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NONCOMMUTATIVE PLURISUBHARMONIC POLYNOMIALS PART I: GLOBAL ASSUMPTIONS JEREMY M. GREENE1†, J. WILLIAM HELTON2, AND VICTOR VINNIKOV3 1 1 0 2 n Abstract. We consider symmetric polynomials, p, in the noncommutative (nc) free vari- a ables {x ,x ,...,x }. We define the nc complex hessian of p as the second directional 1 2 g J derivative (replacing xT by y) 4 1 q(x,xT)[h,hT]:= ∂∂s2∂pt(x+th,y+sk)|t,s=0|y=xT,k=hT. ] A Wecallanncsymmetricpolynomialncplurisubharmonic(nc plush)ifithasannc complex hessianthatis positivesemidefinite whenevaluatedonalltuples ofn×nmatricesforevery O size n; i.e., h. q(X,XT)[H,HT](cid:23)0 at for all X,H ∈(Rn×n)g for every n≥1. m Inthis paper,weclassifyallsymmetricnc plushpolynomialsasconvexpolynomialswith [ anncanalyticchangeofvariables;i.e.,anncsymmetricpolynomialpisncplushifandonly if it has the form 2 v (0.1) p= fjTfj + kjkjT +F +FT 7 X X where the sums are finite and f , k , F are all nc analytic. 0 j j 1 Inthispaper,wealsopresentatheoryofnoncommutativeintegrationforncpolynomials 0 and we prove a noncommutative version of the Frobenius theorem. 1. A subsequent paper, [G10], proves that if the nc complex hessian, q, of p takes positive 0 semidefinitevaluesonan“ncopenset”thenqtakespositivesemidefinitevaluesonalltuples 1 X,H. Thus, p has the form in Equation (0.1). The proof, in [G10], draws on most of the 1 theorems in this paper together with a very different technique involving representationsof : v noncommutative quadratic functions. i X r a 1. Introduction The main findings of this paper, the description of nc plush functions and an integration theory for nc functions, were indicated in the abstract, so the introduction consists of precise definitionsandprecise statements oftheplushresults. Theintroductionislaidoutasfollows. 2000 Mathematics Subject Classification. 47A56, 46L07,32H99, 32A99, 46L89. Key words and phrases. noncommutative analytic function, noncommutative analytic maps, noncommutative plurisubharmonic polynomial. 1Research supported by NSF grants DMS-0700758,DMS-0757212. 2Research supported by NSF grants DMS-0700758,DMS-0757212,and the Ford Motor Co. 3Partially supported by a grant from the Israel Science Foundation. †The material in this paper is part of the Ph.D. thesis of Jeremy M. Greene at UCSD. 1 2 GREENE, HELTON, AND VINNIKOV In Section 1.1, we review noncommutative (nc) polynomials and matrix positivity. In Section 1.2, we review nc directional derivatives and provide some examples as to how to compute them. We also define the nc complex hessian and nc plurisubharmonicity and we compare this to the classical analogue in several complex variables. Finally, we close the introductionwith thestatement ofthemaintheorems inSection 1.3. These theorems classify all nc plurisubharmonic polynomials and then describe the uniqueness of this classification. 1.1. NC Polynomials, Their Derivatives, and Plurisubharmonicity. Now we give basic definitions. Many of the the definitions we shall need sit in the context of an elegant theory of noncommutative analytic functions, such as is developed in the articles [K-VV] and [Voi1, Voi2]; see also [Pop6]. Also, related to our results are those on various classes of noncommutative functions on balls as in [AK, BGM]. Transformations on nc variables with analytic functions are described in [HKM, Pop6]. 