Universität Konstanz
The convex Positivstellensatz in a free algebra
J. William Helton Igor Klep Scott McCullough
Konstanzer Schriften in Mathematik Nr. 288, August 2011
ISSN 1430-3558
© Fachbereich Mathematik und Statistik Universität Konstanz
Fach D 197, 78457 Konstanz, Germany
Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-152850
THE CONVEX POSITIVSTELLENSATZ IN A FREE ALGEBRA
J. WILLIAM HELTON1, IGOR KLEP2, AND SCOTT MCCULLOUGH3
Abstract. The main result of this paper establishes the perfect noncommutative Nicht- negativstellensatz on a convex semialgebraic set: suppose L is a monic linear pencil in g variables and letDL be its positivity domain
DL:= [
n∈N
X ∈(SRn×n)g |L(X)0 .
Then a noncommutative polynomialpispositive semidefinite onDL if and only if it has a weighted sum of squares representation with optimal degree bounds. Namely,
p=s∗s+
finite
X
j
fj∗Lfj,
where s, fj are vectors of noncommutative polynomials of degree no greater than deg(p)2 . This result contrasts sharply with the commutative setting, where the degrees ofs, fj are vastly greater than deg(p) and assuming onlypnonnegative yields a clean Positivstellensatz so seldom that the cases are noteworthy.
The main ingredient of the proof is an analysis of rank preserving extensions of truncated noncommutative Hankel matrices. It is proved that any suchpositive definite matrixMk of
“degreek” has, for eachm ≥0, a positive semidefinite Hankel extension ˜Mk+m of degree k+mand the same rank asMk.
1. Introduction
A Positivstellensatz is an algebraic certificate for a given polynomialpto have a specific positivity property and such theorems date back in some form for over one hundred years for conventional (commutative) polynomials, cf. [BCR, Las, Mar, PD]. Positivstellens¨atze
Date: 20 February 2011.
2010Mathematics Subject Classification. Primary 47A57, 14P10; Secondary 47B35, 13J30, 46N10.
Key words and phrases. Positivstellensatz, Hankel matrix, flat extension, moment problem, rank preserv- ing, noncommutative algebra, free positivity.
1Research supported by NSF grants DMS-0700758, DMS-0757212, and the Ford Motor Co.
2Research supported by the Slovenian Research Agency grants J1-3608 and P1-0288. Partly supported by the Mathematisches Forschungsinstitut Oberwolfach Research in Pairs RiP program. The author thanks Markus Schweighofer for valuable discussions. Part of this research was done while the author held a visiting professorship at the University Konstanz supported by the program “free spaces for creativity”.
3Research supported by the NSF grant DMS-0758306.
1
for polynomials in noncommuting variables are a creature of this century - see [HKM2, HM1, KS1, PNA, DLTW]; for software equipped to dealing with positive noncommutative polynomials we refer to [HOSM,CKP]. Often in the noncommutative setting such theorems have cleaner statements than their commutative counterparts. For instance, a multivariate (commutative) polynomial on Rg which is pointwise nonnegative need not be a sum of squares, but a noncommutative polynomial which is nonnegative (in a sense made precise below) isa sum of squares - a result of the first author [Hel].
Classical commutative Positivstellens¨atze generally require p to be positive - the cases where nonnegative suffices are few and noteworthy, cf. [Sce], and the degrees of the polyno- mials appearing in the representation ofpas a weighted sum of squares, are typically of very high degree compared to that of p.
The main result of [HM1] gave a Positivstellensatz for noncommutative polynomials which was an exact extension, warts and all (the strict positivity assumption and the pos- sibility of high degree weights), of the commutative Putinar Positivstellensatz [Put]. While gratifying, it was not, as in retrospect we have come to expect in the free algebra setting, cleaner than its commutative counterpart. What we find in this paper for noncommuta- tive polynomials is that when the underlying semialgebraic set, defined by a matrix-valued noncommutative polynomialq, is convex, a“perfect” Positivstellensatz holds; namely, a rep- resentation
(1) p=
finite
X
j
s∗jsj +
finite
X
j
fj∗qfj
where sj, fj are noncommutative polynomials of degree no greater than deg(p)+22 holds for any p which is “nonnegative” where q is “nonnegative.” In particular, the main result in this article says under the stronger, compared to that in [HM1], hypothesis that the non- commutative polynomial q is concave, if p nonnegative on the set Dq, the set where q is nonnegative, thenphas a “perfect” Positivstellensatz representation. That is, for convexDq
(or, equivalently, for concave q [HM1]) we have a “perfect” Positivstellensatz. Indeed this is a Nichtnegativstellensatz, as p is only assumed to be nonnegative on Dq. As a corollary when q = 1 and Dq is everything we recover the result mentioned in the first paragraph about nonnegative noncommutative polynomials being sum of squares.
The other line of main results in this paper is a theory of rank preserving extensions of multivariate noncommutative Hankel (moment) matrices. We formulate a notion of non- commutative moments and prove:
(1) a finite sequence is a noncommutative moment sequence if the corresponding trun- cated noncommutative Hankel matrix is positive definite;
(2) every positive definite truncated noncommutative Hankel matrixH has a “flat” ex- tension in a strong sense; namely, there is a noncommutative Hankel matrix extension of any size having rank equal to that of H.
We use (2) to prove (1). These free algebra results are much cleaner than what holds in the commutative case (cf. [CF1, CF2, Las]), where flat extensions are the key, but unfor- tunately may or may not exist, with no simple way to determine which. Moments and Positivstellens¨atze are dual in a sense and we use our flat extension results together with duality techniques to prove our “perfect” Positivstellensatz.
The article also contains information about an extension of our Positivstellensatz to non- convex situations, the latter coming at the expense of a stronger nonnegativity hypothesis on p. In the remainder of this introduction, we state our main Positivstellensatz after providing the needed background and definitions. The subsequent section treats noncommutative mo- ment sequences or more generally multivariate Hankel structure. There we prove the rank preserving extension result. Section3uses the Hankel theory to prove the Positivstellensatz, and Section 4extends our Positivstellensatz to nonconvex situations.
