Operator Theory: Advances and Applications, Vol 207, 223 254 c 2010 Birkhäuser Verlag Basel/Switzerland The Operator Fejér-Riesz Theorem Michael A Dritschel and James Rovnyak To the memory of Paul Richard Halmos Abstract The Fejér-Riesz theorem has inspired numerous generalizations in one and several variables, and for matrix- and operator-valued functions This paper is a survey of some old and recent topics that center around Rosenblum s operator generalization of the classical Fejér-Riesz theorem Mathematics Subject Classification (2000) Primary 47A68; Secondary 60G25, 47A56, 47B35, 42A05, 32A70, 30E99 Keywords Trigonometric polynomial, Fejér-Riesz theorem, spectral factorization, Schur complement, noncommutative polynomial, Toeplitz operator, shift operator 1 Introduction The classical Fejér-Riesz factorization theorem gives the form of a nonnegative trigonometric polynomial on the real line, or, equivalently, a Laurent polynomial that is nonnegative on the unit circle For the statement, we write D = {z : z < 1} and T = {ζ : ζ =1} for the open unit disk and unit circle in the complex plane Fejér-Riesz Theorem A Laurent polynomial q(z) = m k= m q kz k which has complex coefficients and satisfies q(ζ) 0 for all ζ T can be written q(ζ) = p(ζ) 2, ζ T, for some polynomial p(z) =p 0 + p 1 z + + p m z m,andp(z) can be chosen to have no zeros in D The original sources are Fejér [22] and Riesz [47] The proof is elementary and consists in showing that the roots of q(z) occur in pairs z j and 1/ z j with z j 1 Then the required polynomial p(z) is the product of the factors z z j adjusted by a suitable multiplicative constant c Details appear in many places; see, eg, [28, p 20], [34, p 235], or [60, p 26]
224 MA Dritschel and J Rovnyak The Fejér-Riesz theorem arises naturally in spectral theory, the theory of orthogonal polynomials, prediction theory, moment problems, and systems and control theory Applications often require generalizations to functions more general than Laurent polynomials, and, more than that, to functions whose values are matrices or operators on a Hilbert space The spectral factorization problem is to write a given nonnegative matrix- or operator-valued function F on the unit circle in the form F = G G where G has an analytic extension to the unit disk (in a suitably interpreted sense) The focal point of our survey is the special case of a Laurent polynomial with operator coefficients The operator Fejér-Riesz theorem (Theorem 21) obtains a conclusion similar to the classical result for a Laurent polynomial whose coefficients are Hilbert space operators: if Q j, j = m,,m, are Hilbert space operators such that m Q(ζ) = Q k ζ k 0, ζ T, (11) k= m then there is a polynomial P (z) =P 0 +P 1 z + +P m z m with operator coefficients such that Q(ζ) =P (ζ) P (ζ), ζ T (12) This was first proved in full generality in 1968 by Marvin Rosenblum [49] The proof uses Toeplitz operators and a method of Lowdenslager, and it is a fine example of operator theory in the spirit of Paul Halmos Rosenblum s proof is reproduced in 2 Part of the fascination of the operator Fejér-Riesz theorem is that it can be stated in a purely algebraic way The hypothesis (11) on Q(z) isequivalenttothe statement that an associated Toeplitz matrix is nonnegative The conclusion (12) is equivalent to 2m + 1 nonlinear equations whose unknowns are the coefficients P 0,P 1,,P m of P (z) Can it be that this system of equations can be solved by an algebraic procedure? The answer is, yes, and this is a recent development The iterative procedure uses the notion of a Schur complement and is outlined in 3 There is a surprising connection between Rosenblum s proof of the operator Fejér-Riesz theorem and spectral factorization The problem of spectral factorization is formulated precisely in 4, using Hardy class notions A scalar prototype is Szegő s theorem (Theorem 41) on the representation of a positive integrable and log-integrable function w on the unit circle in the form h 2 for some H 2 function h The operator and matrix counterparts of Szegő s theorem, Theorems 45 and 47, have been known for many years and go back to fundamental work in the 1940s and 1950s which was motivated by applications in prediction theory (see the historical notes at the end of 4) We present a proof that is new to the authors and we suspect not widely known It is based on Theorem 43, which traces its origins to Rosenblum s implementation of the Lowdenslager method In 4 wealso state without proof some special results that hold in the matrix case The method of Schur complements points the way to an approach to multivariable factorization problems, which is the subject of 5 Even in the scalar case,
The Operator Fejér-Riesz Theorem 225 the obvious first ideas for multivariable generalizations of the Fejér-Riesz theorem are false by well-known examples Part of the problem has to do with what one might think are natural restrictions on degrees In fact, the restrictions on degrees are not so natural after all When they are removed, we can prove a result, Theorem 51, that can be viewed as a generalization of the operator Fejér-Riesz theorem in the strictly positive case We also look at the problem of outer factorization, at least in some restricted settings In recent years there has been increasing interest in noncommutative function theory, especially in the context of functions of freely noncommuting variables In 6 we consider noncommutative analogues of the d-torus, and corresponding notions of nonnegative trigonometric polynomials In the freely noncommutative setting, there is a very nice version of the Fejér-Riesz theorem (Theorem 61) In a somewhat more general noncommutative setting, which also happens to cover the commutative case as well, we have a version of Theorem 51 for strictly positive polynomials (Theorem 62) Our survey does not aim for completeness in any area In particular, our bibliography represents only a selection from the literature The authors regret and apologize for omissions 2 The operator Fejér-Riesz theorem In this section we give the proof of the operator Fejér-Riesz theorem by Rosenblum [49] The general theorem had precursors A finite-dimensional version was given by Rosenblatt [48], an infinite-dimensional special case by Gohberg [26] We follow standard conventions for Hilbert spaces and operators If A is an operator, A is its adjoint Norms of vectors and operators are written Except where noted, no assumption is made on the dimension of a Hilbert space, and nonseparable Hilbert spaces are allowed Theorem 21 (Operator Fejér-Riesz Theorem) Let Q(z) = m k= m Q kz k be a Laurent polynomial with coefficients in L(G) for some Hilbert space G IfQ(ζ) 0 for all ζ T, then Q(ζ) =P (ζ) P (ζ), ζ T, (21) for some polynomial P (z) =P 0 + P 1 z + + P m z m with coefficients in L(G) The polynomial P (z) can be chosen to be outer The definition of an outer polynomial will be given later; in the scalar case, a polynomial is outer if and only if it has no zeros in D The proof uses (unilateral) shift and Toeplitz operators (see [11] and [29]) By a shift operator here we mean an isometry S on a Hilbert space H such that the unitary component of S in its Wold decomposition is trivial With natural identifications, we can write H = G G for some Hilbert space G and S(h 0,h 1,)=(0,h 0,h 1,)