1.1.1. NC Variables and Monomials. We consider the free semi-group on the 2g noncommuting formal variables x ,...,x ,xT,...,xT. The variables xT are the formal transposes 1 g 1 g j of the variables x . The free semi-group in these 2g variables generates monomials in all of j these variables x ,...,x ,xT,...,xT, often called monomials in x, xT. 1 g 1 g If m is a monomial, then mT denotes the transpose of the monomial m. For example, given the monomial (in the x ’s) xw = x x ...x , the involution applied to xw is (xw)T = j j1 j2 jn xT ...xT xT . jn j2 j1 1.1.2. The Ring of NC Polynomials. Let Rhx ,...,x ,xT,...,xTi denote the ring of non- 1 g 1 g commutative polynomials over R in the noncommuting variables x ,...,x ,xT,...,xT. We 1 g 1 g often abbreviate Rhx ,...,x ,xT,...,xTi by Rhx,xTi. 1 g 1 g Note that Rhx,xTi maps to itself under the involution T. We call a polynomial nc analytic if it contains only the variables x and none of the j transposed variables xT. Similarly, we call a polynomial nc antianalytic if it contains only i the variables xT and none of the variables x . j i We call an nc polynomial, p, symmetric if pT = p. For example, p = x xT + xTx is 1 1 2 2 symmetric. The polynomial p˜ = x x x + x x is nc analytic but not symmetric. Finally, 1 2 4 3 1 the polynomial pˆ= xTxT +4xT is nc antianalytic but not symmetric. 2 1 3 1.1.3. Substituting Matrices for NC Variables. If p is an nc polynomial in the variables x ,...,x ,xT,...,xT and 1 g 1 g X = (X ,X ,...,X ) ∈ (Rn×n)g, 1 2 g the evaluation p(X,XT) is defined by simply replacing x by X and xT by XT. Note that, j j j j forZ = (0 ,0 ,...,0 ) ∈ (Rn×n)2g whereeach0 isthen×nzeromatrix, p(0 ) = I ⊗p(0 ). n n n n n n n 1 Because of this simple relationship, we often simply write p(0) with the size n unspecified. The involution, T, is compatible with matrix transposition, i.e., pT(X,XT) = p(X,XT)T. NONCOMMUTATIVE PLURISUBHARMONIC POLYNOMIALS: PART I 3 1.1.4. Matrix Positivity. We say that an nc symmetric polynomial, p, in the 2g variables x ,...,x ,xT,...,xT, is matrix positive if p(X,XT) is a positive semidefinite matrix when 1 g 1 g evaluated on every X ∈ (Rn×n)g for every size n ≥ 1; i.e., p(X,XT) (cid:23) 0 for all X ∈ (Rn×n)g and all n ≥ 1. In [H02], Helton classified all matrix positive nc symmetric polynomials as sums of squares. We recall Theorem 1.1 from [H02]: Theorem 1.1. Suppose p is a non-commutative symmetric polynomial. If p is a sum of squares, then p is matrix positive. If p is matrix positive, then p is a sum of squares. 1.2. NC Differentiation. First we make some definitions and state some properties about nc differentiation. s In the study of complex variables, we have polynomials in z and z¯. We can then take derivatives with respect to z and z¯; i.e., ∂p(z,z¯) and ∂p(z,z¯). We can also ∂z ∂z¯ make a matrix and fill it with mixed partial derivatives; i.e., the (i,j)-th entry of the matrix is ∂2p . In classical several complex variables, this matrix of mixed partial derivatives is ∂zi∂z¯j called the complex hessian. The noncommutative differentiation of polynomials in x and xT defined in this paper is analogoustoclassicaldifferentiationofpolynomialsinz andz¯fromseveral complexvariables. 1.2.1. Definition of Directional Derivative. Let p be an nc polynomial in the nc variables x = (x ,...,x ) and xT = (xT,...,xT). In order to define a directional derivative, we first 1 g 1 g replace all xT by y . Then the directional derivative of p with respect to x in the i i j direction h is j ∂p dp (1.1) p [h ] := (x,xT)[h ] = (x ,...,x +th ,...,x ,y ,...,y )| | . xj j ∂xj j dt 1 j j g 1 g t=0 yi=xTi The directional derivative of p with respect to xT in the direction k is j j ∂p dp (1.2) p [k ] := (x,xT)[k ] = (x ,...,x ,y ,...,y +tk ,...,y )| | . xTj j ∂xT j dt 1 g 1 j j g t=0 yi=xTi j Often, we take k = hT in Equation (1.2) and we define j j g ∂p dp ∂p p [h] := (x,xT)[h] = (x+th,y)| | = (x,xT)[h ] x ∂x dt t=0 y=xT ∂x i Xi=1 i g ∂p dp ∂p p [hT] := (x,xT)[hT] = (x,y +tk)| | = (x,xT)[hT]. xT ∂xT dt t=0 y=xT,k=hT ∂xT i Xi=1 i Then, we (abusively1) define the ℓth directional derivative of p in the direction h as dℓp p(ℓ)(x)[h] := (x+th,y +tk)| | dtℓ t=0 y=xT,k=hT 1Formoredetail,see[HMV06]. Theideaforcomputingp(ℓ)(x)[h]isthatwefirstnoncommutativelyexpand p(x+th). Then, p(ℓ)(x)[h] is the coefficient of tℓ multiplied by ℓ!; i.e., p(ℓ)(x)[h]=(ℓ!)