Positivstellensatz\"e for noncommutative rational expressions

TL;DR

This paper extends Positivstellensatz"e to noncommutative rational expressions, providing algebraic certificates for positivity on polynomially convex sets, especially when the set is convex, leading to a 'perfect' Positivstellensatz.

Contribution

It generalizes Positivstellensatz"e from noncommutative polynomials to rational expressions, establishing conditions for positivity certificates on convex sets.

Findings

Positivity on polynomially convex sets implies existence of algebraic certificates.

Results are stronger under convexity assumptions, yielding a 'perfect' Positivstellensatz.

Provides a framework for certifying positivity of noncommutative rational expressions.

Abstract

We derive some Positivstellensatz\"e for noncommutative rational expressions from the Positivstellensatz\"e for noncommutative polynomials. Specifically, we show that if a noncommutative rational expression is positive on a polynomially convex set, then there is an algebraic certificate witnessing that fact. As in the case of noncommutative polynomials, our results are nicer when we additionally assume positivity on a convex set-- that is, we obtain a so-called "perfect Positivstellensatz" on convex sets.

Equations

$$\left[\begin{array}{cc}7i& 1000x_1{x}_2{x}_1-x_2^2\\ x_1^2+x_1{x}_2& 0\end{array}\right]$$

$${\left[\begin{array}{cc}7i& 1000x_1{x}_2{x}_1-x_2^2\\ x_1^2+x_1{x}_2& 0\end{array}\right]}^{\ast }=\left[\begin{array}{cc}-7i& x_1^2+x_2{x}_1\\ 1000x_1{x}_2{x}_1-x_2^2& 0\end{array}\right].$$

$${\mathcal{D}}_{\mathcal{P}}=\{X\in B(\mathcal{H}{)}^d|p(X)\ge 0,\forall p\in \mathcal{P},X_i=X_i^{\ast }\}.$$

$$q=\sum _{\text{finite}}{s}_i^{\ast }{s}_i+\sum _{\text{finite}}{r}_j^{\ast }{p}_j{r}_j$$

$$1,x_1{x}_1^{-1},1+x_2(8x_1^3{x}_2{x}_1+8{)}^{-1}.$$

$$q\equiv \sum _{\text{finite}}{s}_i^{\ast }{s}_i+\sum _{\text{finite}}{r}_j^{\ast }{p}_j{r}_j$$

$$\mathcal{O}=\mathcal{P}\cup \{\pm [1-u_j{g}_j(x){]}^{\ast }[1-u_j{g}_j(x)],\pm [1-g_j(x)u_j{]}^{\ast }[1-g_j(x)u_j]\}\cup \{D_j-u_j^{\ast }{u}_j\}$$

$$\hat{q}=\sum s_i^{\ast }{s}_i+\sum r_j^{\ast }{o}_j{r}_j$$

$$o_j(x,g)=D_j-[g_j(x{)}^{-1}{]}^{\ast }{g}_j(x{)}^{-1}=[g_j(x{)}^{-1}{]}^{\ast }[D_j{g}_j(x{)}^{\ast }{g}_j(x)-1]g_j(x{)}^{-1},$$

$$r\equiv \sum _{\text{finite}}{s}_i^{\ast }{s}_i+\sum _{\text{finite}}{r}_j^{\ast }L{r}_j$$

$${\mathcal{M}}_r=\{g_j(x)u_{j}b-b|g_j(x)u_{j}b\in {\tilde{\mathcal{C}}}_r\}.$$

$${\mathcal{Z}}_r=\{(X,U,v)|m(X,U)v=0,m\in {\mathcal{M}}_r,L(X)\ge 0\}.$$

$$⟨r(X)v,v⟩=⟨\tilde{r}(X,U)v,v⟩\ge 0,$$

$$\tilde{r}(x,u)=\sum {\tilde{s}}_i^{\ast }{\tilde{s}}_i+\sum {\tilde{r}}_j^{\ast }L{\tilde{r}}_j+\sum {\iota }_k^{\ast }{m}_k+m_k^{\ast }{\iota }_k$$

$$r(x)\equiv \sum s_i^{\ast }{s}_i+\sum r_j^{\ast }L{r}_j.$$

Full text

Research supported by NSF Mathematical Science Postdoctoral Research Fellowship DMS 1606260.

1. Introduction

We consider the positivity of noncommutative rational functions on polynomially convex sets. The theory on positive noncommutative polynomials has been well-studied [3, 6, 4], essentially inspired by the operator theoretic methods from the theory of positive (commutative) polynomials on polynomially convex sets originating in the work [10, 9]. We note that going from the polynomial to the rational case is less clear than in the noncommutative case because we cannot “clear denominators,” as it were.