226 MA Dritschel and J Rovnyak when the elements of H are written in sequence form Suppose that such a shift S is chosen and fixed If T,A L(H), we say that T is Toeplitz if S TS = T,and that A is analytic if AS = SA An analytic operator A is said to be outer if ran A is a subspace of H of the form F F for some closed subspace F of G As block operator matrices, Toeplitz and analytic operators have the forms T 0 T 1 T 2 A 0 0 0 T 1 T 0 T 1 A 1 A 0 0 T = T 2 T 1 T, A = 0 A 2 A 1 A (22) 0 Here { E 0 S j TE 0 G, j 0, T j = (23) E0 TS j E 0 G, j < 0, where E 0 g =(g, 0, 0,) is the natural embedding of G into H For examples, consider Laurent and analytic polynomials Q(z) = m k= m Q kz k and P (z) = P 0 + P 1 z + + P m z m with coefficients in L(G) Set Q j =0for j >mand P j =0forj>m Then the formulas Q 0 Q 1 Q 2 P 0 0 0 Q 1 Q 0 Q 1 P 1 P 0 0 T Q = Q 2 Q 1 Q, T P = 0 P 2 P 1 P (24) 0 define bounded operators on H Boundedness follows from the identity Q(ζ)f(ζ),g(ζ) G dσ(ζ) = Q j k f k,g j G, (25) T k,j=0 where σ is normalized Lebesgue measure on T and f(ζ) =f 0 + f 1 ζ + f 2 ζ 2 + and g(ζ) =g 0 + g 1 ζ + g 2 ζ 2 + have coefficients in G, all but finitely many of which are zero The operator T Q is Toeplitz, and T P is analytic Moreover, Q(ζ) 0 for all ζ T if and only if T Q 0; Q(ζ) =P (ζ) P (ζ) for all ζ T if and only if T Q = TP T P Definition 22 We say that the polynomial P (z) is outer if the analytic Toeplitz operator A = T P is outer In view of the example (24), the main problem is to write a given nonnegative Toeplitz operator T in the form T = A A,whereA is analytic We also want to know that if T = T Q for a Laurent polynomial Q, thenwecanchoosea = T P for an outer analytic polynomial P of the same degree Lemmas 23 and 24 reduce the problem to showing that a certain isometry is a shift operator
The Operator Fejér-Riesz Theorem 227 Lemma 23 (Lowdenslager s Criterion) Let H be a Hilbert space, and let S L(H) be a shift operator Let T L(H) be Toeplitz relative to S as defined above, and suppose that T 0 LetH T be the closure of the range of T 1/2 in the inner product of H Then there is an isometry S T mapping H T into itself such that S T T 1/2 f = T 1/2 Sf, f H In order that T = A A for some analytic operator A L(H), it is necessary and sufficient that S T is a shift operator In this case, A can be chosen to be outer Proof The existence of the isometry S T follows from the identity S TS = T, which implies that T 1/2 Sf and T 1/2 f have the same norms for any f H If S T is a shift operator, we can view H T as a direct sum H T = G T G T with S T (h 0,h 1,)=(0,h 0,h 1,) Here dim G T dim G Toseethis,notice that a short argument shows that T 1/2 ST and S T 1/2 agree on H T, and therefore T 1/2 (ker ST ) ker S The dimension inequality then follows because T 1/2 is oneto-one on the closure of its range Therefore we may choose an isometry V from G T into G Define an isometry W on H T into H by W (h 0,h 1,)=(Vh 0,Vh 1,) Define A L(H) by mapping H into H T via T 1/2 and then H T into H via W : Af = WT 1/2 f, f H Straightforward arguments show that A is analytic, outer, and T = A A Conversely, suppose that T = A A where A L(H) is analytic Define an isometry W on H T into H by WT 1/2 f = Af, f H ThenWS T = SW, and hence ST n = W S n W for all n 1 Since the powers of S tend strongly to zero, so do the powers of ST, and therefore S T is a shift operator Lemma 24 In Lemma 23, lett = T Q be given by (24) for a Laurent polynomial Q(z) of degree m IfT = A A where A L(H) is analytic and outer, then A = T P for some outer analytic polynomial P (z) of degree m Proof Let Q(z) = m k= m Q kz k Recall that Q j =0for j >m By (23) applied to A, what we must show is that S j AE 0 =0forallj>m It is sufficient to show that S m+1 AE 0 = 0 By (23) applied to T,sinceT = A A and A is analytic, E 0 A S j AE 0 = E 0 Sj TE 0 = Q j =0, j > m It follows that ran S m+1 AE 0 ran AS k E 0 for all k 0, and therefore ran S m+1 AE 0 ran A (26) Since A is outer, ran A reduces S, and so rans m+1 AE 0 S m+1 AE 0 = 0 by (26), and the result follows ran A Therefore The proof of the operator Fejér-Riesz theorem is now easily completed
228 MA Dritschel and J Rovnyak Proof of Theorem 21 Define T = T Q as in (24) Lemmas 23 and 24 reduce the problem to showing that the isometry S T is a shift operator It is sufficient to show that ST n f 0 for every f in H T Claim: If f = T 1/2 h where h H has the form h = (h 0,,h r, 0,), then ST n f = 0 for all sufficiently large n For if u H and n is any positive integer, then ST n f,t 1/2 u = f,st n T 1/2 u = T 1/2 h, T 1/2 S n u = H T H T H Th,Sn u H By the definition of T = T Q, Th has only a finite number of nonzero entries (depending on m and r), and the first n entries of S n u are zero (irrespective of u) The claim follows from the arbitrariness of u In view of the claim, ST n f 0 for a dense set of vectors in H T, and hence by approximation this holds for all f in H T ThusS T is a shift operator, and the result follows A more general result is proved in the original version of Theorem 21 in [49] There it is only required that Q(z)g is a Laurent polynomial for a dense set of g in G (the degrees of these polynomials can be unbounded) We have omitted an accompanying uniqueness statement: the outer polynomial P (z) in Theorem 21 canbechosensuchthatp (0) 0, and then it is unique See [2] and [50] 3 Method of Schur complements We outline now a completely different proof of the operator Fejér-Riesz theorem The proof is due to Dritschel and Woerdeman [19] and is based on the notion of a Schur complement The procedure constructs the outer polynomial P (z) = P 0 + P 1 z + + P m z m one coefficient at a time A somewhat different use of Schur complements in the operator Fejér-Riesz theorem appears in Dritschel [18] The method in [18] plays a role in the multivariable theory, which is taken up in 5 We shall explain the main steps of the construction assuming the validity of two lemmas Full details are given in [19] and also in the forthcoming book [3] by Bakonyi and Woerdeman The authors thank Mihaly Bakonyi and Hugo Woerdeman for advance copies of key parts of [3], which has been helpful for our exposition The book [3] includes many additional results not discussed here Definition 31 Let H be a Hilbert space Suppose T L(H), T 0 Let K be a closed subspace of H, andletp K L(H, K) be orthogonal projection of H onto K Then (see Appendix, Lemma A2) there is a unique operator S L(K), S 0, such that (i) T PK SP K 0; (ii) if S L(K), S 0, and T PK SP K 0, then S S We write S = S(T,K) andcalls the Schur complement of T supported on K