(coefficient of tℓ). 4 GREENE, HELTON, AND VINNIKOV so the first directional derivative of p in the direction h is ∂p ∂p (1.3) p′(x)]h] = (x,xT)[h]+ (x,xT)[hT] ∂x ∂xT (1.4) = p [h]+p [hT]. x xT Itisimportanttonotethatthedirectionalderivativeisanncpolynomialthatishomogeneous degree 1 in h, hT. If p is symmetric, so is p′. 1.2.2. Examples of Differentiation. Here we provide some examples of how to compute directional derivatives. Example 1.2. Let p = x xTx +xTx xT. Then we have 1 2 1 1 2 1 ∂p p [h ] = (x,xT)[h ] = h xTx +x xTh x1 1 ∂x 1 1 2 1 1 2 1 1 ∂p p [hT] = (x,xT)[hT] = x hTx xT2 2 ∂xT 2 1 2 1 2 ∂p p [h] = (x,xT)[h] = h xTx +x xTh +xTh xT x ∂x 1 2 1 1 2 1 1 2 1 and, p′(x)[h] = h xTx +x hTx +x xTh +hTx xT +xTh xT +xTx hT. 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 Example 1.3. Given a general monomial, with c ∈ R, m = cxi1xi2 ···xin j1 j2 jn where each i is either 1 or T, we get that k m′ = chi1xi2 ···xin +cxi1hi2xi3 ···xin +···+cxi1 ···xin−1hin. j1 j2 jn j1 j2 j3 jn j1 jn−1 jn 1.2.3. Hessian and Complex Hessian. Often, one is most interested in the hessian of a polynomial and its positivity; as this determines convexity. However, in this paper, we are most concerned with the complex hessian, since it turns out to be related to “nc analytic” changes of variables. We define the nc complex hessian , q(x,xT)[h,hT], of an nc polynomial p as the nc polynomial in the 4g variables x = (x ,...,x ), xT = (xT,...,xT), h = (h ,...,h ), and 1 g 1 g 1 g hT = (hT,...,hT) 1 g ∂2p (1.5) q(x,xT)[h,hT] := (x+th,y +sk)| | . ∂s∂t t,s=0 y=xT,k=hT The nc complex hessian is an iterated nc directional derivative in the sense that we compute it as follows. We first take the nc directional derivative of p with respect to xT in the direction hT to get p [hT]. Then, we take the nc directional derivative of that with xT respect to x in the direction h to get (p [hT]) [h]. We will see later, in Lemma 2.12, that xT x we can switch the order of differentiation to (p [h]) [hT] and we still get the same answer. x xT Sometimes, we note the nc complex hessian as p [hT,h]. Hence, we have the following xT,x NONCOMMUTATIVE PLURISUBHARMONIC POLYNOMIALS: PART I 5 equivalent notations for the nc complex hessian (and we will use each one when context is convenient): (1.6) q(x,xT)[h,hT] = p [hT,h] = (p [hT]) [h] = (p [h]) [hT]. xT,x xT x x xT Note, an extremely important fact about q(x,xT)[h,hT], which is restated in Theorem 2.16 (P1), is that it is quadratic in h,hT and that each term contains some h and some j hT. The nc complex hessian is actually a piece of the full nc hessian which is k ∂2p ∂2p p′′ = (x+th,y)| | + (x+th,y +sk)| | ∂t2 t=0 y=xT ∂t∂s t,s=0 y=xT,k=hT ∂2p ∂2p + (x+th,y +sk)| | + (x,y +sk)| | ∂s∂t t,s=0 y=xT,k=hT ∂s2 s=0 y=xT,k=hT ∂2p ∂2p = 2q(x,xT)[h,hT]+ (x+th,y)| | + (x,y +sk)| | . ∂t2 t=0 y=xT ∂s2 s=0 y=xT,k=hT We call a symmetric nc polynomial, p, nc plurisubharmonic (or nc plush) if the nc complex hessian, q, of p is matrix positive. In other words, we require that q be positive semidefinite when evaluated on all tuples of n×n matrices for every size n; i.e., q(X,XT)[H,HT] (cid:23) 0 for all X,H ∈ (Rn×n)g for every n ≥ 1. 