Riesz's theorem for orthogonal matrix polynomials

Manuscript: April, 27 2001 RIESZ'S THEOREM FOR ORTHOGONAL MATRIX POLYNOMIALS Pedro Lopez-Rodriguez Universidad de Sevilla Abstract. We describe the image through the Stieltjes transform of the set of solutions Vof a matrix moment problem. We extend Riesz's theorem to the matrix setting, proving that those matrices of measures of Vfor which the matrix polynomials are dense in the corresponding L2space are precisely those whose Stieltjes transform is an extremal point (in the sense of convexity) of the image set. 1. Introduction. For a positive Borel measure ºon Rwith ¯nite moments of any order sn=RRtndº(t) we denote by Vthe set of positive Borel measures ¹on Rsatisfying RRtnd¹(t)=sn,n¸0, that is, the set of solutions to the Hamburger moment problem de¯ned by º.ByVnwe denote the set of positive Borel measures on Rsuch that RRtkd¹(t)=sk,0·k·n,that is, the set of solutions to the truncated moment problem de¯ned by º. We say that the measure ºis determinate if there is no other positive measure having the same moments as those of º,thatis,ifV=fºg, otherwise we say that ºis indeterminate. This alternative is related to the index of de¯ciency of the operator de¯ned on `2by the in¯nite Jacobi matrix J=0 B B @ b0a1 a1b1a2 a2b2a3 ......... 1 C C A; where the coe±cients ai(6=0)andbiare the coe±cients which appear in the three term recurrence relation satis¯ed by the orthogonal polynomials (pn)nassociated to º, tpn(t)=an+1pn+1(t)+bnpn(t)+anpn¡1(t);n¸0: The index of de¯ciency of Jis 0 if the moment problem is determinate and 1 if the moment problem is indeterminate. This work has been partially supported by DGICYT ref. PB-96-1321-CO2. This work was partly carried out at the Mathematics Institute of the University of Copenhagen under a grant of the spanish Ministry of Education and Science of the Programa de becas de formaci¶on de personal investigador en el extranjero. 1991 Mathematics Subject Classi¯cation. 42C05, 44A60. Typeset by A M S-T EX 1 2 P. LOPEZ-RODRIGUEZ In 1922 Nevanlinna proved that for a ¯xed non real ¸the image through the Stieltjes transform of all the measures of Vin the point ¸ I(V)(¸)=½ZR d¹(t) t¡¸:¹2V¾ is either a point if the moment problem is determinate or a circle if the moment problem is indeterminate, and the occurrence of these two cases does not depend on the non real ¸chosen (see [N]or[A]). The measures ¹for which I(¹)(¸) lies in the circumference of this circle I(V)(¸) are called N-extremal (Nevanlinna-extremal). In 1923 M. Riesz proved that in order that the set Pof polynomials in L2(¹)bedense, it is necessary and su±cient that the measure ¹be N-extremal at every non real point ¸, and for this it is su±cient that it should be N-extremal in at least one such point (see [Ri] or [A]). The purpose of this paper is to generalize these two results for a completely indeterminate matrix moment problem. Given º=(ºi;j)1·i;j·Na