The Operator Fejér-Riesz Theorem 229 Schur complements satisfy an inheritance property, namely, if K K + H, then S(T,K )=S(S(T,K + ), K ) If T isspecifiedinmatrixform, ( ) A B T = : K K B C K K, then S = S(T,K) is the largest nonnegative operator in L(K) such that ( ) A S B 0 B C The condition T 0 is equivalent to the existence of a contraction G L(K, K ) such that B = C 1 2 GA 1 2 (Appendix, Lemma A1) In this case, G can be chosen so that it maps ran A into ran C and is zero on the orthogonal complement of ran A, and then S = A 1 2 (I G G)A 1 2 When C is invertible, this reduces to the familiar formula S = A B C 1 B Lemma 32 Let M L(H), M 0, and suppose that ( ) A B M = : K K K K B C for some closed subspace K of H (1) If S(M,K) =P P and C = R R for some P L(K) and R L(K ),then there is a unique X L(K, K ) such that ( P X M = )( P 0 0 R X R ) and ran X ran R (31) (2) Conversely, if (31) holds for some operators P, X, R, thens(m,k) =P P We omit the proof and refer the reader to [3] or [19] for details Proof of Theorem 21 using Schur complements Let Q(z) = m k= m Q kz k satisfy Q(ζ) 0 for all ζ T We shall recursively construct the coefficients of an outer polynomial P (z) =P 0 + P 1 z + + P m z m such that Q(ζ) =P (ζ) P (ζ), ζ T Write H = G G and G n = G G with n summands As before, set Q k =0for k >m, and define T Q L(H) by Q 0 Q 1 Q 2 Q 1 Q 0 Q 1 T Q = Q 2 Q 1 Q 0 For each k =0, 1, 2,, define S(k) =S(T Q, G k+1 ),
230 MA Dritschel and J Rovnyak which we interpret as the Schur complement of T Q on the first k + 1 summands of H = G G ThusS(k) isa(k +1) (k + 1) block operator matrix satisfying S(S(k), G j+1 )=S(j), 0 j<k<, (32) by the inheritance property of Schur complements Lemma 33 For each k =0, 1, 2,, ( ) Y 0 Y1 Y k+1 Y 1 S(k +1)= S(k) Yk+1 for some operators Y 0,Y 1,,Y k+1 in L(G) Fork m 1, ( Y0 Y 1 Y k+1 ) = ( Q0 Q 1 Q k 1 ) Again see [3] or [19] for details Granting Lemmas 32 and 33, we can proceed with the construction Construction of P 0,P 1 Choose P 0 = S(0) 1 2 Using Lemma 33, write ( ) Y0 Y S(1) = 1 Y1 S(0) In Lemma 32(1) take M = S(1) and use the factorizations S(S(1), G 1 ) (32) = S(0) = P 0 P 0 and S(0) = P 0 P 0 Choose P 1 = X where X L(G) is the operator produced by Lemma 32(1) Then ( )( ) P S(1) = 0 P1 P0 0 0 P0 and ran P P 1 P 1 ran P 0 (33) 0 Construction of P 2 Next use Lemma 33 to write ( ) Y 0 Y1 Y 2 S(2) = ( ) Y, S(1) 1 Y 2 and apply Lemma 32(1) to M = S(2) with the factorizations S(S(2), G 1 ) (32) = S(0) = P0 P 0, ( )( ) P S(1) = 0 P1 P0 0 0 P0 P 1 P 0
The Operator Fejér-Riesz Theorem 231 This yields operators X 1,X 2 L(G) such that S(2) = P 0 X1 X2 0 P0 P1 P 0 0 0 X 1 P 0 0, (34) 0 0 P0 X 2 P 1 P 0 ( ) ( ) X1 P0 0 ran ran (35) X 2 P 1 P 0 In fact, X 1 = P 1 To see this, notice that we can rewrite (34) as ( ) P X )( P 0 S(2) = 0 R X R, ( ) P0 0 ( ) P =, X = X2 P X 1 P 1, R = P0 0 By (33) and (35), ran P 1 ran P 0 and ran X 2 ran P 0, and therefore ran X ran P 0 Hence by Lemma 32(2), ( )( ) S(S(2), G 2 )= P P P = 0 X1 P0 0 0 P0 (36) X 1 P 0 Comparing this with ( )( S(S(2), G 2 ) (32) = S(1) (33) P = 0 P1 P0 0 P0 P 1 ) 0, P 0 (37) we get P0 P 1 = P0 X 1 By (35), ran X 1 ran P 0, and therefore X 1 = P 1 Now choose P 2 = X 2 to obtain S(2) = P 0 P1 P2 0 P0 P1 P 0 0 0 P 1 P 0 0, (38) 0 0 P0 P 2 P 1 P 0 ( ) ( ) P1 P0 0 ran ran (39) P 2 P 1 P 0 Inductive step We continue in the same way for all k =1, 2, 3,Atthekth stage, the procedure produces operators P 0,,P k such that P0 Pk P 0 0 S(k) =, (310) 0 P0 P k P 0 P 1 P 0 0 ran ran (311) P k P k P 0
232 MA Dritschel and J Rovnyak By Lemma 33, in the case k m, ( Q 0 Q 1 Q m 0 0 ) Q 1 S(k) = Q m (312) 0 S(k 1) 0 The zeros appear here when k>m, and their presence leads to the conclusion that P k =0fork>m We set then P (z) =P 0 + P 1 z + + P m z m Comparing (310) and (312) in the case k = m, we deduce 2m +1relations which are equivalent to the identity Q(ζ) =P (ζ) P (ζ), ζ T Final step: P (z) is outer Define T P as in (24) With natural identifications, ( ) P 0 0 0 P 1 T P = P 2 T P (313) The relations (311), combined with the fact that P k =0forallk>m,implythat P 1 ran P 2 ran T P Hence for any g G, a sequence f n can be found such that P 1 T P f n P 2 g Then by (313), T P ( g f n ) ( ) P0 g 0 It follows that ran T P contains every vector (P 0 g, 0, 0,)withg L(G), and hence ran T P ran P 0 ran P 0 The reverse inclusion holds because by (311), the ranges of P 1,P 2, are all contained in ran P 0 ThusP(z)isouter
4 Spectral factorization The Operator Fejér-Riesz Theorem 233 The problem of spectral factorization is to write a nonnegative operator-valued function F on the unit circle in the form F = G G where G is analytic (in a sense made precise below) The terminology comes from prediction theory, where the nonnegative function F plays the role of a spectral density for a multidimensional stationary stochastic process The problem may be viewed as a generalization of a classical theorem of Szegő from Hardy class theory and the theory of orthogonal polynomials (see Hoffman [35, p 56] and Szegő [62, Chapter X]) We write H p and L p for the standard Hardy and Lebesgue spaces for the unit disk and unit circle See Duren [20] Recall that σ is normalized Lebesgue measure on the unit circle T Theorem 41 (Szegő s Theorem) Let w L 1 satisfy w 0 ae on T and log w(ζ) dσ > T Then w = h 2 ae on T for some h H 2,andh can be chosen to be an outer function Operator and matrix generalizations of Szegő s theorem are stated in Theorems 45 and 47 below Some vectorial function theory is needed to formulate these and other results We assume familiarity with basic concepts but recall a few definitions For details, see, eg, [30, 61] and [50, Chapter 4] In this section, G denotes a separable