1.2.4. Examples of Complex Hessians. Here we provide some examples of how to compute nc complex hessians. Example 1.4. Let p = x xTx +xTx xT as in Example 1.2. Then, we have 1 2 1 1 2 1 q = h hTx +x hTh +hTh xT +xTh hT. 1 2 1 1 2 1 1 2 1 1 2 1 Example 1.5. Let p = xTxTxx. Then, we have q(x,xT)[h,hT] = hTxThx+hTxTxh+xThThx+xThTxh = (hTxT +xThT)(hx+xh) = (hx+xh)T(hx+xh). We can see that, for any X,H ∈ Rn×n for any size n ≥ 1, we have that q(X,XT)[H,HT] = (HX +XH)T(HX +XH) (cid:23) 0. Hence, this nc polynomial, p = xTxTxx, is nc plush. Example 1.6. The nc complex hessian of any nc analytic polynomial is 0. The nc complex hessian of any nc antianalytic polynomial is 0. Hence, both nc analytic and nc antianalytic polynomials are nc plush. 6 GREENE, HELTON, AND VINNIKOV 1.3. Main Result. In this paper we classify all symmetric nc plush polynomials in g free variables. Theorem 1.7. An nc symmetric polynomial p in free variables is nc plurisubharmonic if and only if p can be written in the form (1.7) p = fTf + k kT +F +FT j j j j X X where the sums are finite and each f , k , F is nc analytic. j j (cid:3) Proof. The proof requires the rest of this paper and culminates in Section 4. The subsequent paper ([G10]) strengthens the result of Theorem 1.7 by weakening the hypothesis whilekeeping thesameconclusion. Specifically, weassumethatthencpolynomial is nc plush on “an nc open set” and conclude that it is nc plush everywhere and hence has the form in Equation (1.7). The proof, in [G10], draws on most of the theorems in this paper together with a very different technique involving representations of noncommutative quadratic functions. The representation in Equation (1.7) is unique up to the natural transformations. Theorem 1.8. Let p be an nc symmetric polynomial in free variables that is nc plurisubharmonic and let N M N := min{N : p = fTf + k kT +F +FT} j j j j Xj=1 Xj=1 e N M M := min{M : p = fTf + k kT +F +FT}. j j j j Xj=1 Xj=1 f Then, we can represent p as e f N M p = f˜Tf˜ + k˜ k˜T +F˜ +F˜T j j j j Xj=1 Xj=1 and if N and M are integers such that N ≥ N, M ≥ M and N Me f p = fTf + k kT +F +FT , j j j j Xj=1 Xj=1 then there exist isometries U : RNe −→ RN and U : RMf −→ RM such that 1 2 f f˜ kT k˜T 1 1 1 1 . . . .  ..  = U1 .. +~c1 and  ..  = U2 .. +~c2  fN   fÑe   kMT   k˜MTf  where ~c ∈ RN and ~c ∈ RM. 1 2 NONCOMMUTATIVE PLURISUBHARMONIC POLYNOMIALS: PART I 7 Proof. Theorem 1.7 gives the desired form of p and nc integration will give the uniqueness. (cid:3) We provide the details of the proof in Section 4. A byproduct of the proof of Theorem 1.7 is noncommutative integration theory of nc polynomials. This includes a Frobenius theorem for nc polynomials and is discussed further in Section 2. 1.4. Guide to the Paper. InSection2, we providea theoryofnoncommutative integration for nc polynomials and in Section 2.4, we state and prove a noncommutative version of the Frobenius theorem. In Section 3, we prove that the nc complex hessian for an nc plush polynomial is the sum of hereditary and antihereditary squares. Finally, in Section 4, we prove the main results. We apply nc integration theory to the sum of squares representation of the nc complex hessian found in Section 3. We also settle the issue of uniqueness of this sum of squares representation. The authors would like to thank Mark Stankus for fruitful discussions on noncommutative integration. 