1.1. Words and NC polynomials. Given positive integers n and g, let (Rn×n)g denote the set of g-tuples of realn×n matrices. A natural norm on (Rn×n)g is given by
kXk2 =X kXjk2
for X = (X1, . . . , Xg)∈(Rn×n)g. We use SRn×n to denote symmetricn×n matrices.
We write hxi for the monoid freely generated by x = (x1, . . . , xg), i.e., hxi consists of words in the g noncommuting letters x1, . . . , xg (including the empty word ∅ which plays the role of the identity). Let Rhxi denote the associative R-algebra freely generated by x, i.e., the elements of Rhxi are polynomials in the noncommuting variables x with coefficients in R. Its elements are called (nc) polynomials. An element of the form aw where 0 6= a ∈ R and w ∈ hxi is called a monomial and a its coefficient. Hence words are monomials whose coefficient is 1. EndowRhxiwith the natural involutionwhich fixes R∪ {x} pointwise, reverses the order of words, and acts linearly on polynomials. For example, (2−3x21x2x3)∗ = 2−3x3x2x21.Polynomials invariant with respect to this involution are symmetric. The length of the longest word in a noncommutative polynomial f ∈Rhxi is thedegree off and is denoted by deg(f). The set of all words of degree≤k is hxik, and Rhxik is the vector space of all noncommutative polynomials of degree at most k.
Matrix-valued noncommutative polynomials – elements of Rj×`hxi; i.e., j ×` matrices with entries fromRhxi– will play a role in what follows. Elements ofRj×`hxiare conveniently
represented using tensor products as P = X
w∈hxi
Bw⊗w∈Rj×`hxi,
whereBw ∈Rj×`, and the sum is finite. Note that the involution∗ extends to matrix-valued polynomials by
P∗ =X
w
B∗w⊗w∗ ∈R`×jhxi.
In the casej =` and P∗ =P, we say P is symmetric.
1.1.1. Polynomial evaluations. If p ∈ Rhxi is a noncommutative polynomial and X ∈ (Rn×n)g, the evaluationp(X)∈Rn×n is defined by simply replacing xi byXi. For example, if p= 3x1x2, then
p "
0 1 1 0
# ,
"
1 0
0 −1
#!
= 3 "
0 1 1 0
# "
1 0 0 −1
#!
=
"
0 −3
3 0
# . Similarly, if p(x) =α∈R and X ∈(Rn×n)g, then p(X) = αIn.
Most of our evaluations will be on tuples ofsymmetric matricesX ∈(SRn×n)g; our invo- lution fixes the variablesxelementwise, so only these evaluations give rise to∗-representations of noncommutative polynomials. Polynomial evaluations extend to matrix-valued polyno- mials by evaluating entrywise. Note that ifP is symmetric, and X ∈(SRn×n)g, then P(X) is a symmetric matrix.
1.2. Linear and concave polynomials. If A1, . . . , Ag are symmetric `×` matrices, then ΛA:=X
Ajxj
is a (homogeneous) linear symmetric matrix-valued polynomial, also called a (homoge- neous) linear pencil. To ΛA we associate the monic linear pencil
LA:=I−ΛA=I`−X Ajxj. A symmetric q∈R`×`hxi is concaveprovided
(2) q tX+ (1−t)Y
tq(X) + (1−t)q(Y), 0≤t≤1
for all X, Y ∈ (SRn×n)g. The main result in [HM2] tells us that if q is scalar-valued (i.e.,
`= 1) symmetric and q(0) = I`, then q is concave if and only if it has the form (3) q(x) = I`−Λ(x)−s∗(x)s(x)
for some linear polynomial Λ ∈ Rhxi and linear (column) vector-valued s ∈ R`×1hxi. This result remains true, with the obvious modifications, forq matrix-valued. A proof is given in Subsection 1.3.3.
1.3. The Positivstellensatz. An element of the form g∗g will be called a (hermitian) square. Let Σ denote the cone of sums of squares of polynomials, and, given a nonnegative integer N, let ΣN denote the cone of sums of squares of polynomials of degree at most N. Thus elements of ΣN have degree at most 2N, i.e., ΣN ⊆ Rhxi2N. Conversely, since the highest order terms in a sum of squares cannot cancel, Rhxi2N ∩Σ = ΣN.
Fix a symmetric matrix-valued q ∈ R`×`hxi. Let Rhxi`k denote `-vectors with entries fromRhxik. Thus Rhxi`k =R`×1hxik. Given α, β ∈N, set
(4) Mα,β(q) := Σα+nfiniteX
j
fj∗qfj | fj ∈Rhxi`βo
⊆ Rhximax[2α,2β+a],
where a is the degree ofq. Let
Dnq :={X ∈(SRn×n)g |q(X)0} and Dq := [
n∈N
Dqn.
We often abbreviate Mα,β(q) to Mα,β and likewise for D. Obviously, if f ∈ Mα,β then f|D 0. We call Mα,β the truncated quadratic module and D the noncommutative semialgebraic set generated byq. If q is degree 1, then D is also called an LMI set.
If q(0) =I (q is monic), the noncommutative set D contains a nontrivial noncommu- tative neighborhood of 0; i.e., there exists > 0 such that for each n, if X ∈ (SRn×n)g and kXk< , thenX ∈ D.
Definition 1.1. Let Pn denote those polynomials f ∈ Rhxi satisfying f(X) 0 for all X ∈ Dn. We say Mα,β has thed−PosSS(Positivstellensatz)property ifdeg(p)≤2d+1 and p∈ Pn for all n implies p∈Mα,β. Equivalently, P ∩Rhxi2d+1 ⊆Mα,β.
Let σ#(r) := Pr
j=0gj denote the number of words of degree at most r. The smallest positive integer r such that p ∈ Pr and p has degree at most 2d+ 1 implies implies p∈ Pn for all n is the test rank of test rank of P.
The following is the free convex Positivstellensatz, one of the main results in this paper.