Hilbert space Functions f and F on the unit circle with values in G and L(G), respectively, are called weakly measurable if f(ζ),v and F (ζ)u, v are measurable for all u, v G Nontangential limits for analytic functions on the unit disk are taken in the strong (norm) topology for vector-valued functions, and in the strong operator topology for operator-valued functions We fix notation as follows: (i) We write L 2 G and L L(G) for the standard Lebesgue spaces of weakly measurable functions on the unit circle with values in G and L(G) (ii) Let HG 2 and H L(G) be the analogous Hardy classes of analytic functions on the unit disk We identify elements of these spaces with their nontangential boundary functions, and so the spaces may alternatively be viewed as subspaces of L 2 G and L L(G) (iii) Let N + L(G) be the space of all analytic functions F on the unit disk such that ϕf belongs to HL(G) for some bounded scalar outer function ϕ The elements of N + L(G) are also identified with their nontangential boundary functions A function F HL(G) is called outer if FH2 G is dense in H2 F for some closed subspace F of G A function F N + L(G) is outer if there is a bounded scalar outer function ϕ such that ϕf HL(G) and ϕf is outer in the sense just defined The definition of an outer function given here is consistent with the previously defined notion for polynomials in 2
234 MA Dritschel and J Rovnyak A function A HL(G) is called inner if multiplication by A on H2 G is a partial isometry In this case, the initial space of multiplication by A is a subspace of H 2 G of the form H 2 F where F is a closed subspace of G Toprovethis,noticethatboth the kernel of multiplication by A and the set on which it is isometric are invariant under multiplication by z Therefore the initial space of multiplication by A is a reducing subspace for multiplication by z, and so it has the form H 2 F where F is a closed subspace of G (see [29, p 106] and [50, p 96]) Every F HL(G) has an inner-outer factorization F = AG, wherea is an inner function and G is an outer function This factorization can be chosen such that the isometric set HF 2 for multiplication by A on H2 G coincides with the closure of the range of multiplication by G The inner-outer factorization is extended in an obvious way to functions F N + L(G) Details are given, for example, in [50, Chapter 5] The main problem of this section can now be interpreted more precisely: Factorization Problem Given a nonnegative weakly measurable function F on T, find a function G in N + L(G) such that F = G G ae on T If such a function exists, we say that F is factorable If a factorization exists, the factor G can be chosen to be outer by the innerouter factorization Moreover, an outer factor G can be chosen such that G(0) 0, and then it is unique [50, p 101] By the definition of N + L(G), a necessary condition for F to be factorable is that log + F (ζ) dσ <, (41) T where log + x is zero or log x according as 0 x 1or1<x<, andsoweonly need consider functions which satisfy (41) In fact, in proofs we can usually reduce to the bounded case by considering F/ ϕ 2 for a suitable scalar outer function ϕ The following result is another view of Lowdenslager s criterion, which we deduce from Lemma 23 A direct proof is given in [61, pp 201 203] Lemma 42 Suppose F L L(G) and F 0 ae on T LetK F be the closure of F 1 2 HG 2 in L2 G,andletS F be the isometry multiplication by ζ on K F ThenF is factorable if and only if S F is a shift operator, that is, ζ n F 1 2 HG 2 = {0} (42) n=0 Proof In Lemma 23 take H = HG 2 viewed as a subspace of L2 G,andletS be multiplication by ζ on H Define T L(H) bytf = PFf, f H, wherep is the projection from L 2 G onto H2 G One sees easily that T is a nonnegative Toeplitz operator, and so we can define H T and an isometry S T as in Lemma 23 In fact, S T is unitarily equivalent to S F via the natural isomorphism W : H T K F such that W (T 1 2 f) =F 1 2 f for every f in H ThusS F is a shift operator if and only if S T is a shift operator, and by Lemma 23 this is the same as saying that T = A A where A L(H) is analytic, or equivalently F is factorable [50, p 110]
The Operator Fejér-Riesz Theorem 235 We obtain a very useful sufficient condition for factorability Theorem 43 Suppose F L L(G) and F 0 ae For F to be factorable, it is sufficient that there exists a function ψ in L L(G) such that (i) ψf HL(G) ; (ii) for all ζ T except at most a set of measure zero, ψ(ζ) F (ζ)g is one-to-one If these conditions are met and F = G G ae with G outer, then ψg HL(G) Theorem 43 appears in Rosenblum [49] with ψ(ζ) = ζ m (viewed as an operator-valued function) The case of an arbitrary inner function was proved and applied in a variety of ways by Rosenblum and Rovnyak [50, 51] V I Matsaev first showed that more general functions ψ can be used Matsaev s result is evidently unpublished, but versions were given by DZ Arov [1, Lemma to Theorem 4] and AS Markus [41, Theorem 343 on p 199] Theorem 43 includes all of these versions We do not know if the conditions (i) and (ii) in Theorem 43 are necessary for factorability It is not hard to see that they are necessary in the simple cases dim G =1anddimG = 2 (for the latter case, one can use [50, Example 1, p 125]) The general case, however, is open Proof of Theorem 43 Let F satisfy (i) and (ii) Define a subspace M of L 2 G by M = ζ n F 1 2 HG 2 = ζ n F 1 2 HG 2 n=0 We show that M = {0} By(i), ψf 1 1 2 M = ψf 2 ζ n F 1 2 HG 2 ζ n ψfhg 2 ζ n HG 2 = {0} (43) n=0 n=0 Thus ψf 1 2 M = {0} Nowifg M, thenψf 1 2 g =0aeby(43)HenceF 1 2 g =0 ae by (ii) By the definition of M, g F 1 2 HG 2, and standard arguments show from this that g(ζ) F (ζ) 1 2 G ae Therefore g = 0 ae It follows that M = {0}, and so F is factorable by Lemma 42 Let F = G G ae with G outer We prove that ψg HL(G) by showing that ψg HG 2 H2 G SinceG is outer, GH2 G = H2 F for some closed subspace F of G By (i), ψg (GHG 2 )=ψfh2 G H2 G Therefore ψg HF 2 H2 G Suppose f H2 G F, and consider any h L2 G Then G f,h L 2 = f(ζ),g(ζ)h(ζ) G G dσ =0, T because ran G(ζ) F ae Thus ψg f = 0 ae It follows that ψg HG 2 H2 G,and therefore ψg HL(G) n=0 n=0
