Manusc ip : Ap il, 27 2001
RIESZ'S THEOREM FOR ORTHOGONAL MATRIX POLYNOMIALS
Ped o Lopez-Rod iguez
Uni e sidad de Se illa
Abs ac . We desc ibe he image h ough he S iel jes ans o m o he se o solu ions
Vo a ma ix momen p oblem. We ex end Riesz's heo em o he ma ix se ing, p o ing
ha hose ma ices o measu es o V o which he ma ix polynomials a e dense in he
co esponding L2space a e p ecisely hose whose S iel jes ans o m is an ex emal poin (in
he sense o con exi y) o he image se .
1. In oduc ion.
Fo a posi i e Bo el measu e ºon Rwi h ¯ni e momen s o any o de sn=RR ndº( )
we deno e by V he se o posi i e Bo el measu es ¹on Rsa is ying RR nd¹( )=sn,n¸0,
ha is, he se o solu ions o he Hambu ge momen p oblem de¯ned by º.ByVnwe
deno e he se o posi i e Bo el measu es on Rsuch ha RR kd¹( )=sk,0·k·n, ha
is, he se o solu ions o he unca ed momen p oblem de¯ned by º.
We say ha he measu e ºis de e mina e i he e is no o he posi i e measu e ha ing he
same momen s as hose o º, ha is,i V= ºg, o he wise we say ha ºis inde e mina e.
This al e na i e is ela ed o he index o de¯ciency o he ope a o de¯ned on `2by he
in¯ni e Jacobi ma ix
J=0
B
B
@
b0a1
a1b1a2
a2b2a3
.........
1
C
C
A;
whe e he coe±cien s ai(6=0)andbia e he coe±cien s which appea in he h ee e m
ecu ence ela ion sa is¯ed by he o hogonal polynomials (pn)nassocia ed o º,
pn( )=an+1pn+1( )+bnpn( )+anpn¡1( );n¸0:
The index o de¯ciency o Jis 0 i he momen p oblem is de e mina e and 1 i he momen
p oblem is inde e mina e.
This wo k has been pa ially suppo ed by DGICYT e . PB-96-1321-CO2.
This wo k was pa ly ca ied ou a he Ma hema ics Ins i u e o he Uni e si y o Copenhagen unde a
g an o he spanish Minis y o Educa ion and Science o he P og ama de becas de o maci¶on de pe sonal
in es igado en el ex anje o.
1991 Ma hema ics Subjec Classi¯ca ion. 42C05, 44A60.
Typese by A
M
S-T
EX
1
2 P. LOPEZ-RODRIGUEZ
In 1922 Ne anlinna p o ed ha o a ¯xed non eal ¸ he image h ough he S iel jes
ans o m o all he measu es o Vin he poin ¸
I(V)(¸)=½ZR
d¹( )
¡¸:¹2V¾
is ei he a poin i he momen p oblem is de e mina e o a ci cle i he momen p oblem
is inde e mina e, and he occu ence o hese wo cases does no depend on he non eal
¸chosen (see [N]o [A]). The measu es ¹ o which I(¹)(¸) lies in he ci cum e ence o
his ci cle I(V)(¸) a e called N-ex emal (Ne anlinna-ex emal).
In 1923 M. Riesz p o ed ha in o de ha he se Po polynomials in L2(¹)bedense,
i is necessa y and su±cien ha he measu e ¹be N-ex emal a e e y non eal poin ¸,
and o his i is su±cien ha i should be N-ex emal in a leas one such poin (see [Ri]
o [A]).
The pu pose o his pape is o gene alize hese wo esul s o a comple ely inde e mi-
na e ma ix momen p oblem.
Gi en º=(ºi;j)1·i;j·Na posi i e de¯ni e ma ix o measu es ( o any Bo el se A he
nume ical ma ix º(A) is posi i e semide¯ni e) wi h ¯ni e ma ix momen s
Sk=ZR
kdº( )
o any o de k¸0, we deno e by V hese o posi i ede¯ni ema iceso measu esha ing
he same ma ix momen s as hose o º,andbyVn he se o posi i e ma ices o measu es
whose momen s up o deg ee na e he same as hose o º.
We say ha he posi i e de¯ni e ma ix o measu es ºis de e mina e i no o he posi i e
de¯ni e ma ix o measu es has he same momen s as hose o º, ha is, he posi i e de¯ni e
ma ix o measu es ºis uniquely de e mined by he momen s RR ndº( ), n¸0.
By (Pn)1
n=0 we deno e he sequence o o hono mal ma ix polynomials wi h espec o
º,Pno deg ee nand wi h non-singula leading coe±cien .