positive de¯nite matrix of measures (for any Borel set Athe numerical matrix º(A) is positive semide¯nite) with ¯nite matrix moments Sk=ZR tkdº(t) of any order k¸0, we denote by Vthesetofpositivede¯nitematricesofmeasureshaving the same matrix moments as those of º,andbyVnthe set of positive matrices of measures whose moments up to degree nare the same as those of º. We say that the positive de¯nite matrix of measures ºis determinate if no other positive de¯nite matrix of measures has the same moments as those of º, that is, the positive de¯nite matrix of measures ºis uniquely determined by the moments RRtndº(t), n¸0. By (Pn)1 n=0 we denote the sequence of orthonormal matrix polynomials with respect to º,Pnof degree nand with non-singular leading coe±cient. These polynomials (Pn)nsatisfy a three term recurrence relation of the form (1.1) tPn(t)=An+1Pn+1(t)+BnPn(t)+A¤ nPn¡1(t);n¸0; (Anand Bnbeing N£Nmatrices such that det(An)6= 0 and B¤ n=Bn), with initial condition P¡1(t)=µ(here and in the rest of this paper, we write µfor the null matrix, the dimension of which can be determined from the context. For instance, here µis the N£Nnull matrix). It is well-known that this recurrence relation is equivalent to the orthogonality with respect to a positive de¯nite matrix of measures: this is the matrix version of Favard's Theorem (see [AN], [D1]and[DL1]). We denote by Qn(t) the corresponding sequence of polynomials of the second kind, Qn(t)=ZR Pn(t)¡Pn(x) t¡xdº(x);n¸0; which also satisfy the recurrence relation (1.1), with initial conditions Q0(t)=µand Q1(t)=A¡1 1. RIESZ'S THEOREM FOR ORTHOGONAL MATRIX POLYNOMIALS 3 In the matrix case the determinacy or indeterminacy of the matrix moment problem is also related to the index of de¯ciency of the operator Jde¯ned by the in¯nite N-Jacobi matrix J=0 B B @ B0A1 A¤ 1B1A2 A¤ 2B2A3 ......... 1 C C A on the space `2,whereAnand Bnare the coe±cients which appear in the three term recurrence relation (1.1). In this case the index of de¯ciency can be any natural number from 0 to N, 0 in the determinate case and Nin the completely indeterminate case. In the latter case the two series 1 X k=0 Q¤ k(¸)Pk(´)and 1 X k=0 P¤ k(¸)Pk(´) converge uniformly in the variables ¸and ´on every bounded set of the complex plane (see [K]). In [B]itisprovedthattherankofthelimitmatrixR(¸)= lim n!1 Rn(¸) exists and is thesameforeverynonreal¸,where (1.2) Rn(¸)=Ãn X k=0 P¤ k(¸)Pk(¸)!¡1 : This result is also mentioned by Krein in [K], who refers to [Na] for a proof. In this paper we assume the rank of this matrix is Nand consequently the matrix 1 X k=0 P¤ k(¸)Pk(¸)is invertible, for every non real ¸, and equal to R(¸)¡1. As in the scalar case, for a ¯xed non real ¸, we denote by I(V)(¸) the image through the Stieltjes transform of all the matrices of measures of Vin the point ¸ I(V)(¸)=½ZR d¹(t) t¡¸:¹2V¾: Firstly, for a ¯xed non real ¸, we describe the set I(V)(¸). This is the set of N£N complex matrices !satisfying the matrix inequality (1.3) [!