236 MA Dritschel and J Rovnyak For a simple application of Theorem 43, suppose that F is a Laurent polynomial of degree m, andchooseψ to be ζ m I In short order, this yields another proof of the operator Fejér-Riesz theorem (Theorem 21) Another application is a theorem of Sarason [55, p 198], which generalizes the factorization of a scalar-valued function in H 1 as a product of two functions in H 2 (see [35, p 56]) Theorem 44 Every G in N + L(G) can be written G = G 1G 2,whereG 1 and G 2 belong to N + L(G) and G 2 G 2 =[G G] 1/2 and G 1 G 1 = G 2 G 2 ae Proof Suppose first that G HL(G) Foreachζ T, write G(ζ) =U(ζ)[G (ζ)g(ζ)] 1 2, where U(ζ) is a partial isometry with initial space ran [G (ζ)g(ζ)] 1 2 Itcanbe shown that U is weakly measurable We apply Theorem 43 with F =[G G] 1 2 and ψ = U Conditions (i) and (ii) of Theorem 43 are obviously satisfied, and so we obtain an outer function G 2 HL(G) such that G 2 G 2 =[G G] 1/2 ae and UG 2 HL(G) SetG 1 = UG 2ByconstructionG 1 HL(G), G = U(G G) 1 2 =(UG 2 )G 2 = G 1 G 2, and G 1 G 1 = G 2 U UG 2 = G 2G 2 ae The result follows when G H L(G) The general case follows on applying what has just been shown to ϕ 2 G,where ϕ is a scalar-valued outer function such that ϕ 2 G HL(G) The standard operator generalization of Szegő s theorem also follows from Theorem 43 Theorem 45 Let F be a weakly measurable function on T having invertible values in L(G) satisfying F 0 ae If log + F (ζ) dσ < and log + F (ζ) 1 dσ <, T then F is factorable Proof Since log + F (ζ) is integrable, we can choose a scalar outer function ϕ 1 such that F 1 = F/ ϕ 1 2 L L(G) Since log + F (ζ) 1 is integrable, so is log + F 1 (ζ) 1 Hence there is a bounded scalar outer function ϕ such that T ϕf 1 1 L L(G)
The Operator Fejér-Riesz Theorem 237 We apply Theorem 43 to F 1 with ψ = ϕf1 1 Condition (i) is satisfied because ψf 1 = ϕi Condition (ii) holds because the values of ψ are invertible ae Thus F 1 is factorable, and hence so is F Theorem 43 has a half-plane version, the scalar inner case of which is given in [50, p 117] This has an application to the following generalization of Akhiezer s theorem on factoring entire functions [50, Chapter 6] Theorem 46 Let F be an entire function of exponential type τ, having values in L(G), such that F (x) 0 for all real x and log + F (t) 1+t 2 dt < Then F (x) =G(x) G(x) for all real x where G is an entire function with values in L(G) such that exp( iτz/2)g is of exponential type τ/2 and the restriction of G to the upper half-plane is an outer function Matrix case We end this section by quoting a few results for matrix-valued functions The matrix setting is more concrete, and one can do more Statements often require invertibility assumptions We give no details and leave it to the interested reader to consult other sources for further information Our previous definitions and results transfer in an obvious way to matrixvalued functions For this we choose G = C r for some positive integer r and identify operators on C r with r r matrices The operator norm of a matrix is denoted WewriteL r r,h r r in place of L L(G),H L(G) and for the norms on these spaces Theorem 45 is more commonly stated in a different form for matrix-valued functions Theorem 47 Suppose that F is an r r measurable matrix-valued function having invertible values on T such that F 0 ae and log + F is integrable Then F is factorable if and only if log det F is integrable Recall that when F is factorable, there is a unique outer G such that F = G G and G(0) 0 It makes sense to inquire about the continuity properties of the mapping Φ: F G with respect to various norms For example, see Jacob and Partington [37] We cite one recent result in this area Theorem 48 (Barclay [5]) Let F, F n, n =1, 2,,ber r measurable matrixvalued functions on T having invertible values ae and integrable norms Suppose that F = G G and F n = G n G n,whereg, G n are r r matrix-valued outer functions such that G(0) 0 and G n (0) 0, n =1, 2,Then if and only if lim n T G(ζ) G n (ζ) 2 dσ =0
238 MA Dritschel and J Rovnyak (i) lim F (ζ) F n (ζ) dσ =0,and n T (ii) the family of functions {log det F n } n=0 is uniformly integrable A family of functions {ϕ α } α A L 1 is uniformly integrable if for every ε>0 there is a δ>0 such that E ϕ α dσ < ε for all α A whenever σ(e) <δsee[5] for additional references and similar results in other norms A theorem of Bourgain [9] characterizes all functions on the unit circle which are products hg with g, h H : Afunctionf L has the form f = hg where g, h H if and only if log f is integrable This resolves a problem of Douglas and Rudin [17] The problem is more delicate than spectral factorization; when f = 1 ae, the factorization cannot be achieved in general with inner functions Bourgain s theorem was recently generalized to matrix-valued functions Theorem 49 (Barclay [4, 6]) Suppose F L r r has invertible values ae Then F has the form F = H G ae for some G, H in Hr r if and only if log det F is integrable In this case, for every ε>0 such a factorization can be found with G H < F + ε The proof of Theorem 49 in [6] is long and technical In fact, Barclay proves an L p -version of this result for all p, 1 p Another type of generalization is factorization with indices We quote one result to illustrate this notion Theorem 410 Let F be an r r matrix-valued function with rational entries Assume that F has no poles on T and that det F (ζ) 0for all ζ in T Then there exist integers κ 1 κ 2 κ r such that F (z) =F (z)diag {z κ1,,z κr }F + (z), where F ± are r r matrix-valued functions with rational entries such that (i) F + (z) has no poles for z 1 and det F + (z) 0for z 1; (ii) F (z) has no poles for z 1 including z = and det F (z) 0for z 1 including z = The case in which F is nonnegative on T can be handled using the operator Fejér-Riesz theorem (the indices are all zero in this case) The general case is given in Gohberg, Goldberg, and Kaashoek [27, pp 236 239] This is a large subject that includes, for example, general theories of factorization in Bart, Gohberg, Kaashoek, and Ran [7] and Clancey and Gohberg [13] Historical remarks Historical accounts of spectral factorization appear in [2, 30, 50, 52, 61] Briefly, the problem of factoring nonnegative matrix-valued functions on the unit circle rose to prominence in the prediction theory of multivariate stationary stochastic processes The first results of this theory were announced by Zasuhin [65] without complete proofs; proofs were supplied by MG Kreĭn in lectures Modern accounts