Theorem 1.2 (Convex Positivstellensatz). Supposeq ∈R`×`hxi is a matrix-valued symmet- ric noncommutative polynomial.
(1) If q is concave and monic then Md+1,d(q) has the d−PosSS property.
(2) If q is a monic linear pencil, then Md,d(q) has the d−PosSS property.
In either case, the test rank is no greater than σ#(d+ 1).
Remark 1.3. The conclusion of Theorem1.2 may fail ifq is not assumed to be monic. For instance, if
q=
"
x 1 1 0
#
∈R2×2hxi1,
thenM0,0 does not have the 0−PosSS property. Indeed,Dq =∅, so−1∈ P, but −16∈M0,0. The example can also be modified to obtain Dq is nonempty (and bounded). For details and more on the study of empty LMI sets we refer the reader to [KS2]. One of the main results there states thatDq is empty (for a nonhomogeneous linear pencilq) if and only if the truncated quadratic module Md,d (in the ring R[x] of polynomials in commuting variables) contains −1 for some (explicitly computable) d∈N.
Remark 1.4. In [HKM1] we studied LMI sets and their inclusions. The linear Positivstel- lensatz there [HKM1, Theorem 1.1] states the following: Suppose q, r are two monic linear pencils withDq bounded. Then Dq ⊆ Dr if and only ifr is in the (matrix-valued) truncated quadratic module M0,0(q). For r scalar-valued this is a very special case of Theorem 1.2.
Furthermore, the Positivstellensatz [HKM1, Theorem 5.1] is a weak form of Theorem 1.2.
Remark 1.5. The main result of [HM3] says that if q is matrix-valued, symmetric, and monic and the component of 0 of
Bq := [
n∈N
{X ∈(SRn×n)g |q(X)0}
is convex, then there is a monic linear pencil Lsuch that this component of 0 is of the form BL. In particular, if Bq is itself convex, then its closure is DL for some L. In this sense, Theorem 1.2 covers the case that the underlying nc semialgebraic set is convex.
The difficult part in proving Theorem 1.2 is showing that Md+1,d(q) has the d−PosSS property in the case that q is a monic linear pencil. The argument occupies the bulk of this article. The passages from q linear to q concave and from Md+1,d to Md,d are rather simple and the details are in the following two subsubsections, §1.3.1 and §1.3.2, here in the introduction. The proof of Theorem 1.2 culminates in Section 3.3, using the results on noncommutative multivariate Hankel matrices from Section 2. What can be said in the absence of concavity of q is the topic of Section 4.
1.3.1. From linear to concave. The following lemma reduces the proof of Theorem1.2 for q concave to the case of q linear.
Lemma 1.6. Suppose Md+1,d(q) has the d−PosSS property whenever q is a monic linear pencil. Then Md+1,d(q) has the d−PosSS property whenever q is concave and monic.
Proof. By Proposition1.8below, it may be assumed thatq∈R`×`hxiis described by equation (3) for some linear pencil ΛA ∈R`×`hxi and linear s∈R`
0×`hxi. Let
Q=
"
I`0 s s∗ I −ΛA
#
∈R(`+`
0)×(`+`0)hxi1.
HenceQis a monic linear pencil and, as is easily checked using Schur complements,Dq =DQ. Thus, a given symmetricpis positive semidefinite onDq if and only if it positive semidefinite onDQ.
Let Q=LDL∗ be the LDU decomposition of Q, that is L=
"
I 0 s∗ I
#
, and D =
"
I 0
0 I−Λ−s∗s
# .
By hypothesis,Md+1,d(Q) has thed−PosSS property and we are to show thatMd+1,d(q) does too. To this end suppose p has degree at most 2d+ 1 and is positive semidefinite on Dq =DQ. Hence phas a representation as
p=G+X
j
h fj∗ gj∗
i Q
"
fj
gj
# ,
with gj ∈ Rhxi`d, fj ∈ Rhxi`d0 and G ∈ Σd+1 a sum of squares of polynomials of degree at most d+ 1. Since
L∗
"
fj gj
#
=
"
fj+sgj gj
# , it follows that
(5) p=G+X
(fj+sgj)∗(fj+sgj) +X
g∗j(1−Λ−s∗s)gj.
Observing thatfj+sgj has degree at most d+ 1, (5) shows thatp∈Md+1,d(q) and completes the proof.
1.3.2. From Md+1,d to Md,d. It turns out that in the case q is monic linear, Md+1,d has the d−PosSS property if and only if Md,d does.
Lemma 1.7. Suppose q is a monic linear pencil. If p has degree at most 2d+ 1 and p ∈ Md+1,d(q), then p∈Md,d(q).
Proof. Ifp∈Md+1,d then
p=X
gj∗gj +X fj∗qfj,
for polynomials gj of degree at most d+ 1 and fj of degree at most d. Any degree 2d+ 2 terms in P
g∗jgj appear as (positively weighted) squares and can not be canceled by terms
inP
fj∗qfj, since the latter have degree at most 2d+ 1. Hence eachgj must have degree at most 2d.
1.3.3. Concave polynomials. The structure of symmetric concave matrix-valued polynomials is quite rigid.
Proposition 1.8. Ifq is a symmetric concave matrix-valued polynomial withq(0) =I, then there exists a linear pencils L and Λ such that L is symmetric and
q =I −L−Λ∗Λ.
Proof. Suppose q is an `×` matrix-valued symmetric polynomial. Thus, using the tensor product notation,
q=X
w
Qw ⊗w,
for some`×` matricesQw with Q∗w =Qw∗. By hypothesisQ∅ =q(0) =I`, the`×` identity.
Given vectors γ, η∈R`, consider the scalar-valued polynomial, qγ,η =X
w
hQwγ, ηiw.
By polarization,
qγ,η = 1
2[qγ+η,γ+η −qγ,γ −qη,η].