These polynomials (Pn)nsa is y a h ee e m ecu ence ela ion o he o m
(1.1) Pn( )=An+1Pn+1( )+BnPn( )+A¤
nPn¡1( );n¸0;
(Anand Bnbeing N£Nma ices such ha de (An)6= 0 and B¤
n=Bn), wi h ini ial
condi ion P¡1( )=µ(he e and in he es o his pape , we w i e µ o he null ma ix,
he dimension o which can be de e mined om he con ex . Fo ins ance, he e µis he
N£Nnull ma ix). I is well-known ha his ecu ence ela ion is equi alen o he
o hogonali y wi h espec o a posi i e de¯ni e ma ix o measu es: his is he ma ix
e sion o Fa a d's Theo em (see [AN], [D1]and[DL1]).
We deno e by Qn( ) he co esponding sequence o polynomials o he second kind,
Qn( )=ZR
Pn( )¡Pn(x)
¡xdº(x);n¸0;
which also sa is y he ecu ence ela ion (1.1), wi h ini ial condi ions Q0( )=µand
Q1( )=A¡1
1.
RIESZ'S THEOREM FOR ORTHOGONAL MATRIX POLYNOMIALS 3
In he ma ix case he de e minacy o inde e minacy o he ma ix momen p oblem is
also ela ed o he index o de¯ciency o he ope a o Jde¯ned by he in¯ni e N-Jacobi
ma ix
J=0
B
B
@
B0A1
A¤
1B1A2
A¤
2B2A3
.........
1
C
C
A
on he space `2,whe eAnand Bna e he coe±cien s which appea in he h ee e m
ecu ence ela ion (1.1). In his case he index o de¯ciency can be any na u al numbe
om 0 o N, 0 in he de e mina e case and Nin he comple ely inde e mina e case. In
he la e case he wo se ies
1
X
k=0
Q¤
k(¸)Pk(´)and
1
X
k=0
P¤
k(¸)Pk(´)
con e ge uni o mly in he a iables ¸and ´on e e y bounded se o he complex plane
(see [K]).
In [B]i isp o ed ha he anko helimi ma ixR(¸)= lim
n!1 Rn(¸) exis s and is
hesame o e e ynon eal¸,whe e
(1.2) Rn(¸)=Ãn
X
k=0
P¤
k(¸)Pk(¸)!¡1
:
This esul is also men ioned by K ein in [K], who e e s o [Na] o a p oo . In his pape
we assume he ank o his ma ix is Nand consequen ly he ma ix
1
X
k=0
P¤
k(¸)Pk(¸)is
in e ible, o e e y non eal ¸, and equal o R(¸)¡1.
As in he scala case, o a ¯xed non eal ¸, we deno e by I(V)(¸) he image h ough
he S iel jes ans o m o all he ma ices o measu es o Vin he poin ¸
I(V)(¸)=½ZR
d¹( )
¡¸:¹2V¾:
Fi s ly, o a ¯xed non eal ¸, we desc ibe he se I(V)(¸). This is he se o N£N
complex ma ices !sa is ying he ma ix inequali y
(1.3) [!+C(¸)]R(¸)¡1[!+C(¸)]¤·j¸¡¸j¡2R(¸);
whe e C(¸)=B(¸; ¸)D(¸; ¸)¡1(see he P elimina ies o he de¯ni ions o B(¸; ¸)and
D(¸; ¸). A·Bmeans ha B¡Ais posi i e semide¯ni e.
The ex emal poin s (in he sense o con exi y) o he se I(V)(¸) a e he ma ices ! o
which equali y is a ained in (1.3). I ¹is a ma ix o measu es in V o which I(¹)(¸)is
4 P. LOPEZ-RODRIGUEZ
aex emalpoin o I(V)(¸), we call his ma ix o measu es N-ex emal, as in he scala
case.
Finally, we gene alize Riesz's heo em o he ma ix se ing by p o ing ha he ma ices
o measu es o V o which he se Po ma ix polynomials is dense in he co esponding
space L2(¹) a e p ecisely he N-ex emal ma ices o measu es, and ha he N-ex emali y
o a ma ix o measu es does no depend on he non eal ¸chosen.
Fo he case N= 1 one eco e s he e y well known classical o mulas exposed in [A,
Ch. 1].
2. P elimina ies.
In wha ollows, i P(¸) is a ma ix polynomial, we deno e by P¤(¸) he polynomial
ob ained om P(¸) by eplacing each o i s ma ix coe±cien s by i s he mi ian conjuga e,
so ha P(¸)¤=P¤(¸). Fo a ma ix polynomial o wo a iables P(¸; ´) he de¯ni ion is
he same, so ha we ha e P(¸; ´)¤=P¤(¸; ´). I F(¸) is a holomo phic ma ix unc ion
we de¯ne F¤(¸)=F(¸)¤.