+C(¸)]R(¸)¡1[!+C(¸)]¤·j¸¡¸j¡2R(¸); where C(¸)=B(¸; ¸)D(¸; ¸)¡1(see the Preliminaries for the de¯nitions of B(¸; ¸)and D(¸; ¸). A·Bmeans that B¡Ais positive semide¯nite. The extremal points (in the sense of convexity) of the set I(V)(¸) are the matrices !for which equality is attained in (1.3). If ¹is a matrix of measures in Vfor which I(¹)(¸)is 4 P. LOPEZ-RODRIGUEZ aextremalpointofI(V)(¸), we call this matrix of measures N-extremal, as in the scalar case. Finally, we generalize Riesz's theorem to the matrix setting by proving that the matrices of measures of Vfor which the set Pof matrix polynomials is dense in the corresponding space L2(¹) are precisely the N-extremal matrices of measures, and that the N-extremality of a matrix of measures does not depend on the non real ¸chosen. For the case N= 1 one recovers the very well known classical formulas exposed in [A, Ch. 1]. 2. Preliminaries. In what follows, if P(¸) is a matrix polynomial, we denote by P¤(¸) the polynomial obtained from P(¸) by replacing each of its matrix coe±cients by its hermitian conjugate, so that P(¸)¤=P¤(¸). For a matrix polynomial of two variables P(¸; ´) the de¯nition is the same, so that we have P(¸; ´)¤=P¤(¸; ´). If F(¸) is a holomorphic matrix function we de¯ne F¤(¸)=F(¸)¤. The set of positive de¯nite matrices of measures is endowed with the vague and weak topologies. If ºa positive de¯nite matrix of measures, the set Vof matrices of measures having the same moments as ºis a compact and convex set for these topologies which coincide on V(see [DL2]). For ¹a positive de¯nite matrix of measures, the space L2(¹)isde¯nedasthesetof N£Nmatrix functions f:R!MN£N(C) such that ¿(f(t)M(t)f(t)¤)2L1(¿¹), where M(t) is the Radon-Nikodym derivative of ¹with respect to its trace (¿¹)(foramatrix A=(ai;j)1·i;j·N, we denote ¿A for its trace, i. e. ¿A =PN i=1 ai;i): M=(mi;j)N i;j=1 =µd¹i;j d¿¹ ¶1·i;j·N : The space L2(¹) is endowed with the norm kfk2;¹ =k¿(f(t)M(t)f(t)¤)1 2k2;¿¹ =µZR ¿(f(t)M(t)f(t)¤)d¿¹(t)¶1 2 and is a Hilbert space. The duality works as for the scalar case (see [R]or[DL2]formore details. For the de¯nition of the Lpspaces associated to ¹,1·p<1,see[DL2]). We stress that since we only impose the matrices of measures in V2nto have ¯nite moments up to degree 2n,for¹2V2nwe can guarantee only that the polynomials up to degree nbelong to the corresponding space L2(¹). In any case, the polynomials (Pk)k=0;:::;n are orthonormal with respect to any measure in V2n. We include here the matrix version of some classical formulas for orthonormal scalar polynomials. The proofs are easily veri¯ed using the three term recurrence relation (1.1). (2.1) An(u; v)=(v¡u) n¡1 X k=0 Q¤ k(u)Qk(v)=Q¤ n¡1(u)AnQn(v)¡Q¤ n(u)A¤ nQn¡1(v);for u; v 2C; RIESZ'S THEOREM FOR ORTHOGONAL MATRIX POLYNOMIALS 5 (2.2) Bn(u; v)=¡I+(v¡u) n¡1 X k=0 Q¤ k(u)Pk(v)=Q¤ n¡1(u)AnPn(v)¡Q¤ n(u)A¤ nPn¡1(v);for u; v 