Foundations of Free Noncommutative Function Theory

Kaliuzhnyi-Verbovetskyi, Dmitry S.; Vinnikov, Victor

doi:10.1090/surv/199

Cited by 136 publications

(225 citation statements)

References 63 publications

Supporting

Mentioning

214

Contrasting

Order By: Relevance

“…Noncommutative functions and transforms. In this subsection we largely follow [6,21] in describing the noncommutative extensions of the analytic transforms introduced in the previous subsection, and [18] in the definition of noncommutative sets and functions. We refer to these three articles and [28,29] for details on, and proofs of, the statements below.…”

Section: Noncommutative Functions Distributions and Convolutionsmentioning

confidence: 99%

See 1 more Smart Citation

Some Geometric Properties of the Subordination Function Associated to an Operator-Valued Free Convolution Semigroup

Belinschi

2017

Complex Anal. Oper. Theory

View full text Add to dashboard Cite

Abstract. In his article On the free convolution with a semicircular distribution, Biane found very useful characterizations of the boundary values of the imaginary part of the Cauchy-Stieltjes transform of the free additive convolution of a probability measure on R with a Wigner (semicircular) distribution. Biane's methods were recently extended by Huang to measures which belong to the partial free convolution semigroups introduced by Nica and Speicher. This note further extends some of Biane's methods and results to free convolution powers of operator-valued distributions and to free convolutions with operator-valued semicirculars. In addition, it investigates properties of the Julia-Carathéodory derivative of the subordination functions associated to such semigroups, extending results from [7].

show abstract

Section: Noncommutative Functions Distributions and Convolutionsmentioning

confidence: 99%

“…A remarkable result (see [18,Theorem 7.2]) states that, under very mild conditions on Ω, local boundedness for f implies each f n is analytic as a map between Banach spaces. As mentioned in the previous section, the function G µ encodes only the symmetric part of the distribution µ.…”

Section: Noncommutative Functions Distributions and Convolutionsmentioning

confidence: 99%

Some Geometric Properties of the Subordination Function Associated to an Operator-Valued Free Convolution Semigroup

Belinschi

2017

Complex Anal. Oper. Theory

View full text Add to dashboard Cite

show abstract

“…Related notions. Noncommutative polynomials and noncommutative rational functions arise in several contexts including systems theory [BGM06,BGM05] and is a part of the new field of free (freely noncommutative) analysis [SV06,KV+,AM+]. Sums of squares representations are a theme in both the commutative and noncommutative settings [Pu13].…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…Trace-convexity is a notion frequently used in free probability and free analysis [SV06,KV+], where e.g. the trace of certain potentials is assumed to be convex; see [Gui06,Car10,GS09] or the references therein.…”

Section: Introductionmentioning

confidence: 99%

On trace-convex noncommutative polynomials

Klep¹,

McCullough²,

Nelson³

2016

Michigan Math. J.

View full text Add to dashboard Cite

Abstract. To each continuous function f : R → R there is an associated trace function on n × n real symmetric matrices Tr f . The classical Klein lemma states that f is convex if and only if Tr f is convex. In this note we present an algebraic strengthening of this lemma for univariate polynomials f : Tr f is convex if and only if the noncommutative second directional derivative of f is a sum of hermitian squares and commutators in the free algebra. We also give a localized version of this result.

show abstract

“…In contexts when E is finite dimensional, matricial sets are essentially the noncommutative sets that Bill Helton, Igor Klep, Scott McCullough and others study in the setting of linear matrix inequalities [HKM11b,HKM11a,HKMS09]. They are also closely connected to the noncommutative sets in the work of Dmitry Kaliuzhnyȋ-Verbovetskyȋ and Victor Vinnikov devoted to noncommutative function theory [KVV09, KV10,KVV]. There is another property that the discs also enjoy, viz.…”

Section: Introductionmentioning

confidence: 99%

Tensorial Function Theory: From Berezin Transforms to Taylor’s Taylor Series and Back

Muhly¹,

Solel²

2013

Integr. Equ. Oper. Theory

View full text Add to dashboard Cite

Abstract. Let H ∞ (E) be the Hardy algebra of a W * -correspondence E over a W * -algebra M . Then the ultraweakly continuous completely contractive representations of H ∞ (E) are parametrized by certain sets AC(σ) indexed by N Rep(M ) -the normal * -representations σ of M . Each set AC(σ) has analytic structure, and each element F ∈ H ∞ (E) gives rise to an analytic operator-valued function F σ on AC(σ) that we call the σ-Berezin transform of F . The sets {AC(σ)} σ∈Σ and the family of functions { F σ } σ∈Σ exhibit "matricial structure" that was introduced by Joeseph Taylor in his work on noncommutative spectral theory in the early 1970s. Such structure has been exploited more recently in other areas of free analysis and in the theory of linear matrix inequalities. Our objective here is to determine the extent to which the matricial structure characterizes the Berezin transforms.

show abstract

Foundations of Free Noncommutative Function Theory

Cited by 136 publications

References 63 publications

Some Geometric Properties of the Subordination Function Associated to an Operator-Valued Free Convolution Semigroup

Some Geometric Properties of the Subordination Function Associated to an Operator-Valued Free Convolution Semigroup

On trace-convex noncommutative polynomials

Tensorial Function Theory: From Berezin Transforms to Taylor’s Taylor Series and Back

Contact Info

Product

Resources

About