The Operator Fejér-Riesz Theorem 239 of prediction theory and matrix generalizations of Szegő s theorem are based on fundamental papers of Helson and Lowdenslager [31, 32], and Wiener and Masani [63, 64] The general case of Theorem 45 is due to Devinatz [15]; other proofs are given in [16, 30, 50] For an engineering view and computational methods, see [38, Chapter 8] and [56] The original source for Lowdenslager s Criterion (Lemmas 23 and 42) is [40]; an error in [40] was corrected by Douglas [16] There is a generalization, given by Sz-Nagy and Foias [61, pp 201 203], in which the isometry may have a nontrivial unitary component and the shift component yields a maximal factorable summand Lowdenslager s Criterion is used in the construction of canonical models of operators by de Branges [10] See also Constantinescu [14] for an adaptation to Toeplitz kernels and additional references 5 Multivariable theory It is natural to wonder to what extent the results for one variable carry over to several variables Various interpretations of several variables are possible The most straightforward is to consider Laurent polynomials in complex variables z 1,,z d that are nonnegative on the d-torus T d The method of Schur complements in 3 suggests an approach to the factorization problem for such polynomials Care is needed, however, since the first conjectures for a multivariable Fejér-Riesz theorem that might come to mind are false, as explained below Multivariable generalizations of the Fejér-Riesz theorem are thus necessarily weaker than the one-variable result One difficulty has to do with degrees, and if the condition on degrees is relaxed, there is a neat result in the strictly positive case (Theorem 51) By a Laurent polynomial in z =(z 1,,z d ) we understand an expression Q(z) = m 1 k 1= m 1 m d k d = m d Q k1,,k d z k1 1 zk d d (51) We assume that the coefficients belong to L(G), where G is a Hilbert space With obvious interpretations, the scalar case is included By an analytic polynomial with coefficients in L(G) we mean an analogous expression, of the form P (z) = m 1 k 1=0 m d k d =0 P k1,,k d z k1 1 zk d d (52) The numbers m 1,,m d in (51) and (52) are upper bounds for the degrees of the polynomials in z 1,,z d, which we define as the smallest values of m 1,,m d that can be used in the representations (51) and (52) Suppose that Q(z) has the form (51) and satisfies Q(ζ) 0 for all ζ T d,thatis,forallζ =(ζ 1,,ζ d )with ζ 1 = = ζ d = 1 Already in the scalar case, one cannot always find an analytic polynomial P (z) such that Q(ζ) = P (ζ) P (ζ), ζ T d This was first explicitly shown by Lebow and Schreiber [39] There are also difficulties in writing Q(ζ) = r j=1 P j(ζ) P j (ζ), ζ T d,forsome
240 MA Dritschel and J Rovnyak finite set of analytic polynomials, at least if one requires that the degrees of the analytic polynomials do not exceed those of Q(z) as in the one-variable case (see Naftalevich and Schreiber [44], Rudin [53], and Sakhnovich [54, 36])The example in [44] is based on a Cayley transform of a version of a real polynomial over R 2 called Motzkin s polynomial, which was the first explicit example of a nonnegative polynomial in R d, d>1, which is not a sum of squares of polynomials What is not mentioned in these sources is that if we loosen the restriction on degrees, the polynomial in [44] can be written as a sum of squares (see [19]) Nevertheless, for three or more variables, very general results of Scheiderer [57] imply that there exist nonnegative, but not strictly positive, polynomials which cannot be expressed as such finite sums regardless of degrees Theorem 51 Let Q(z) be a Laurent polynomial in z =(z 1,,z d ) with coefficients in L(G) for some Hilbert space G Suppose that there is a δ>0 such that Q(ζ) δi for all ζ T d Then Q(ζ) = r P j (ζ) P j (ζ), ζ T d, (53) j=1 for some analytic polynomials P 1 (z),,p r (z) in z =(z 1,,z d ) which have coefficients in L(G) Furthermore, for any fixed k, the representation (53) can be chosen such that the degree of each analytic polynomial in z k is no more than the degree of Q(z) in z k The scalar case of Theorem 51 follows by a theorem of Schmüdgen [58], which states that strictly positive polynomials over compact semialgebraic sets in R n (that is, sets which are expressible in terms of finitely many polynomial inequalities) can be written as weighted sums of squares, where the weights are the polynomials used to define the semialgebraic set (see also [12]); the proof is nonconstructive On the other hand, the proof we sketch using Schur complements covers the operator-valued case, and it gives an algorithm for finding the solution One can also give estimates for the degrees of the polynomials involved, though we have not stated these We provetheorem51 for the cased = 2, following Dritschel [18] The general case is similar The argument mimics the method of Schur complements, especially in its original form used in [18] In place of Toeplitz matrices whose entries are operators, in the case of two variables we use Toeplitz matrices whose entries are themselves Toeplitz matrices The fact that the first level Toeplitz blocks are infinite in size causes problems, and so we truncate these blocks to finite size Then everything goes through, but instead of factoring the original polynomial Q(z), the result is a factorization of polynomials Q (N) (z) thatareclosetoq(z) When Q(ζ) δi on T d for some δ>0, there is enough wiggle room to factor Q(z) itself We isolate the main steps in a lemma
The Operator Fejér-Riesz Theorem 241 Lemma 52 Let Q(z) = m 1 j= m 1 m 2 k= m 2 Q jk z j 1 zk 2 be a Laurent polynomial with coefficients in L(G) such that Q(ζ) 0 for all ζ =(ζ 1,ζ 2 ) in T 2 Set Q (N) (z) = m 1 j= m 1 m 2 N +1 k N +1 k= m 2 Then for each N m 2, there are analytic polynomials F l (z) = m 1 N j=0 k=0 with coefficients in L(G) such that Q (N) (ζ) = Q jk z j 1 zk 2 F (l) jk zj 1 zk 2, l =0,,N, (54) N F l (ζ) F l (ζ), ζ T 2 (55) l=0 Proof Write m 1 ( m2 Q(z) = Q jk z2 )z k j1 = m 1 R j (z 2 ) z j 1, j= m 1 k= m 2 j= m 1 andextendallsumstorunfrom to by setting Q jk =0andR j (z 2 )=0if j >m 1 or k >m 2 Introduce a Toeplitz matrix T whose entries are the Toeplitz matrices T j corresponding to the Laurent polynomials R j (z 2 ), that is, T 0 T 1 T 2 Q j0 Q j, 1 Q j, 2 T 1 T 0 T 1 Q j1 Q j0 Q j, 1 T = T 2 T 1 T, T j = 0 Q j2 Q j1 Q, j0 j =0, ±1, ±2,NoticethatT is finitely banded, since T j =0for j >m 1 The identity (25) has the following generalization: Th,h = T q p h p,h q = Q(ζ)h(ζ),h(ζ) G dσ 2 (ζ) T 2 p=0 q=0 Here ζ =(ζ 1,ζ 2 )anddσ 2 (ζ) =dσ(ζ 1 )dσ(ζ 2 ) Also, h(ζ) = p=0 q=0 h pq ζ p 1 ζq 2,