2C; (this is Green's formula), (2.3) Cn(u; v)=I+(v¡u) n¡1 X k=0 P¤ k(u)Qk(v)=P¤ n¡1(u)AnQn(v)¡P¤ n(u)A¤ nQn¡1(v);for u; v 2C; (2.4) Dn(u; v)=(v¡u) n¡1 X k=0 P¤ k(u)Pk(v)=P¤ n¡1(u)AnPn(v)¡P¤ n(u)A¤ nPn¡1(v);for u; v 2C; (this a Christo®el-Darboux formula). We will also use the Liouville-Ostrogradsky formula (2.5) Qn(¸)P¤ n¡1(¸)¡Pn(¸)Q¤ n¡1(¸)=A¡1 n;for ¸2C; and the relations (2.6) Pn(¸)Q¤ n(¸)=Qn(¸)P¤ n(¸);for ¸2C; (2.7) An(u; v)D¤ n(u; v)¡B n(u; v)C¤ n(u; v)=I; for u; v 2C; and ¯nally (2.8) Cn(u; v)D¤ n(u; v)=Dn(u; v)C¤ n(u; v);for u; v 2C: We will also use that (2.9) Cn(¸; ¸)=¡Bn(¸; ¸)¤;for ¸2C; and that (2.10) Dn(¸; ¸)=(¸¡¸)Rn¡1(¸)¡1and D¤ n(¸; ¸)=(¸¡¸)Rn¡1(¸)¡1;for ¸2C: By A(u; v), B(u; v), C(u; v)andD(u; v) we denote the limit matrix functions de¯ned from An(u; v), Bn(u; v), Cn(u; v)andDn(u; v)whenntends to in¯nity. 3. The main theorems. For any non real ¸,wede¯nethesetBn(¸)tobethesetofN£Ncomplex matrices ! such that 6 P. LOPEZ-RODRIGUEZ (3.1) [!+Cn(¸)]Rn¡1(¸)¡1[!+Cn(¸)]¤·j¸¡¸j¡2Rn¡1(¸); where Cn(¸)=Bn(¸; ¸)Dn(¸; ¸)¡1. The calculations in Lemma 1 (see below) show that Bn(¸)isalsothesetofN£N complex matrices !satisfying the matrix inequality (3.2) n¡1 X k=0 (Q¤ k(¸)+!P¤ k(¸))(Qk(¸)+Pk(¸)!¤)·!¡!¤ ¸¡¸: We put B1(¸) for the intersection of all the sets Bn(¸). B1(¸) is clearly the set of N£Ncomplex matrices !such that (3.3) [!+C(¸)]R(¸)¡1[!+C(¸)]¤·j¸¡¸j¡2R(¸); where C(¸)=B(¸; ¸)D(¸; ¸)¡1. Similarly, B1(¸)isalsothesetofN£Ncomplex matrices !such that (3.4) 1 X k=0 (Q¤ k(¸)+!P¤ k(¸))(Qk(¸)+Pk(¸)!¤)·!¡!¤ ¸¡¸: Looking at (3.1) and (3.3) it is immediate that upon a linear matrix transformation, any of the sets Bn(¸)orB1(¸) is in a one to one correspondence with the set of N£N complex matrices Tsatisfying TT¤·I, which is a convex set whose extremal points are the matrices verifying TT¤=I, that is, the unitary matrices (this is a well-known result in operator theory which can be proved for example with the aid of the singular value decomposition of matrices). This implies that these sets Bn(¸)andB1(¸) are convex sets whose extremal points (ExtBn(¸) and ExtB1(¸)) are those for which equality is attained in (3.1) and (3.2) or (3.3) and (3.4) respectively. By using formulas (2.1), (2.2), (2.3) and (2.4) in (3.2) it is straightforward to see that an equivalent condition for !to be an extremal point of Bn(¸) is that the matrix (3.5) (!P¤ n(¸)+Q¤ n(¸))A¤ n(Pn¡1(¸)!¤+Qn¡1(¸)) is hermitian. It is clear that for all n¸1wehaveB1(¸)µBn+1(¸)µBn(¸). It is also clear that !belongs to the set of interior points of Bn(¸)orB1(¸)(IntBn(¸)andIntB1(¸)) if a strict inequality is attained in (3.1) and (3.2) or (3.3) and (3.4) respectively. We have the following results: Theorem 1. Let Vdenote the set of solutions to a completely indeterminate matrix moment problem de¯ned by a natrix of measures ºand let ¸2CnR.Thenwehave B1(¸)=I(V)(¸): RIESZ'S THEOREM FOR ORTHOGONAL MATRIX POLYNOMIALS 7 The key to prove this theorem will be