2. NC Integration In this section, we introduce a natural notion of noncommutative (nc) integration and then give some basic properties. We say that an nc polynomial p in x = (x ,...,x ) and h 1 g j is integrable in x if there exists an nc polynomial f(x) such that f [h ] = p. We say that j xj j an nc polynomial p in x = (x ,...,x ) and h = (h ,...,h ) is integrable if there exists an 1 g 1 g nc polynomial f(x) such that f′(x)[h] = p. 2.1. Notation. Let m be a monomial containing only the variables x , x , ..., x . When 1 2 g we write m| , we mean the set of monomials that are degree one in h where one x in xi→hi i i m has been replaced by h . For example, if m = x x x x , then i 1 2 1 2 m| = {h x x x , x x h x } x1→h1 1 2 1 2 1 2 1 2 and m| = {x h x x , x x x h }. x2→h2 1 2 1 2 1 2 1 2 We also define a double substitution as follows. When we write m| := (m| )| , xi→hi,xj→hj xi→hi xj→hj we mean the set of monomials that are degree one in h and degree one in h where one x i j i in m has been replaced by h and one x in m has been replaced by h . Note that we have i j j (2.1) m| = m| . xi→hi,xj→hj xj→hj,xi→hi Using m = x x x x , we have that 1 2 1 2 m| = m| x1→h1,x2→h2 x2→h2,x1→h1 = {h h x x , h x x h , x h h x , x x h h }. 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 8 GREENE, HELTON, AND VINNIKOV Sometimes we will start with a monomial m that is degree 1 in h and we wish to replace i this h by x . When we write m| , the set we get contains just one monomial so we abuse i i hi→xi notation and use m| to represent the actual monomial in this set. hi→xi 2.2. Differentially Wed Monomials. For γ either 1 or T, two monomials m and m˜ are called 1-differentially wed with respect to xγ if both m and m˜ contain the same hγ to j j exactly first order and if m has an xγ where m˜ has an hγ and if m˜ has an xγ where m has j j j an hγ. Thus interchanging hγ and this xγ in m produces m˜; i.e., j j j m|hγ→xγ = m˜|hγ→xγ j j j j Moregenerally, forα,β either1orT,twomonomialsmandm˜ arecalled1-differentially wed if both are degree one in h or hT and if m| = m˜| hαi→xαi hβj→xβj From these definitions, if m and m˜ are 1-differentially wed with respect to a particular variable then m and m˜ are 1-differentially wed but not the other way around (which we demonstrate below). Example 2.1. The monomials m = h xTx and m˜ = x xTh are 1-differentially wed with 1 2 1 1 2 1 respect to x . 1 Example 2.2. The monomials m = h xTx and m˜ = x hTx are 1-differentially wed (but 1 2 1 1 2 1 not with respect to a particular variable). Example 2.3. The monomials m = x h x and m˜ = x x h are not 1-differentially wed 2 2 2 1 2 2 with respect to any variable. Theorem 2.4. A polynomial p in x = (x ,...,x ),h = (h ,...,h ) is integrable if and only 1 g 1 g if each monomial in p has degree one in h (i.e., contains some h ) and whenever a monomial j m occurs in p, each monomial which is 1-differentially wed to m also occurs in p and has the same coefficient. Proof. First suppose the polynomial p in x,h can be integrated in x. Then there exists a polynomial, f(x), such that f′(x)[h] = p. Write f as N f = m i Xi=1 where each m is a monomial in x. Then, by applying Example 1.3 to each monomial m , if i i a monomial m˜ occurs in p = f′, then every 1-differentially wed monomial to m˜ also occurs in p = f′ with the same coefficient. Now suppose each monomial in p has degree 1 in h (i.e., contains some h ) and if m is j a monomial in p, then each monomial which is 1-differentially wed also occurs in p with the same coefficient. We will show that p can be integrated in x NONCOMMUTATIVE PLURISUBHARMONIC POLYNOMIALS: PART I 9 Write N p = m i Xi=1 where m is a monomial in x and degree 1 in h. Now we will change the order of summation i of these monomials so that we group together all monomials that are 1-differentially wed. We do this in the following way. Let w be the polynomial that contains m and all 1-differentially wed monomials to 1 1 m . 