242 MA Dritschel and J Rovnyak where the coefficients are vectors in G and all but finitely many are zero, and h 0 h p0 h = h 1, h p = h p1, p =0, 1, 2, It follows that T acts as a bounded operator on a suitable direct sum of copies of G SinceQ(ζ) 0onT 2, T 0 Fix N m 2 Set T 0 T 1 T 2 T T 1 T 0 T 1 = T 2 T 1 T 0, where T j is the upper (N +1) (N +1)blockofT j with a normalizing factor: Q j0 Q j, 1 Q j, N T j = 1 Q j1 Q j0 Q j, N+1 N +1, j =0, ±1, ±2, Q jn Q j,n 1 Q j0 Then T is the Toeplitz matrix corresponding to the Laurent polynomial m 1 Ψ(w) = T j w j j= m 1 Moreover, T 0 since it is a positive constant multiple of a compression of T Thus Ψ(w) 0for w = 1 By the operator Fejér-Riesz theorem (Theorem 21), Ψ(w) =Φ(w) Φ(w), w =1, (56) for some analytic polynomial Φ(w) = m 1 j=0 Φ jw j whose coefficients are (N +1) (N + 1) matrices with entries in L(G) Write Φ j = ( ) Φ jn Φ j,n 1 Φ j0, where Φ jk is the kth column in Φ j Set F (z) = m 1 j=0 k=0 N Φ jk z j 1 zk 2 The identity (56) is equivalent to 2m 1 + 1 relations for the coefficients of Ψ(w) The coefficients of Ψ(w) are constant on diagonals, there being N + 1 k terms in the kth diagonal above the main diagonal, and similarly below If these terms are summed, the result gives 2m 1 + 1 relations equivalent to the identity Q (N) (ζ) = F (ζ) F (ζ), ζ T 2 (57)
The Operator Fejér-Riesz Theorem 243 We omit the calculation, which is straightforward but laborious To convert (57) to the form (55), write F (0) jk F (1) Φ jk = jk, j =0,,m 1 and k =0,,N F (N) jk Then F 0 (z) F 1 (z) F (z) =, F N (z) where F 0 (z),,f N (z) are given by (54), and so (57) takes the form (55) Proof of Theorem 51 for the case d =2 Suppose N m 2,andset Q(z) = m 1 j= m 1 m 2 k= m 2 N +1 N +1 k Q jk z j 1 zk 2 The values of Q(z) are selfadjoint on T 2,and Q(z) =Q(z)+S(z), where S(z) = m 1 m 2 j= m 1 k= m 2 k N +1 k Q jk z j 1 zk 2 Now choose N large enough that S(ζ) <δ, ζ T 2 Then Q(ζ) 0onT 2,and the result follows on applying Lemma 52 to Q(z) Further details can be found in [18], and a variation on this method yielding good numerical results is given in Geronimo and Lai [23] While, as we mentioned, there is in general little hope of finding a factorization of a positive trigonometric polynomial in two or more variables in terms of one or more analytic polynomials of the same degree, it happens that there are situations where the existence of such a factorization is important In particular, Geronimo and Woerdeman consider this question in the context of the autoregressive filter problem [24, 25], with the first paper addressing the scalar case and the second the operator-valued case, both in two variables They show that for scalarvalued polynomials in this setting there exists a factorization in terms of a single stable (so invertible in the bidisk D 2 ) analytic polynomial of the same degree if and only if a full rank condition holds for certain submatrices of the associated Toeplitz matrix ([24, Theorem 113]) The condition for operator-valued polynomials is similar, but more complicated to state We refer the reader to the original papers for details
244 MA Dritschel and J Rovnyak Stable scalar polynomials in one variable are by definition outer, so the Geronimo and Woerdeman results can be viewed as a statement about outer factorizations in two variables In [19], a different notion of outerness is considered As we saw in 3, in one variable outer factorizations can be extracted using Schur complements The same Schur complement method in two or more variables gives rise to a version of outer factorization which in general does not agree with that coming from stable polynomials In [19], this Schur complement version of outerness is used when considering outer factorizations for polynomials in two or more variables As in the Geronimo and Woerdeman papers, it is required that the factorization be in terms of a single analytic polynomial of the same degree as the polynomial being factored Then necessary and sufficient conditions for such an outer factorization under these constraints are found ([19, Theorem 41]) The problem of spectral factorization can also be considered in the multivariable setting Blower [8] has several results along these lines for bivariate matrix-valued functions, including a matrix analogue of Szegő s theorem similar to Theorem 47 His results are based on a two-variable matrix version of Theorem 51, and the arguments he gives coupled with Theorem 51 can be used to extend these results to polynomials in d>2 variables as well 6 Noncommutative factorization We now present some noncommutative interpretations of the notion of several variables, starting with the one most frequently considered, and for which there is an analogue of the Fejér-Riesz theorem It is due to Scott McCullough and comes very close to the one-variable result Further generalizations have been obtained by Helton, McCullough and Putinar in [33] For a broad overview of the area, two nice survey articles have recently appeared by Helton and Putinar [34] and Schmüdgen [59] covering noncommutative real algebraic geometry, of which the noncommutative analogues of the Fejér-Riesz theorem are one aspect In keeping with the assumptions made in [42], all Hilbert spaces in this section are taken to be separable Fix Hilbert spaces G and H, and assume that H is infinite dimensional Let S be the free semigroup with generators a 1,,a d ThusS is the set of words w = a j1 a jk, j 1,,j k {1,,d}, k =0, 1, 2,, (61) with the binary operation concatenation The empty word is denoted e The length of the word (61) is w = k (so e =0)LetS m be the set of all words (61) of length at most m The cardinality of S m is l m =1+d + d 2 + + d m We extend S to a free group G We can think of the elements of G as words in a 1,,a d,a 1, with two such words identified if one can be obtained 1,,a 1 d from the other by cancelling adjacent terms of the form a j and a 1 j The binary operation in G is also concatenation Words in G of the form h = v 1 w with v, w S play a special role and are called hereditary Notice that a hereditary
The Operator Fejér-Riesz Theorem 245 word h has many representations h = v 1 w with v, w S LetH m be the set of hereditary words h which have at least one representation in the form h = v 1 w with v, w S m We can now introduce the noncommutative analogues of Laurent and analytic polynomials A hereditary polynomial is a formal expression Q = h Q h, (62) h H m where Q h L(G) for all h Analytic polynomials are hereditary polynomials of the special form P = w P w, (63) w S m where P w L(G) for all w Theidentity is defined to mean that Q h = v,w S m h=v 1 w Q = P P P v P w, h H d Next we give meaning to the expressions Q(U) andp (U) for hereditary and analytic polynomials (62) and (63) and any tuple U =(U 1,,U d ) of unitary operators on H First define U w L(H) for any w S by writing w in the form (61) and setting U w = U j1 U jk By convention, U e = I is the identity operator on H Ifh G is a hereditary word, set U h =(U v ) U w for any representation h = v 1 w with v, w S; this definition does not depend on the choice of representation Finally, define Q(U),P(U) L(H G) by Q(U) = U h Q h, P(U) = U w P w h H m w S m The reader is referred to, for example, Murphy [43, 63] for the construction of tensor products of Hilbert spaces and algebras, or Palmer, [45, 110] for a more detailed account Theorem 61 (McCullough [42]) Let Q = h H m h Q h be a hereditary polynomial with coefficients in L(G) such that Q(U) 0 for every tuple U =(U 1,,U d ) of unitary operators on H Then for some l l m,there