the inclusions (3.6) IntBn(¸)µI(V2n¡2)(¸)µBn(¸): The proofs of these inclusions present more di±culties than in the scalar case. We will prove then later in Lemmas 1 to 7. Indeed, in the scalar case the set I(V2n¡2)(¸)isgiven by I(V2n¡2)(¸)=Bn(¸)n½¡qn¡1(¸) pn¡1(¸)¾: The point ¡qn¡1(¸)=pn¡1(¸) lies on the border of the circle Bn(¸). When amoves along the real axis, the quotient ¡qn(¸)¡aqn¡1(¸) pn(¸)¡apn¡1(¸) describes all the points of the circumference of the closed disk Bn(¸) except for the limit point ¡qn¡1(¸)=pn¡1(¸). The well known quadrature formula (see [A,p. 20])givesthat every point de¯ned by the former quotient for a2Rbelongs to I(V2n¡2)(¸). It is easy to see that ¡qn¡1(¸)=pn¡1(¸)=2I(V2n¡2)(¸), but this is of no importance because taking into account that I(V2n¡2)(¸) is a convex set and the simple geometry of the circles Bn(¸) it is immediate to deduce that IntBn(¸)µI(V2n¡2)(¸). This inclusion is not at all so immediate in the matrix case. We prove it in Lemmas 2 to 7 by means of new ideas. Proof of Theorem 1 Suppose ¯rst that !2B1(¸). B1(¸)iscontainedinBn(¸) for all nand thus we can put !=!n, for all n,being!nin Bn(¸). Since the interior set IntBn(¸)isdenseinBn(¸), we can ¯nd ´nin IntBn(¸)suchthatlim n!1 k!n¡´nk=0. SinceIntBn(¸)µI(V2n¡2)(¸), there exists a matrix of measures ¾nin V2n¡2such that ´n=I(¾n)(¸), for all n.Sincethe set f¹¸µ:¿¹(R)·cgis vaguely compact, where cis a positive constant (see Lemma 3.8 in [DL2]) and ¾n(R)=S0for n¸0, there exists ¾a vague accumulation point of asubsequence(¾np)of(¾n). Like in the proof of Lemma 3.10 of [DL2]wehave¾2V. We have ¾n(R)=¾(R)forn¸1, so by virtue of Theorem 3.1 of [DL2], (¾n)converges weakly to ¾.Inparticular I(¾)(¸) = lim p!1 I(¾np)(¸) = lim p!1 ´np=lim p!1 !np=! andwehaveprovedthatB1(¸)µI(V)(¸). Since I(V2n¡2)(¸)µBn(¸), for every n, the reverse inclusion is clear. ¥ Theorem 2. (Riesz's theorem for orthogonal matrix polynomials) Let ¹be a positive de¯nite matrix of measures corresponding to a completely indeterminate matrix moment problem. Then the following conditions are equivalent: (1) There exists ¸02CnRsuch that I(¹)(¸0)is an extremal point (in the sense of convexity) of the set B1(¸0). (2) For any ¸2CnR,I(¹)(¸)is an extremal point (in the sense of convexity) of the set B1(¸) (3) Pis dense in L2(¹),equivalently(Pn(t))1 n=0 is an orthonormal basis for the Hilbert space L2(¹). 8 P. LOPEZ-RODRIGUEZ Proof of Theorem 2 (3) )(2) The polynomials are dense in L2(¹) if for any function fin the space L2(¹) we have equality in Bessel's inequality, which is equivalent to (3.7) 1 X k=0 (f;Pk)(f;Pk)¤=ZR f(t)M(t)f¤(t)d¿¹(t): In particular, for f¸(t)= I t¡¸2L 2(¹)wehave (f¸;P k)=ZR I t¡¸d¹(t)P¤ k(t) =ZR d¹(t)P¤ k(t)¡P¤ k(¸) (t¡¸)+ZR d¹(t) t¡¸P¤ k(¸) =Q¤ k(¸)+I(¹)(¸)P¤ k(¸); being I(¹)(¸)=ZR d¹(t) t¡¸the Stieltjes transform of ¹in the point ¸,and ZR f(t)d¹(t)f¤(t)=ZR d¹(t) jt¡¸j2=I(¹)(¸)¡I(¹)(¸)¤ ¸¡¸=ImI(¹)(¸) Im¸; so equality in (3.7) is 1 X k=0 (Q¤ k(¸)+!