Now suppose q is concave. It follows, for each unit vector γ ∈ R`, that qγ,γ ∈ Rhxi is concave. By a main result in [HM2], qγ,γ has degree at most two and moreover, there exists lγ,γ and λγ,γ linear so that
(6) qγ,γ = 1−lγ,γ −λ∗γ,γλγ,γ.
Note thatλγ,γ is vector-valued, so the last term on the right hand side is a sum of squares.
From polarization, we conclude that q itself has degree at most two so that q =I−L−Q,
where L is linear and Q is homogeneous of degree two. Moreover, from equation (6), Q is positive semidefinite. Since a nonnegative polynomial which is homogeneous of degree two has the form Λ∗Λ, for some (not necessarily square) linear matrix-valued Λ [McC], the conclusion follows.
2. Hankel matrices and their flat extensions
A main ingredient in the proof of Theorem 1.2 is the solution of a noncommutative mo- ment problem via rank preserving extensions of multivariate Hankel matrices. It is described in this section, which is organized as follows. Hankel matrices are introduced in the first subsection; the second subsection exposits the existence and construction of one-step rank preserving extensions; the passage from positive linear functionals to tuples of matrices via the Gelfand-Naimark-Segal (GNS) construction is the topic of the third subsection. The proof of the second main result of this article, that a positive definite noncommutative Han- kel matrix has a rank preserving extension to an infinite Hankel matrix, as stated formally in the second subsection, is given in the last subsection.
2.1. Hankel matrices. LetHk denoteRhxik with Hilbert space inner product determined by declaring the words in hxik to be an orthonormal basis. Its dimension is σ#(k). There is an intimate connection between positive linear functionals on Rhxi2k and positive definite matrices on Hk. We summarize the interplay in the next proposition.
Proposition 2.1. If L:Rhxi2k →R is linear, then there exists a unique linear mapping H on Hk such that, for words v, w∈ hxik,
(7) L(w∗v) = hHv, wi.
Further,L is positive onΣk if and only ifH is positive definite. Conversely, if H is a linear map on Hk such that
(8) hHv, wi=hHα, βi
whenever v, w, α, β ∈Rhxik and w∗v =β∗α, then there is a linear mapping L:Rhxi2k →R such that (7) holds.
Abusing notation slightly, the form H(u, v) = hHu, vi is called a Hankel matrix. In preparation for the proof of Theorem 1.2, set δ =d+ 1 and
Mδ :=Mδ,d(I`−ΛA).
Recall A is a g-tuple of symmetric `×` matrices.
Positivity of a linear functional L on Mδ can evidently be rephrased in terms of the corresponding Hankel matrix H. Let H ⊗I` denote the block diagonal matrix with H as each diagonal entry.
Lemma 2.2. A linear functional L:Rhxi2δ →R is nonnegative on Mδ if and only if H is positive semidefinite and
(9)
(H⊗I`)f, f
≥
(H⊗I`)ΛA(x)f, f .
for all f ∈Rhxi`d.
Proof. That L nonnegative is equivalent to H positive semidefinite is a consequence of Proposition 2.1. Assuming L is nonnegative, then L is nonnegative on Mδ if and only if L(f∗(I −ΛA(x)f)) ≥ 0 for each polynomial f ∈ Rhxi` of degree at most d. On the other hand,
L(f∗(I−ΛA(x)f) =L(f∗f)−X
fj∗ΛA(x)j,kfk
=X
hHfj, fji −X
hHΛA(x)j,kfk, fji
=h(H⊗I`)f, fi − h(H⊗I`)ΛA(x)f, fi.
2.2. The nc world is flat. Now we focus on the rank of Hankel matrices, beginning with the second main result of this paper.
Theorem 2.3(Existence of Rank Preserving Extensions).Any given positive definite Hankel operator A on Hk, has an extension to a Hankel operator Am on Hk+m whose rank is the same as the rank of A. There are many rank preserving Hankel extensions A1, however all Am with m >1 are uniquely determined by A1.
The proof will be given in Section 2.5. The next several subsections develop the needed background.
The classical commutative case gives good perspective. First of all, Theorem2.3holds in one variable (g = 1) but fails in the commutative multivariate setting (cf. [CF1,CF2, Las]).
There, what one seeks is an m and an Am whose rank equals rank(Am−1); in this case there are uniquely determined positive semidefinite Hankel matrices Am+j extending Am with rank(Am+j) = rank(Am) for all j ≥ 0. The matrix Am is called a flat extension of A, indeed anm-step flat extension. In this language we have shown thatevery positive definite noncommutative Hankel matrix has a “1-step flat” extension. Thus we are led, in the spirit of synchronizing terminology, to refer to a rank preserving 1-step noncommutative Hankel extension as aflat extension. Now we begin to build machinery needed to prove Theorem 2.3.
Definition 2.4. Let A ∈Rn×n be a symmetric matrix. A (symmetric) extension of A is a symmetric matrix E ∈R(n+`)×(n+`) of the form
E =
"
A B B∗ C
#
for some B ∈ Rn×` and C ∈ R`×`. Such an extension is rank preserving if rankA = rankE.
IfA is positive semidefinite and B is given, thenE is a rank preserving extension if and only if there is a matrix Z such that B =AZ and C =Z∗AZ. If A is positive definite, not just semidefinite, then this is in turn equivalent to C=B∗A−1B.
Let H be a positive semidefinite Hankel operator on Hk. Express H, as a block 2×2 matrix in terms ofA, B, Cas in the definition above with respect to the orthogonal decompo- sition ofHk =Hk−1⊕K(whereKis the subspaceRhxi=kof polynomials of degree exactlyk).
The matrixH being positive semidefinite, implies there is a matrixZ such thatB =AZ and C−B∗A†B is positive semidefinite, with A† denoting the standard pseudoinverse, cf. [Dym, Theorem 11.10]. Define
H˜ :=
"
A B
B∗ Z∗AZ
# .
Note: if A is positive definite (e.g. H is positive definite), then Z∗AZ =B∗A−1B.
The following proposition embodies a most critical free algebra fact which fails in the commutative multivariate case.
