space is polynomially convex, the set of free functions satisfies a Oka-Weil Kaplansky density type theorem-contractive functions can be approximated by contractive polynomials.

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“…In that paper, the authors relied on the additional assumption that the operator NC function was sequentially continuous in the strong operator topology (SOT). Structure in the SOT in the context of operatorial noncommutative analysis has also been studied in [10,12]. The main result in [2] on algebraicity was recently refined by Augat and McCarthy in [7] who were able to omit the sequential SOT-continuity hypothesis.…”

Section: Preliminariesmentioning

confidence: 99%

“…Recently [2,7,10,12], it has become beneficial (and interesting in its own right) to consider a completion of the aforementioned matricial setting. By this we mean that we want to study noncommutative functions that are defined on domains Σ ⊆ B(H) d , where H is an infinite-dimensional separable Hilbert space.…”

Section: Introductionmentioning

confidence: 99%

Noncommutative functions are weakly algebraic on operatorial polynomial polyhedra

Mancuso

2021

Preprint

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An operatorial polynomial polyhedron is a set of the formwhere B(H) denotes the space of bounded operators on a separable Hilbert space H, and δ is a matrix of polynomials in d noncommuting variables. These sets appear throughout the literature on noncommutative function theory. While much of what has been written involves matricial polynomial polyhedra, there do exist δ such that the associated B δ (B(H)) is non-empty but contains no matrix points. Algebraicity of operatorial noncommutative functions has been established in the case that the domain B δ (B(H)) is a balanced set (hence contains the matrix point 0). In this paper, we dispense of such assumptions on the domain and prove that an operatorial noncommutative function on any B δ (B(H)) is weakly algebraic in the sense that its value at each operator tuple Z lies in the weak operator topology closure of the unital algebra generated by the coordinates of Z.

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Noncommutative Partial Convexity Via $$\Gamma $$-Convexity

Cited by 5 publications

References 22 publications

Noncommutative partially convex rational functions

Noncommutative partially convex rational functions

Invariant structure preserving functions and an Oka-Weil Kaplansky density type theorem

Noncommutative functions are weakly algebraic on operatorial polynomial polyhedra

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