(¸)P¤ k(¸))(Qk(¸)+Pk(¸)!¤(¸)) = ImI(¹)(¸) Im¸; that is I(¹)(¸)2ExtB1(¸). (2) )(1) is obvious. (1) )(3) We suppose (1) holds and we claim fn ¸0=I (t¡¸0)n2 P,forn¸1. The assertion for n= 1 is the assumption. We now prove that fn+1 ¸02 P under the assumption fn ¸02 P, so that the claim is established by induction. For given ²>0, there exists a matrix polynomial P2Psuch that kfn ¸0¡Pk2·²jIm¸0j. Dividing Pby (x¡¸0)Iwe get P(x)=(x¡¸0)Q(x)+A,withQanother polynomial of degree n¡1andAaN£N complex matrix. We have kfn+1 ¸0¡Af¸0¡Qk2 2= =¿ZRµI (t¡¸0)n+1 ¡A t¡¸0 ¡Q(t)¶M(t)µI (t¡¸0)n+1 ¡A t¡¸0 ¡Q(t)¶¤ d¿¹(t) =¿ZR 1 jt¡¸0j2µI (t¡¸0)n¡A¡Q(t)(t¡¸0)¶M(t)µI (t¡¸0)n¡A¡Q(t)(t¡¸0)¶¤ d¿¹(t) RIESZ'S THEOREM FOR ORTHOGONAL MATRIX POLYNOMIALS 9 ·1 jIm¸0j2kI (t¡¸0)n¡P(t)k2 2·²2; and since A t¡¸0 belongs to the closure of Pwe deduce that I (t¡¸0)n+1 also does. Now, if f2L 2(¹) is orthogonal to Pand we consider the Stieltjes transform I(f¹)(z)=ZR f(t) t¡zd¹(t);z2CnR using that fn ¸0;fn ¸02 P for n¸1, we see that I(f¹)(n)(z)=µ; for z=¸0; ¸0;and n¸0; but then we have an analytic function I(f¹)inadomainDsuch that I(f¹)(n)(z0)=µ for n¸0andacertainz02D.SoI(f¹)isequaltoµin D.WeconcludethatI(f¹) is identically zero in each of the two half planes CnR,andthenf=µ,¹a.e., hence the polynomials are dense in L2(¹). ¥ We also have the following Theorem: Theorem 3. If ¹2V2n¡2is such that I(¹)(¸)2ExtBn(¸),thenPn¡1=L2(¹) Proof of Theorem 3 The hypothesis means that equality is attained in (3.2), that is, the function f¸(t)= I t¡¸ can be approximated by matrix polynomials up to degree n¡1. Now the proof ¯nishes exactly in the same way as the proof of Theorem 2. ¥ 4. Proofs of the inclusions. In this last section we study in detail the set Bn(¸) and other related sets, with the purpose of proving the inclusions (3.6). We remark that these inclusions are valid without supposing the matrix moment problem to be completely indeterminate. We ¯rst prove the second inclusion of (3.6) Lemma 1. I(V2n¡2)(¸)µBn(¸) Proof Let's suppose ¹is a measure in V2n¡2. We know that the ¯rst northonormal matrix polynomials P0;:::;P n¡1form an orthonormal system in the space L2(¹). After the above calculations, from Bessel's inequality for the function f¸(t)= I t¡¸,wededucethat n¡1 X k=0 (Q¤ k(¸)+I(¹)(¸)P¤ k(¸))(Qk(¸)+Pk(¸)I(¹)(¸)¤)·ImI(¹)(¸) Im¸: 16 P. LOPEZ-RODRIGUEZ Since vibelongs to Ker(!P¤ n¡1(¸)+Q¤ n¡1(¸)), and using that Tis hermitian, we have that for any m+1·i·Nand for any 1 ·j·m, vi(!P¤ n(¸)+Q¤ n(¸))A¤ nu¤ j=vi(!P¤ n(¸)+Q¤ n(¸))A¤ n(Pn¡1(¸)!¤+Qn¡1(¸)v¤ j =vi(!P¤ n¡1(¸)+Q¤ n¡1(¸))An(Pn(¸)!