Realizations of non-commutative rational functions around a matrix centre, I: synthesis, minimal realizations and evaluation on stably finite algebras

Porat, Motke; Vinnikov, Victor

doi:10.48550/arxiv.1905.11304

Cited by 3 publications

(10 citation statements)

References 57 publications

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“…Classically, the inner factor of any polynomial in D is a finite Blaschke product, and hence a rational analytic function with poles outside of the open disk. Rational functions have been studied extensively in the NC setting by several authors [55,28,56,25,43,17].…”

Section: Discussionmentioning

confidence: 99%

Blaschke-Singular-Outer factorization of free non-commutative functions

Jury¹,

W.²,

Shamovich³

2020

Preprint

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By classical results of Herglotz and F. Riesz, any bounded analytic function in the complex unit disk has a unique inner-outer factorization. Here, a bounded analytic function is called inner or outer if multiplication by this function defines an isometry or has dense range, respectively, as a linear operator on the Hardy Space, H 2 , of analytic functions in the complex unit disk with square-summable Taylor series. This factorization can be further refined; any inner function θ decomposes uniquely as the product of a Blaschke inner function and a singular inner function, where the Blaschke inner contains all the vanishing information of θ, and the singular inner factor has no zeroes in the unit disk.We prove an exact analogue of this factorization in the context of the full Fock space, identified as the Non-commutative Hardy Space of analytic functions defined in a certain multi-variable non-commutative open unit ball.

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Section: Discussionmentioning

confidence: 99%

Blaschke-Singular-Outer factorization of free non-commutative functions

Jury¹,

W.²,

Shamovich³

2020

Preprint

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“…A realization theory for such expressions (and hence functions) is required in particular for all of the applications mentioned above. Such a theory is presented in our first paper [64] and continues here, using the ideas of the general theory of nc functions. Other types of realizations of nc rational functions that are not necessarily regular at 0 have been considered (see [26,27], [83], and also the recent papers [69,70,71,72]).…”

Section: Introductionmentioning

confidence: 97%

“…This is the second in a series of papers with the goal of generalizing the theory of Fornasini-Marchesini realizations centred at 0 (or any other scalar point), to the case of Fornasini-Marchesini realizations centred at an arbitrary matrix point in the domain of regularity of a nc rational function. In the first paper of the series ( [64]), we proved the following main result: Theorem 2 ([64], Corollary 2.18 and Theorem 3.3). If R is a nc rational function of x 1 , .…”

Section: Introductionmentioning

confidence: 99%

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Realizations of non-commutative rational functions around a matrix centre, II: The lost-abbey conditions

Porat¹,

Vinnikov²

2020

Preprint

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In a previous paper the authors generalized classical results of minimal realizations of non-commutative (nc) rational functions, using nc Fornasini-Marchesini realizations which are centred at an arbitrary matrix point. In particular, it was proved that the domain of regularity of a nc rational function is contained in the invertibility set of a corresponding pencil of any minimal realization of the function. In this paper we prove an equality between the domain of a nc rational function and the domain of any of its minimal realizations. As for evaluations over stably finite algebras, we show that the domain of the realization w.r.t any such algebra coincides with the so called matrix domain of the function w.r.t the algebra. As a corollary we show that the domain of regularity and the stable extended domain coincide. In contrary to both the classical case and the scalar case -where every matrix coefficients which satisfy the controllability and observability conditions can appear in a minimal realization of a nc rational function -the matrix coefficients in our case have to satisfy certain equations, called linearized lost-abbey conditions, which are related to Taylor-Taylor expansions in nc function theory. Contents 24 4.1. Linearized lost-abbey conditions w.r.t algebras 24 4.2. Conclusions and main results 29 4.3. Evaluations over algebras 33 5. Stable Extended Domain 37 References 41

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“…where ⊗ is Kronecker's product and E ı ∈ M n (C) are the standard matrix units. For applications of related ideas to noncommutative rational functions see [Vol18,PV].…”

Section: Introductionmentioning

confidence: 99%

Factorization of noncommutative polynomials and Nullstellensätze for the free algebra

Helton,

Klep,

Volčič

2019

Preprint

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This article gives a class of Nullstellensätze for noncommutative polynomials. The singularity set of a noncommutative polynomial f = f (x 1 , . . . , x g ) isThe first main theorem of this article shows that the irreducible factors of f are in a natural bijective correspondence with irreducible components of Z n (f ) for every sufficiently large n.With each polynomial h in x and x * one also associates its real singularity set Z re (h) = {X : det h(X, X * ) = 0}. A polynomial f which depends on x alone (no x * variables) will be called analytic. The main Nullstellensatz proved here is as follows: for analytic f but for h dependent on possibly both x and x * , the containment Z (f ) ⊆ Z re (h) is equivalent to each factor of f being "stably associated" to a factor of h or of h * .For perspective, classical Hilbert type Nullstellensätze typically apply only to analytic polynomials f, h, while real Nullstellensätze typically require adjusting the functions by sums of squares of polynomials (sos). Since the above "algebraic certificate" does not involve a sos, it seems justified to think of this as the natural determinantal Hilbert Nullstellensatz. An earlier paper of the authors (Adv. Math. 331 (2018) 589-626) obtained such a theorem for special classes of analytic polynomials f and h. This paper requires few hypotheses and hopefully brings this type of Nullstellensatz to near final form.Finally, the paper gives a Nullstellensatz for zeros V(f ) = {X : f (X, X * ) = 0} of a hermitian polynomial f , leading to a strong Positivstellensatz for quadratic free semialgebraic sets by the use of a slack variable.

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Realizations of non-commutative rational functions around a matrix centre, I: synthesis, minimal realizations and evaluation on stably finite algebras

Cited by 3 publications

References 57 publications

Blaschke-Singular-Outer factorization of free non-commutative functions

Blaschke-Singular-Outer factorization of free non-commutative functions

Realizations of non-commutative rational functions around a matrix centre, II: The lost-abbey conditions

Factorization of noncommutative polynomials and Nullstellensätze for the free algebra

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