Proposition 2.5. If A is positive definite, then the matrix H˜ is a positive semidefinite Hankel and is a rank preserving extension of A.
Proof. Note
H˜ =
"
A B
B∗ Z∗AZ
#
=
"
A AZ
Z∗A Z∗AZ
#
= h
I Z i∗
A h
I Z i
,
so ˜H is a rank preserving extension of Aand ˜H 0. Note and H H, since˜ C B∗A†B = Z∗AZ.
Now we set about to show ˜H is Hankel. Suppose v, w, α, β are words of length at most k and u=w∗v =β∗α. If u has length at most 2k−1, then
hHv, wi=hHv, wi˜
and similarly for α, β in place ofv, w respectively. Since H is Hankel, it follows that hHv, wi˜ =hHα, βi.˜
On the other hand, if u has length 2k+ 2, then α=v and β =w, thus (2.2) still holds.
A bilinear function of the form f 7→ L f∗(I −ΛA(x))f
is an analog of the localizing matrixcommon in classical moment theory. The following proposition links flat extensions and localizing matrices.
Proposition 2.6. Suppose L:Rhxi2δ →R is positive (on Σδ). Let H be the Hankel matrix for L. If H˜ is as above and L˜ is the resulting linear functional(so that the Hankel matrix of L˜ is H), then˜ H˜ is positive semidefinite and L˜ is nonnegative.
Moreover, if L is nonnegative on Mδ, then so is L.˜
Proof. That ˜H is positive semidefinite, and therefore ˜L is nonnegative on Σδ, is immediate from the above discussion. Thus, to complete the proof it suffices to show, if f ∈ Rhxi`d, then ˜L(f∗(1−ΛA(x))f)≥0. To this end estimate,
L(f˜ ∗(1−ΛA(x))f) =h( ˜H⊗I)f, fi − h( ˜H⊗I)ΛA(x)f, fi
=h(H⊗I)f, fi − h(H⊗I)ΛA(x)f, fi
=L(f∗(I−ΛA(x))f)≥0,
where the second equality follows from the fact thatf has degree at mostd.
2.3. Making tuples X: Gelfand-Naimark-Segal (GNS) construction. The following proposition is a version of the GNS construction. It is the solution to a noncommutative moment problem and is well-known (cf. [McC, Theorem 2.1]). We include the simple proof for the sake of completeness.
Proposition 2.7. If L:Rhxi2k+2 →R is a linear functional which is nonnegative on Σk+1 and positive on Σk, then there exists a tuple X = (X1, . . . , Xg) of symmetric operators on a Hilbert space X of dimension σ#(k) = dimRhxik and a cyclic vector ζ ∈ X such that
L(f) = hf(X)ζ, ζi
for f ∈Rhxi2k+1.
Conversely, if X = (X1, . . . , Xg) is a tuple of symmetric operators on a Hilbert space X of dimension N, the vector ζ is in X, and k is a positive integer, then the linear functional L:Rhxi2k+2 →R defined by
L(f) = hf(X)ζ, ζi is nonnegative on Σk+1.
Proof. First suppose that L : Rhxi2k+2 → R is nonnegative on Σk+1 and positive on Σk. Let H denote the corresponding Hankel matrix; and define ˜H and ˜L as in Proposition 2.6.
Expressing H as
H=
"
A B B∗ C
# ,
the positivity assumption on Limplies A is positive definite. Thus, H˜ =
"
A B
B∗ B∗A−1B
# .
The positive semidefinite matrix ˜H determines the symmetric bilinear form hp, qiH˜ =hHp, qi˜
onHk+1 A standard use of Cauchy-Schwarz inequality shows that the set of nullvectors N :={f ∈Rhxik+1 | hf, fiH˜ = 0}
is a vector subspace of Hk+1. Whence one can endow the quotient ˜H := Hk+1/N with a positive definite hermitian form making it a Hilbert space. Because ˜H is a rank preserving extension of A and A is positive definite, each equivalence class in ˜H has a unique repre- sentative which is a polynomial of degree at most k. Equivalently, for each polynomial r of degree k + 1, there is exactly one polynomial s of degree k such that [r] = [s], where [·] represents the equivalence class. In particular, the dimension of ˜H is at most σ#(k).
The Hilbert space ˜H carries the multiplication operators Xj : ˜H → H:˜ Xjf =xjf, f ∈H,˜ 1≤j ≤g.
One verifies from the definition, the positive definiteness of A, and the fact that ˜H is a rank preserving extension of A, that each Xj is well defined and
hXjp, qi=hxjp, qi=hp, xjqi=hp, Xjqi for all p, q ∈H. In particular,˜ Xj is symmetric.
Finally, given words v ∈ hxik+1 and w∈ hxik, let f =w∗v and observe, withγ equal to [∅], the class of the empty word in ˜H, that
hf(X)γ, γiH˜ =hw(X)∗v(X)γ, γi
=hv(X)γ, w(X)γi
=hHv, wi˜ =L(w∗v) =L(f).
Since any f ∈ Rhxi2k+1 can be written as a linear combination of words of the form w∗v (with w∈ hxik+1 and v ∈ hxik), the proof of the first part of the proposition is complete.
The proof of the converse is routine and is used only in the following subsection as an ingredient in the proof of Theorem 2.3.
2.4. Noncommutative moment sequences. A sequence of real numbers (yw) indexed by words w∈ hxi satisfying
(10) yw =yw∗ for all w,
and y∅ = 1, is called a noncommutative (normalized) moment sequence.
Example 2.8. Given n ∈ N, symmetric matrices A1, . . . , Ag ∈SRn×n, and a (unit) vector v ∈Rn, the sequence given by
(11) yw =hw(A1, . . . , An)v, vi is a noncommutative (normalized) moment sequence.
The noncommutative moment problem asks for the converse of Example 2.8: For which sequences (yw) do there exist n∈N, A1, . . . , Ag ∈SRn×n, and v ∈Rn such that yw satisfies (11)? We then say that (yw) has a moment representation.