¤+Qn(¸))v¤ j=µ: This means that vi(!P¤ n(¸)+Q¤ n(¸))A¤ nbelongs to Im(!P¤ n¡1(¸)+Q¤ n¡1(¸))?, and consequently (4.9) holds if we prove that for any vector uin Im(!P¤ n¡1(¸)+Q¤ n¡1(¸))?we have lim p!1 u(AnPn(¸)Pn¡1(¸)¡1¡Hp)¡1P¤ n¡1(¸)¡1=µ: Since P¤ n¡1(¸)isaninvertiblematrixandfum+1;:::;u Ngis a basis of Im(!P¤ n¡1(¸)+ Q¤ n¡1(¸))?it is enough to prove that for m+1·i·Nwe have lim p!1 ui[AnPn(¸)Pn¡1(¸)¡1¡Hp]¡1=µ: Observe that ui[AnPn(¸)Pn¡1(¸)¡1¡Hp]¡1 =ui[AnPn(¸)Pn¡1(¸)¡1¡C¤MpC]¡1 =uiC¤[CAnPn(¸)Pn¡1(¸)¡1C¤¡Mp]¡1C =ei[CAnPn(¸)Pn¡1(¸)¡1C¤¡Mp]¡1C; where ei=(0;:::;1;:::;0), being the 1 in the position i. The matrix CAnPn(¸)Pn¡1(¸)¡1C¤¡Mpis of the form 0 B B B B B B B B @ ®1;1::: ® 1;m ®1;m+1 ::: ® 1;N . . ..... . .. . ..... . . ®m;1::: ® m;m ®m;m+1 ::: ® m;N ®m+1;1::: ® m+1;m ®m+1;m+1 ¡p::: ® m+1;N . . ..... . .. . ..... . . ®N;1::: ® N;m ®N;m+1 ::: ® N;N ¡p 1 C C C C C C C C A : The determinant of this matrix is a polynomial in the variable pof degree N¡m,whereas the principal minors Ai;j for ior jbigger than mare polynomials in the variable pof degree N¡m¡1. For this reason, any entry Ei;j of [CAnPn(¸)Pn¡1(¸)¡1C¤¡Mp]¡1 with ior jbigger than mtends to 0 when ptends to in¯nity, and consequently lim p!1 ei[CAnPn(¸)Pn¡1(¸)¡1C¤¡Mp]¡1=µ; for m+1·i·N which proves the result. ¥ We ¯nish by proving that the convex hull of the set ¡n(¸)containsthesetIntBn(¸)of interior points of Bn(¸). RIESZ'S THEOREM FOR ORTHOGONAL MATRIX POLYNOMIALS 17 Lemma 7. IntBn(¸)µco(¡n(¸)) Proof Since ¡n(¸)isdenseinExtBn(¸), that is we have ¡n(¸)=ExtBn(¸), we deduce that co ¡n(¸)=co¡n(¸)=co(ExtBn(¸)) = Bn(¸) the last equality by virtue of Krein-Millman's theorem. Now, applying well-known arguments of convexity we have that Int(co(¡n(¸)) = co ¡n(¸)=Bn(¸): As we have previously mentioned, there exists an invertible linear operator Lde¯ned on the N£Ncomplex matrices transforming Bn(¸) bijectively onto the set B=fT:TT¤·Ig. It is immediate that the image set L(¡n(¸)) is dense in ExtB=fT:TT¤=Ig,andthat Int(co(L(¡n(¸))) = B. To prove that IntBn(¸)µco(¡n(¸)) it is enough to prove that IntBµco(L(¡n(¸))). For this, if there exists x2IntBnInt(co(L(¡n(¸))), we can separate xand Int(co(L(¡n(¸))) with a linear operator ¤ such that ¤(x)=1and¤(z)·1 for any zin Int(co(L(¡n(¸))). Since Int(co(L(¡n(¸))) = Bwe have that ¤(z)·1 for any zin Band thus k¤k·1, but this is in contradiction with ¤(x)=1becausexin an interior point of B. ¥ Acknowledgements The author expresses his gratitude to Professor Antonio J. Dur¶an for proposing the problem and for helpful suggestions for the ¯nal draft, and to Professor Luis R. Piazza for fruitful discussions about questions of convexity. References [A]. N. I. Akhiezer, The classical moment problem and some related questions in analysis, english translation, Oliver and Boyd, Edinburgh, 1965. [AN]. A.I. Aptekarev and E.M. Nikishin, The scattering problem for a discrete Sturm-Liouville operator, Math. USSR-Sb. 49 (1984), 325-355. [B]. Ju. M. Berezanskii, Expansions in eigenfunctions of selfadjoint operators, Translations of Mathematical Monographs, American Mathematical Society, Providence, Rhode Island, 1968. [D1]. A. J. Dur¶an, On orthogonal polynomials with respect to a positive de¯nite matrix of measures, Can. J. 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