246 MA Dritschel and J Rovnyak exist analytic polynomials P j = w P j,w, j =1,,l, w S m with coefficients in L(G) such that Q = P1 P 1 + + Pl P l Moreover, for any tuple U =(U 1,,U d ) of unitary operators on H, Q(U) =P 1 (U) P 1 (U)+ + P l (U) P l (U) In these statements, when G is infinite dimensional, we can choose l =1 As noted by McCullough, when d = 1, Theorem 61 gives a weaker version of Theorem 21 However, Theorem 21 can be deduced from this by a judicious use of Beurling s theorem and an inner-outer factorization McCullough s theorem uses one of many possible choices of noncommutative spaces on which some form of trigonometric polynomials can be defined We place this, along with the commutative versions, within a general framework, which we now explain The complex scalar-valued trigonometric polynomials in d variables form a unital -algebra P, the involution taking z n to z n,whereforn =(n 1,,n d ), n =( n 1,, n d ) If instead the coefficients are in the algebra C = L(G) for some Hilbert space G, then the unital involutive algebra of trigonometric polynomials with coefficients in C is P C The unit is 1 1 A representation of P C is a unital algebra -homomorphism from P C into L(H) for a Hilbert space H Thekey thing here is that z 1,,z d generate C, and so assuming we do not mess with the coefficient space, a representation π is determined by specifying π(z k ), k =1,,d Firstnotethatsincezk z k =1,π(z k ) is isometric, and since zk = z 1 k,we then have that π(z k ) is unitary Assuming the variables commute, the z k s generate a commutative group G which we can identify with Z d under addition, and the irreducible representations of commutative groups are one dimensional This essentially follows from the spectral theory for normal operators (see, for example, Edwards [21, p 718]) However, the one-dimensional representations are point evaluations on T d Discrete groups with the discrete topology are examples of locally compact groups Group representations of locally compact groups extend naturally to the algebraic group algebra, which in this case is P, and then on to the algebra P C by tensoring with the identity representation of C Soaseem- ingly more complex way of stating that a commutative trigonometric polynomial P in several variables is positive / strictly positive is to say that for each (topologically) irreducible unitary representation π of G, the extension of π to a unital -representation of the algebra P C, also called π, has the property that π(p ) 0 / π(p ) > 0 By the way, since T d is compact, π(p ) > 0 implies the existence of some ɛ>0 such that π(p ɛ1 1) = π(p ) ɛ1 0 What is gained through this perspective is that we may now define noncommutative trigonometric polynomials over a finitely generated discrete (so locally
The Operator Fejér-Riesz Theorem 247 compact) group G in precisely the same manner These are the elements of the algebraic group algebra P generated by G; that is, formal complex linear combinations of elements of G endowed with pointwise addition and a convolution product (see Palmer [45, Section 19]) Then a trigonometric polynomial in P C is formally a finite sum over G of the form P = g g P g where P g C for all g We also introduce an involution by setting g = g 1 for g G A trigonometric polynomial P is selfadjoint if for all g, P g = Pg There is an order structure on selfadjoint elements defined by saying that a selfadjoint polynomial P is positive / strictly positive if for every irreducible unital -representation π of G, the extension as above of π to the algebra P C (again called π), satisfies π(p ) 0/ π(p ) > 0; where by π(p ) > 0 we mean that there exists some ɛ>0 independent of π such that π(p ɛ(1 1)) 0 Letting Ω represent the set of such irreducible representations, we can in a manner suggestive of the Gel fand transform define ˆP (π) =π(p ), and in this way think of Ω as a sort of noncommutative space on which our polynomial is defined The Gel fand-raĭkov theorem (see, for example, Palmer [46, Theorem 1246]) ensures the existence of sufficiently many irreducible representations to separate G, soinparticular,ω For a finitely generated discrete group G with generators {a 1,,a d },let S be a fixed unital subsemigroup of G containing the generators The most interesting case is when S is the subsemigroup generated by e (the group identity) and {a 1,,a d } As an example of this, if G is the noncommutative free group in d generators, then the unital subsemigroup generated by {a 1,,a d } consists of group elements w of the form e (for the empty word) and those which are an arbitrary finite product of positive powers of the generators, as in (61) We also need to address the issue of what should play the role of Laurent and analytic trigonometric polynomials in the noncommutative setting The hereditary trigonometric polynomials are defined as those polynomials of the form P = j w j1 w j2 P j,wherew j1,w j2 S We think of these as the Laurent polynomials Trigonometric polynomials over S are referred to as analytic polynomials The square of an analytic polynomial Q is the hereditary trigonometric polynomial Q Q Squares are easily seen to be positive As a weak analogue of the Fejér-Riesz theorem, we prove a partial converse below We refer to those hereditary polynomials which are selfadjoint as real hereditary polynomials, and denote the set of such polynomials by H While these polynomials do not form an algebra, they are clearly a vector space Those which are finite sums of squares form a cone C in H (that is, C is closed under sums and positive scalar multiplication) Any real polynomial is the sum of terms of the form 1 A or w2 w 1 B + w1 w 2 B,wherew 1,w 2 S and A is selfadjoint The first of these is obviously the difference of squares Using w w =1foranyw G, we also have w 2 w 1 B + w 1 w 2 B =(w 1 B + w 2 1) (w 1 B + w 2 1) 1 (1 + B B) Hence H = C C