A noncommutative polynomial (over $\mathbb{C}$) in $d$-variables is an element of the free associative algebra over $\mathbb{C}$ in the noncommuting letters $x_{1},\ldots,x_{d}.$ For example $1000x_{1}x_{2}x_{1}-x_{2}^{2}$ and $x_{1}^{2}+x_{1}x_{2}$ are noncommutative polynomials in two variables. A matricial noncommutative polynomial is a matrix with noncommutative polynomial entries. For example,

[TABLE]

is a matricial noncommutative polynomial. We define an involution $*$ on matricial noncommutative polynomials to be the involution which treats each $x_{i}$ as a self-adjoint variable. For example,

[TABLE]

We say a collection $\mathcal{P}$ of square matricial noncommutative polynomials is Archimedian if $\mathcal{P}$ contains elements of the form $C_{i}-x_{i}^{2}$ for some real numbers $C_{i}$ and each element of $\mathcal{P}$ is self-adjoint.

Let $\mathcal{H}$ be the infinite dimensional separable Hilbert space. For a self-adjoint operator $T$, we say $T\geq 0$ if $T$ is positive semidefinite, we say $T>0$ if $T$ is strictly positive definite in the sense that the spectrum of $T$ is contained in $(0,\infty)$. We define

[TABLE]

Previously, Helton and McCullough showed the following Positivstellensatz for matricial noncommutative polynomials.

Theorem 1.1** (Helton, McCullough [6]).**

Let $\mathcal{P}$ be an Archimedian collection of matricial noncommutative polynomials. Let $q$ be a square matricial noncommutative polynomial. If $q>0$ on $\mathcal{D}_{\mathcal{P}},$ then

[TABLE]

where $s_{i},r_{j}$ are all matricial noncommutative polynomials and $p_{j}\in\mathcal{P}.$

2. The rational positivstellensatz

A noncommutative rational expression is a syntactically correct expression involving $+,(,),^{-1}$ the letters $x_{1},\ldots,x_{d}$ and scalar numbers. We say two nondegenerate expressions are equivalent if they agree on the intersection of their domains. (Nondegeneracy means that the expression is defined for at least one input, or equivalently that the domain is a dense set with interior. That is, examples such as $0^{-1}$ are disallowed.) Examples of noncommutative rational expressions include

[TABLE]

We note that the first two are equivalent.

A matricial noncommutative rational expression is a matrix with noncommutative rational expression entries.

We show the following theorem.

Theorem 2.1**.**

Let $\mathcal{P}$ be an Archimedian collection of noncommutative polynomials. Let $q$ be a square matricial noncommutative rational expression defined on all of $\mathcal{D}_{\mathcal{P}}$. If the noncommutative rational expression $q>0$ on $\mathcal{D}_{\mathcal{P}},$ then

[TABLE]

where $s_{i},r_{j}$ are all matricial noncommutative rational expressions defined on $\mathcal{D}_{\mathcal{P}}$ and $p_{j}\in\mathcal{P}.$

Proof.

We let $g_{j}(x)$ be such that the term $g_{j}(x)^{-1}$ occurs in $q.$ The proof will go by strong induction on the number of such terms. Define

[TABLE]

where $D_{j}$ are positive real scalars chosen to be large enough so that $D_{j}-[g_{j}(x)^{-1}]^{*}g_{j}(x)^{-1}$ is positive on $\mathcal{D}_{\mathcal{P}}.$

We now define a self-adjoint noncommutative polynomial $\hat{q}(x,u)$ so that $\hat{q}(x,g)=q(x).$ Now $\hat{q}$ is a noncommutative polynomial in terms of $x_{i}$ and $u_{j}.$ Moreover, in terms of the $x_{i}$ and $u_{j},$ we see that $q(x,u)$ is positive on $\mathcal{D}_{\mathcal{O}},$ so by Theorem 1.1,

[TABLE]

for some $o_{j}\in\mathcal{O}.$ We now analyze each term of the form $t_{j}=r_{j}^{*}o_{j}r_{j}.$ We need to show that $t_{j}(x,g)$ is of the form (2.1). If $o_{j}\in\mathcal{P}$, we are fine. If $o_{j}=\pm[1-u_{j}g_{j}(x)]^{*}[1-u_{j}g_{j}(x)],$ we are also fine, since $t_{j}(x,g)=0,$ and similarly for the reversed case. If $o_{j}=D_{j}-[u_{j}]^{*}u_{j}$ we note that

[TABLE]

and since $D_{j}g_{j}(x)^{*}g_{j}(x)-1>0$ on $\mathcal{D}_{\mathcal{P}},$ by induction it is of the form (2.1), so we are done. ∎

We note that the same proof can be adapted for the hereditary case in [6]. Moreover, we note that this implies the Agler model theory for rational functions on polynomially convex sets established variously in [2, 1].

