Representation of free Herglotz functions

Pascoe, J. E.; Passer, Benjamin; Tully‐Doyle, Ryan

doi:10.1512/iumj.2019.68.7730

Cited by 13 publications

(10 citation statements)

References 28 publications

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“…(ii) Study of rational functions of non-commuting variables in general and positive real is praticular, has been flourishing recently, as sample references see e.g. [1], [7], [8], [9], [33], [34], [35], [30]. Nevertheless many properties of these functions are yet to be explored.…”

Section: Future Workmentioning

confidence: 99%

Composition of rational functions: State-space realization and applications

Alpay

Lewkowicz

2019

Linear Algebra and its Applications

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We define two versions of compositions of matrix-valued rational functions of appropriate sizes and whenever analytic at infinity, offer a set of formulas for the corresponding state-space realization, in terms of the realizations of the original functions. Focusing on positive real functions, the first composition is applied to electrical circuits theory along with introducing a connection to networks of feedback loops. The second composition is applied to Stieltjes functions.AMS Classification: 08A02 26C15 37F10 47B33 47N70 94C05

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Section: Future Workmentioning

confidence: 99%

Composition of rational functions: State-space realization and applications

Alpay

Lewkowicz

2019

Linear Algebra and its Applications

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“…The following construction is motivated by the dilation theory introduced by A. Frazho [6], [7] and G. Popescu [12], [13] and was privately communicated to the author by J. E. Pascoe. Similar ideas in a different setting were utilized in [11].…”

Section: Strong Nc Functions and The Shift Formmentioning

confidence: 99%

Inverse and Implicit Function Theorems for Noncommutative Functions on Operator Domains

Mancuso¹

2018

Preprint

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Classically, a noncommutative function is defined on a graded domain of tuples of square matrices. In this note, we introduce a notion of a noncommutative function defined on a domain Ω ⊂ B(H) d , where H is an infinite dimensional Hilbert space. Inverse and implicit function theorems in this setting are established. When these operatorial noncommutative functions are suitably continuous in the strong operator topology, a noncommutative dilation-theoretic construction is used to show that the assumptions on their derivatives may be relaxed from boundedness below to injectivity.

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“…Agler [1] developed this representation theory for an appropriate class in the setting of several variables based on operator theory. Recently, Pascoe, Passer and Tully-Doyle [13] proved an nc analog of Herglotz representation theorem. Motivated by these works, we will give a polynomial approximation type characterization of regular free Herglotz functions.…”

Section: Introductionmentioning

confidence: 99%

A polynomial approximation result for free Herglotz-Agler functions

Kojin¹

2022

Preprint

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In this paper, we prove a noncommutative (nc for short) analog of Schwarz lemma for the nc Schur-Agler class and prove that the regular nc Schur-Agler class and the regular free Herglotz class are homeomorphic. Moreover, we give a characterization of regular free Herglotz functions. As an application, we will show that any regular free Herglotz functions can uniformly be approximated by regular Herglotz free polynomials.

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Representation of free Herglotz functions

Cited by 13 publications

References 28 publications

Composition of rational functions: State-space realization and applications

Composition of rational functions: State-space realization and applications

Inverse and Implicit Function Theorems for Noncommutative Functions on Operator Domains

A polynomial approximation result for free Herglotz-Agler functions

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