1 Let 1 ≤ α be the smallest integer such that m is not a term in w . Then let w be 2 α2 1 2 the polynomial that contains m and all 1-differentially wed monomials to m . α2 α2 Let 1 ≤ α be the smallest integer such that m is not a term in w and not a term in 3 α3 1 w . Then let w be the polynomial that contains m and all 1-differentially wed monomials 2 3 α3 to m . α3 We continue this process until it stops (it stops since p is a finite sum of monomials). Then we have written p as ℓ p = w . i Xi=1 It is important to note that with this construction, each w is a homogeneous polynomial i of some fixed degree where each monomial in w is degree 1 in h. i Now define α = 1 and 1 f (x) := m | , 1 ≤ i ≤ ℓ. i αi h→x Then we have, by properties of differentiation and construction of w , that f′ = w . i i i Finally, define ℓ f(x) := f (x) i Xi=1 and notice that f′ = p. (cid:3) Corollary 2.5. A polynomial p in x,h is integrable in x if and only if each monomial in j j p has degree one in h and whenever a monomial m occurs in p, each monomial which is j 1-differentially wed with respect to x also occurs in p and has the same coefficient. j 2.3. Uniqueness of Noncommutative Integration. In this subsection, we explore the uniqueness of noncommutative integration. In classical calculus, integrating produces constants of integration. Here, we provide the noncommutative analogue. Proposition2.6. Suppose mand m˜ are distinctmonomialsinthe variablesx = (x ,...,x ). 1 g Then, we have that (1) (m) [h ] and (m˜) [h ] have no terms in common and hence xi i xi i (m) [h ] 6= (m˜) [h ] xi i xi i 10 GREENE, HELTON, AND VINNIKOV provided x is contained in either m or m˜; and i (2) we have that m′ = (m) [h] 6= (m˜) [h] = m˜′. x x If m and m˜ are distinct monomials in the variables x = (x ,...,x ) and y = (y ,...,y ), 1 g 1 s then m [h] 6= m˜ [h]. x x Proof. If m and m˜ have different degree, then so do their derivatives and we are done. Suppose m and m˜ have the same degree. If m contains x and m˜ does not, then (m˜) [h ] = 0 i xi i while (m) [h ] is a nonzero nc polynomial. xi i Suppose both m and m˜ contain x and are the same degree. Then, write i m = x x ···x and m˜ = x x ···x j1 j2 js k1 k2 ks wherethetupleofintegers(j ,j ,...,j )isnotthesameasthetupleofintegers(k ,k ,...,k ). 1 2 s 1 2 s This forces (m) [h ] and (m˜) [h ] to have no terms in common. This completes the proof xi i xi i of (1). To prove (2), note that g g m′ = (m) [h] = (m) [h ] and m˜′ = (m˜) [h] = (m˜) [h ] x xi i x xi i Xi=1 Xi=1 and if m′ = m˜′, then we must have (m) [h ] = m˜ [h ] for each i. However, (1) implies that xi i xi i this is impossible. If m and m˜ are distinct monomials in the variables x = (x ,...,x ) and y = (y ,...,y ), 1 g 1 g (cid:3) the proof follows exactly the way the proof of (2) does. Lemma 2.7. Suppose p is an nc polynomial in the variables x = (x ,...,x ) such that 1 g p [h ] = 0. Then, p(x ,...,x ) = f(x ,...,x ,x ,...,x ) is an nc polynomial in the xi i 1 g 1 i−1 i+1 g variables x ,...,x ,x ,...,x . 1 i−1 i+1 g Proof. First, if m is a monomial in the variables x = (x ,...,x ) that contains x , then 1 g i m [h ] is a sum of terms where each instance of x is replaced by h (see Example 1.2). Note xi i i i that each term in m [h ] has a different number of variables to the left of h ; hence, the xi i i terms can not cancel. Thus, m [h ] 6= 0. xi i Now suppose p is an nc polynomial in the variables x = (x ,...,x ). We write the nc 1 g polynomial p as N (2.2) p = α m j j Xj=1 where the α are nonzero real constants and the m are distinct monomials. Then, we have j j that N (2.3) 0 = p [h ] = α (m ) [h ]. xi i j j xi i Xj=1