3. The convex perfect rational positivstellensatz

It is important to note that in Theorem 1.1 and Theorem 2.1, the complexity of the sum of squares representation is unbounded and we needed strict inequality. Specifically, in (2.1), the number of terms in each sum and the degree of each $s_{i}$ and $r_{j}$ are not bounded in the statement of the theorem. However, Helton, Klep and McCullough [4] showed that bounds do exist when we additionally assume that $\mathcal{D}_{\mathcal{P}}$ is convex and contains [math] and moreover that $\mathcal{P}$ consists of a single monic linear pencil, $L$, a self-adjoint linear matrix polynomial such that $L(0)$ is the identity. We note that for any finite set $\mathcal{P}$ of noncommutative polynomials such that $\mathcal{D}_{\mathcal{P}}$ is convex and contains [math], there exists such an $L$ [7].

Our goal is to prove the following:

Theorem 3.1**.**

Let $L$ be a monic linear pencil. Suppose $\mathcal{D}_{\{L\}}$ is convex. Let $r$ be a square matricial noncommutative rational expression defined on all of $\mathcal{D}_{\{L\}}$. The noncommutative rational expression $r\geq 0$ on all of $\mathcal{D}_{\{L\}}$ if and only if

[TABLE]

where $s_{i},r_{j}$ are all matricial noncommutative rational expressions defined on all of $\mathcal{D}_{\{L\}}.$

Proof.

Given an expression $r(x),$ we consider the expression $\tilde{r}(x,u)$ where each $g_{j}(x)^{-1}$ occurring in $r$ has been replaced by $u_{j}$ as in the proof of Theorem 2.1.

First we consider the minimal set $\mathcal{C}_{r}$ of rational expressions such that:

1. (1)

$ab\in\mathcal{C}_{r}\Rightarrow b\in\mathcal{C}_{r},$ 2. (2)

$(a+b)c\in\mathcal{C}_{r}\Rightarrow ac\in\mathcal{C}_{r},bc\in\mathcal{C}_{r},$ 3. (3)

$a+b\in\mathcal{C}\Rightarrow a\in\mathcal{C}_{r},b\in\mathcal{C}_{r},$ 4. (4)

$a^{-1}b\in\mathcal{C}_{r}\Rightarrow aa^{-1}b\in\mathcal{C}_{r}.$

From $\mathcal{C}_{r}$, form a set $\tilde{\mathcal{C}}_{r}$ by replacing each occurence of $g_{j}(x)^{-1}$ in elements of $\mathcal{C}_{r}$ with a new symbol $u_{j}.$ We define the set of $\mathcal{M}_{r}$ to be

[TABLE]

Define

[TABLE]

We note that for $(X,U,v)\in\mathcal{Z}_{r}$ and $\tilde{a}(x,u)\in\tilde{\mathcal{C}_{r}},$ one can show we have that $\tilde{a}(X,U)v=\tilde{a}(x,g(X)^{-1})v$ via a recursive argument. We see that $\tilde{r}(x,u)$ satisfies

[TABLE]

on $\mathcal{Z}_{r}$ since $\tilde{r}(X,U)v=r(X)v$ on $\mathcal{Z}_{r}$ by construction. Now, we apply the Helton-Klep-Nelson convex Positivstellensatz[5, Theorem 1.9], where the variety is given by $\mathcal{Z}_{r}$ and the convex set is $\{(X,U)|L(X)\geq 0\}$, to get that:

[TABLE]

where each $\iota_{k}$ is in the real radical of the ideal generated by the elements of $\mathcal{M}_{r}.$ That is, each $\iota_{k}(X,U)v$ vanishes on $\mathcal{Z}_{r}.$ So, substituting $g_{j}(x)^{-1}$ for $u_{j}$ we get that

[TABLE]

∎

We note that we could have proved a bit more: that on the variety $\mathcal{Z}_{r}$ that $\tilde{r}$ is positive and given by a sum of squares. This would essentially correspond to the so-called Moore-Penrose evaluation in [8]. Moreover, we note that the main result on positive rational functions, the noncommutative analogue of Artin’s solution to Hilbert’s seventeenth problem, that regular positive rational expressions are sums of squares[8], follows from our present theorem by taking an empty monic linear pencil, in fact, we obtain a slightly better matricial version of that result. Moreover, one has size bounds inherited from the Helton-Klep-Nelson convex Positivstellensatz [5], that is, checking that a noncommutative rational expression is effective using the algorithms given in[5].