The truncated moment problem is the study of (finite) moment sequences (yw)≤k where w is constrained by degw ≤ k for some k ∈ N, and property (10) hold for these w. For instance, which sequences (yw)≤k have a moment representation, i.e., when does there exist a representation of the values yw as in (11) for degw ≤ k? If this is the case, then the sequence (yw)≤k is called a truncated moment sequence.
The (infinite) Hankel matrix H(y) of a moment sequence y= (yw) is defined by H(y) = (yu∗v)u,v.
This matrix is symmetric due to the condition (10). As is easy to see, a necessary condition for y to be a moment sequence is positive semidefiniteness of H(y) which, beyond the one variable case, is in general not sufficient.
The Hankel matrix of order k is the Hankel matrix Hk(y) indexed by words u, v with degu,degv ≤k. If y is a truncated moment sequence, then Hk(y) is positive semidefinite.
Remark 2.9. In this terminology our results in this section can be rephrased as follows:
(1) every positive definite truncated Hankel matrix H has a “flat” extension in a strong sense: there is a noncommutative Hankel matrix extension of any size having rank equal to that ofH (Theorem 2.3);
(2) (solution to the truncated moment problem) a finite sequence is a noncommutative mo- ment sequence if the corresponding truncated Hankel matrix is positive definite (combine Theorem 2.3 with Proposition 2.7).
We leave it as an exercise for the reader to deduce:
(3) (solution to the moment problem) an infinite sequence y is a noncommutative moment sequence if and only if (a) there exists a C such that for each 1 ≤ j ≤ g matrices C2(yu∗v)−(yu∗x2
jv) are positive semidefinite and there is a bound on the rank of Hk(y) independent ofk.
2.5. Proof of Theorem 2.3. Fix a positive definite Hankel matrix A onHk and choose B and C so that the matrix (block matrix with respect to the decomposition Hk+1 =Hk⊕ K)
H =
"
A B B∗ C
#
is a positive semidefinite Hankel matrix. Thus, the condition on B is, for v of length k+ 1 and uof length at mostk, that Bu∗v :=A(u, v) =A(α, β) if α, β both have length at mostk and u∗v =α∗β. There are no constraints onBu∗v coming from A and the Hankel condition for u∗v of length 2k + 1. Once B is fixed, the only constraint on C is that it is positive semidefinite and enough so to ensure H is also positive semidefinite.
LetLdenote the linear functionalL:Rhxi2k+2 →Rcorresponding toH. An application of Proposition 2.7 produces a finite-dimensional Hilbert space X of dimension σ#(k) and a vector γ ∈ X such that
L(f) = hf(X)γ, γi
for all f ∈Rhxi2k+1. In particular, ifw∈Rhxik+1 and v ∈Rhxik, then H(v, w) =L(w∗v) = hv(X)γ, w(X)γi.
Moreover, X ={g(X)γ |g ∈Rhxik}. Fix m ≥1 and note that the matrix K(v, w) =hv(X)γ, w(X)γi
defined for v, w∈ hxik+m is Hankel and agrees withH for w∈Rhxik+1 and v ∈Rhxik. Finally, the rank of K is, like H, the dimension of X.
3. Proof of Theorem 1.2
By the results of§1.3.1and§1.3.2, Theorem1.2follows from the following a priori weaker statement.
Theorem 3.1. If q = I −ΛA ∈ R`×`hxi is a monic linear pencil, then Md+1,d(q) has the d−PosSS property. Its test rank is no greater than σ#(d+ 1).
This section is devoted to the proof of Theorem 3.1. Thus, throughout q =I−ΛA and d are fixed,δ=d+ 1, and` is the size ofA; i.e.,A is a g-tuple of symmetric`×` matrices.
3.1. The truncated quadratic module is closed. Recall, given a natural number k, Rhxik is the vector space of polynomials of degree at most k and its dimension is σ#(k).
Fix positive integers α, β and let k denote the larger of 2α and 2β+ 1. In particular, the quadratic module Mα,β of equation (4) is a cone in Rhxik (recall we are now taking the degree of q to be one).
There is an > 0 such that if kXk ≤ , then I`−ΛA(X) 12. Using this we norm Rhxi`k by
kpk:= max
kp(X)k | kXk ≤ .
Note that if f ∈Rhxi`β and if kf∗(1−ΛA(x))fk ≤N2, then kf∗fk ≤2N2. Proposition 3.2. The truncated quadratic module Mα,β ⊆Rhxik is closed.
Proof. This result is a consequence of Caratheodory’s theorem. Suppose (pn) is a sequence fromMα,β which converges to some p of degree at most k. Because the sequence converges, it is bounded, in norm, by say N2. The set C equal to Mα,β intersect the ball of radius N2 in Rhxik is convex and contains the sequence (pn). By Caratheodory’s theorem, there is an M such that for each n there exists vector polynomials rn,j ∈Rhxi`α and tn,j ∈ Rhxi`β such that
pn=
M
X
j=1
r∗n,jrn,j+
M
X
j=1
t∗n,j(I−ΛA(x))tn,j.
Since kpnk ≤ N2, it follows that krn,jk ≤ N and likewise kt∗n,j(1−ΛA(x))ts,n,jk ≤ N2. In view of the remarks preceding the proposition, we obtain ktn,jk ≤√
2N for all j, n. Hence for each j, the sequences (rn,j) and (tn,j) are bounded in n. They thus have convergent subsequences. Tracking down these subsequential limits finishes proof.
3.2. Existence of a positive linear functional. Recallδ =d+ 1 andMδ =Md+1,d. Lemma 3.3. There exists a positive linear functional Lˆ:Rhxi2δ→R which is nonnegative on Mδ.
Proof. It is not hard to see that there is a positive L : Rhxi2δ+2 → R. By Proposition 2.7 there exists a tuple of matrices X and a vectorγ such that
L(f) = hf(X)γ, γi
for f ∈Rhxi2δ. Let H denote the Hankel matrix corresponding toL|Rhxi2δ and express H in block form H = (Hj,k) whereHj,k is the matrix (H(u, v))u,v where u∈ hxij and v ∈ hxik.
Fort >0, consider the tuple tX. It corresponds to a linear functional Lt determined by Lt(f) = hf(tX)γ, γi.
Let Ht denote the corresponding Hankel matrix so that Ht(u, v) = Lt(v∗u). Let T denote the block diagonal matrix with tjImj in the (j, j) position. (Here Imj is the identity matrix of size matching the size of Hj,j.) Routine calculations show that T HT =Ht. Hence Ht is positive definite.
Finally, because D contains a neighborhood of 0, for t > 0 any sufficiently small tX must belong to D. Thus, for such t, Lt is positive and nonnegative on Mδ.
3.3. Separation. The final ingredient in the proof Theorem3.1is a Hahn-Banach separation argument. Accordingly, let p∈ P(q)∩Rhxi2d+1 be given. We are to show p∈Mδ.
If the conclusion is false, then by Proposition 3.2 and the Hahn-Banach theorem there is a linear functional L:Rhxi2δ →R that is nonnegative on Mδ and negative on p. Adding, if necessary, a small positive multiple of the linear functional ˆL produced by Lemma 3.3 to L, we can assume that L is positive (not just nonnegative) on Σδ, nonnegative on Mδ, and still negative on p.
LetH denote the Hankel matrix forL. Let ˜H and ˜Ldenote the linear functional and its noncommutative Hankel matrix constructed in Proposition 2.6. By Proposition 2.7, there is a tuple of symmetric matrices X acting on a Hilbert space X and a vector ζ such that
X ={g(X)ζ |g ∈Rhxid} and
L(g) =˜ hg(X)ζ, ζi for all g ∈Rhxi2d+1. In particular, for f ∈Rhxi`d
0≤L(f∗(1−ΛA)f) = ˜L(f∗(1−ΛA)f)
=
f(X)∗(I−ΛA(X))f(X)ζ, ζ (12) .
Since forf ∈Rhxi`d, vectors of the formf(X)ζare all of⊕`1X, it follows thatI−ΛA(X) 0 and thereforeX ∈ Dq. On the other hand, using the assumption that the degree of pis at most 2d+ 1,
(13) 0> L(p) = ˜L(p) = hp(X)ζ, ζi,
contradicting the hypothesis that p ∈ P(q) and the proof of Theorem 3.1, and hence of Theorem 1.2, is complete.
4. Beyond convexity: a harsher positivity test
The Positivstellensatz in [HM1] has no restrictions on the underlying semialgebraic set, whereas Theorem1.2 assumes the set is convex. In this section we consider a case which lies in between. Given a setS, of symmetric noncommutative polynomials whose degrees are at most a, letQ={1−s∗s |s∈S}. We will develop a positivity condition for a polynomialp of degree at most 2d to lie in the cone
Md+a,d(Q) = Σd+a+ X
q∈Q
X
j
fj,q∗ qfj,q | fj,q ∈Rhxid .
Note, in the case that S is finite, if q denotes the diagonal matrix with diagonal entries 1−s∗s, then Md+a,d(Q) =Md+a,d(q).
LetV be a finite-dimensional (real) Hilbert space. Given a vectorv ∈V, natural number η, and a tuple X of symmetric linear maps onV, let OηX,ζ denote the subspace
OX,ζη :={f(X)ζ |f ∈Rhxiη}
of V and PX,ζη be the selfadjoint projection onto this space. Generically, dim OηX,ζ is σ#(η).
The following is a free nonconvex Positivstellensatz with degree bounds.
Theorem 4.1 (Beyond Concave). Assume that DQ contains a nontrivial nc neighborhood of 0 and that p∈Rhxi2d is symmetric. If for any Hilbert space V of dimension σ#(d+a−1), g-tuple of matrices X acting on V and vector ζ ∈V,
PX,ζd 1−s∗(X)s(X)
PX,ζd 0 for all s∈S implies
hp(X)ζ, ζi ≥0, then p∈Md+a,d.
The converse obviously is true.
In other words a clean Positivstellensatz holds without concavity of q (the collectionS), provided we test positivity ofp on a sufficiently large class of matrices and vectors.
Remark 4.2.
(1) If a = 1, then generically dimension counting tells us OdX,ζ isV, and we are back in the setting of Theorem 1.2.
(2) The condition: hp(X)ζ, ζi ≥0 provided
ζ∗(1−s∗(X)s(X))ζ ≥0
is a strong condition converted to a Positivstellensatz in [HMP].
As a start of an outline of the proof of Theorem 4.1, set δ = d+a and proceed with the separation argument producing L in Section 3.3 as before. Now modify the argument in Section 2as follows. Decompose the Hankel matrix corresponding to a separating linear functionalL as a block (d+ 1 +a)×(d+ 1 +a) matrix. (The upper (d+ 1)×(d+ 1) block corresponding to polynomials of degree at most d.) Call the bottom diagonal block F and change it to ˜F as in the argument to get Hankel ˜H whose upper left hand corner A is the same as that of H, but with rank ˜H equal to that of A.
The analog of Proposition 2.6 in this setting is:
Proposition 4.3. Suppose L : Rhxi2δ → R is positive on Σδ. Let H be the Hankel matrix forL. Define H˜ as above and let L˜ denote the resulting linear functional (so that the Hankel matrix of L˜ is H). Since˜ H˜ is positive semidefinite, L˜ is positive.
Moreover, if L is nonnegative on Mδ, then so is L.˜
Proof. It suffices to show, iff ∈Rhxid, then ˜L(f∗(1−s∗s)f)≥0. To this end estimate, L(f˜ ∗(1−s∗s)f) = hHf, f˜ i − hHsf, sfi˜ =hHf, fi − hHsf, sfi˜
≥ hHf, fi − hHsf, sfi=L(f∗(1−s∗s)f)≥0,
where the second equality follows from the fact that f has degree at most d and the first inequality follows from −H˜ −H.
Proof of Theorem 4.1. The proof follows the lines of the proof of Theorem 3.1. With the notation from above, the GNS works just as before with hf, fiH˜ := hHf, f˜ i on f ∈ Hk+a and
N :={f ∈Rhxik+a | hf, fiH˜ = 0}.
The quotient ˜H := Hk+a+1/N with form hf, fiH˜ is a Hilbert space and with X, ζ the pair constructed on ˜Has before we still get (12). The only trouble is that rather than concluding 1−s∗(X)s(X) is positive semidefinite, one only finds
(1−s∗(X)s(X))v, v
≥0 for v ∈OdX,ζ.
However, by hypothesis, this last inequality implies hp(X)v, vi ≥ 0 which gives the same contradiction as found in (13).
References
[BCR] J. Bochnack, M. Coste, M.-F. Roy: Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete3, Springer, 1998.1
[CF1] R. Curto, L. Fialkow: Solution of the truncated complex moment problem for flat data, Mem.
Amer. Math. Soc. 119(1996).3,10
[CF2] R. Curto, L. Fialkow: Flat extensions of positive moment matrices: Recursively generated relations, Mem. Amer. Math. Soc.136(1998).3,10
[CKP] K. Cafuta, I. Klep, J. Povh: NCSOStools: a computer algebra system for symbolic and numerical computation with noncommutative polynomials. To appear in Optim. Methods Softw., available from http://ncsostools.fis.unm.si2
[DLTW] A.C. Doherty, Y.-C. Liang, B. Toner, S. Wehner: The quantum moment problem and bounds on entangled multi-prover games, Twenty-Third Annual IEEE Conference on Computational Com- plexity199–210, IEEE Computer Soc., 2008.2
[Dym] H. Dym: Linear Algebra in Action, Graduate Studies in Mathematics78, American Mathematical Society, 2007.11
[Hel] J.W. Helton: “Positive” noncommutative polynomials are sums of squares,Ann. of Math. (2)156 (2002) 675–694.2
[HKM1] J.W. Helton, I. Klep, S. McCullough: The matricial relaxation of a linear matrix inequality, preprinthttp://arxiv.org/abs/1003.09086
[HKM2] J.W. Helton, I. Klep, S. McCullough: Convexity and Semidefinite Programming in dimension-free matrix unknowns. To appear in theHandbook of Semidefinite, Cone and Polynomial Optimization edited by M. Anjos and J. B. Lasserre, Springer, 2011. 2
[HM1] J.W. Helton, S. McCullough: A Positivstellensatz for noncommutative polynomials,Trans. Amer.
Math. Soc. 356(2004) 3721–3737.2,17
[HM2] J.W. Helton, S. McCullough: Convex noncommutative polynomials have degree two or less,SIAM J. Matrix Anal. Appl. 25(2004) 1124–1139.4,8
[HM3] J.W. Helton, S. McCullough: Every free basic convex semi-algebraic set has an LMI representation, preprinthttp://arxiv.org/abs/0908.43526
[HMP] J.W. Helton, S. McCullough, M. Putinar: Strong majorization in a free∗-algebra, Math. Z.255 (2007) 579–596.18
[HOSM] J.W. Helton, M.C. de Oliveira, M. Stankus, R.L. Miller: NCAlgebra, 2010 release edition. Available from http://math.ucsd.edu/~ncalg2
[KS1] I. Klep, M. Schweighofer: A nichtnegativstellensatz for polynomials in noncommuting variables, Israel J. Math.161(2007) 17–27.2
[KS2] I. Klep, M. Schweighofer: Infeasibility certificates for linear matrix inequalities, in preparation.6 [Las] J.B. Lasserre: Moments, positive polynomials and their applications, Imperial College Press Opti-
mization Series1, 2010.1, 3,10
[Mar] M. Marshall: Positive polynomials and sums of squares, Mathematical Surveys and Monographs 146x American Mathematical Society, 2008. 1
[McC] S. McCullough: Factorization of operator-valued polynomials in several noncommuting variables, Linear Algebra Appl. 326(2001) 193–203.8,12
[PNA] S. Pironio, M. Navascu´es, A. Ac´ın: Convergent relaxations of polynomial optimization problems with noncommuting variables,SIAM J. Optim.20 (2010) 2157–2180.2
[PD] A. Prestel, C.N. Delzell: Positive polynomials. From Hilberts 17th problem to real algebra, Springer Monographs in Mathematics, 2001.1
[Put] M. Putinar: Positive polynomials on compact semi-algebraic sets,Indiana Univ. Math. J.42(1993) 969–984. 2
[Sce] C. Scheiderer: Positivity and sums of squares: a guide to recent results. In: Emerging applications of algebraic geometry271–324, IMA Vol. Math. Appl.149, Springer, 2009.2
J. William Helton, Department of Mathematics, University of California, San Diego E-mail address: helton@math.ucsd.edu
Igor Klep, Univerza v Ljubljani, Fakulteta za matematiko in fiziko, and Univerza v Mariboru, Fakulteta za naravoslovje in matematiko, Slovenia
E-mail address: igor.klep@fmf.uni-lj.si
Scott McCullough, Department of Mathematics, University of Florida, Gainesville E-mail address: sam@math.ufl.edu
NOT FOR PUBLICATION Contents
1. Introduction 1
1.1. Words and NC polynomials 3
1.1.1. Polynomial evaluations 4
1.2. Linear and concave polynomials 4
1.3. The Positivstellensatz 5
1.3.1. From linear to concave 6
1.3.2. FromMd+1,d toMd,d 7
1.3.3. Concave polynomials 8
2. Hankel matrices and their flat extensions 9
2.1. Hankel matrices 9
2.2. The nc world is flat 10
2.3. Making tuplesX: Gelfand-Naimark-Segal (GNS) construction. 12
2.4. Noncommutative moment sequences 13
2.5. Proof of Theorem 2.3 15
3. Proof of Theorem 1.2 15
3.1. The truncated quadratic module is closed 15
3.2. Existence of a positive linear functional 16
3.3. Separation 17
4. Beyond convexity: a harsher positivity test 17
References 19