The se o posi i e de¯ni e ma ices o measu es is endowed wi h he ague and weak
opologies. I ºa posi i e de¯ni e ma ix o measu es, he se Vo ma ices o measu es
ha ing he same momen s as ºis a compac and con ex se o hese opologies which
coincide on V(see [DL2]).
Fo ¹a posi i e de¯ni e ma ix o measu es, he space L2(¹)isde¯nedas hese o
N£Nma ix unc ions :R!MN£N(C) such ha ¿( ( )M( ) ( )¤)2L1(¿¹), whe e
M( ) is he Radon-Nikodym de i a i e o ¹wi h espec o i s ace (¿¹)( o ama ix
A=(ai;j)1·i;j·N, we deno e ¿A o i s ace, 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 wi h he no m
k k2;¹ =k¿( ( )M( ) ( )¤)1
2k2;¿¹ =µZR
¿( ( )M( ) ( )¤)d¿¹( )¶1
2
and is a Hilbe space. The duali y wo ks as o he scala case (see [R]o [DL2] o mo e
de ails. Fo he de¯ni ion o he Lpspaces associa ed o ¹,1·p<1,see[DL2]).
We s ess ha since we only impose he ma ices o measu es in V2n o ha e ¯ni e
momen s up o deg ee 2n, o ¹2V2nwe can gua an ee only ha he polynomials up o
deg ee nbelong o he co esponding space L2(¹). In any case, he polynomials (Pk)k=0;:::;n
a e o hono mal wi h espec o any measu e in V2n.
We include he e he ma ix e sion o some classical o mulas o o hono mal scala
polynomials. The p oo s a e easily e i¯ed using he h ee e m ecu ence ela ion (1.1).
(2.1)
An(u; )=( ¡u)
n¡1
X
k=0
Q¤
k(u)Qk( )=Q¤
n¡1(u)AnQn( )¡Q¤
n(u)A¤
nQn¡1( ); o u; 2C;
RIESZ'S THEOREM FOR ORTHOGONAL MATRIX POLYNOMIALS 5
(2.2)
Bn(u; )=¡I+( ¡u)
n¡1
X
k=0
Q¤
k(u)Pk( )=Q¤
n¡1(u)AnPn( )¡Q¤
n(u)A¤
nPn¡1( ); o u; 2C;
( his is G een's o mula),
(2.3)
Cn(u; )=I+( ¡u)
n¡1
X
k=0
P¤
k(u)Qk( )=P¤
n¡1(u)AnQn( )¡P¤
n(u)A¤
nQn¡1( ); o u; 2C;
(2.4)
Dn(u; )=( ¡u)
n¡1
X
k=0
P¤
k(u)Pk( )=P¤
n¡1(u)AnPn( )¡P¤
n(u)A¤
nPn¡1( ); o u; 2C;
( his a Ch is o®el-Da boux o mula). We will also use he Liou ille-Os og adsky o mula
(2.5) Qn(¸)P¤
n¡1(¸)¡Pn(¸)Q¤
n¡1(¸)=A¡1
n; o ¸2C;
and he ela ions
(2.6) Pn(¸)Q¤
n(¸)=Qn(¸)P¤
n(¸); o ¸2C;
(2.7) An(u; )D¤
n(u; )¡B
n(u; )C¤
n(u; )=I; o u; 2C;
and ¯nally
(2.8) Cn(u; )D¤
n(u; )=Dn(u; )C¤
n(u; ); o u; 2C:
We will also use ha
(2.9) Cn(¸; ¸)=¡Bn(¸; ¸)¤; o ¸2C;
and ha
(2.10) Dn(¸; ¸)=(¸¡¸)Rn¡1(¸)¡1and D¤
n(¸; ¸)=(¸¡¸)Rn¡1(¸)¡1; o ¸2C:
By A(u; ), B(u; ), C(u; )andD(u; ) we deno e he limi ma ix unc ions de¯ned om
An(u; ), Bn(u; ), Cn(u; )andDn(u; )whenn ends o in¯ni y.
3. The main heo ems.
Fo any non eal ¸,wede¯ne hese Bn(¸) obe hese o N£Ncomplex ma ices !
such ha

6 P. LOPEZ-RODRIGUEZ
(3.1) [!+Cn(¸)]Rn¡1(¸)¡1[!+Cn(¸)]¤·j¸¡¸j¡2Rn¡1(¸);
whe e Cn(¸)=Bn(¸; ¸)Dn(¸; ¸)¡1.
The calcula ions in Lemma 1 (see below) show ha Bn(¸)isalso hese o N£N
complex ma ices !sa is ying he ma ix inequali y
(3.2)
n¡1
X
k=0
(Q¤
k(¸)+!P¤
k(¸))(Qk(¸)+Pk(¸)!¤)·!¡!¤
¸¡¸:
We pu B1(¸) o he in e sec ion o all he se s Bn(¸). B1(¸) is clea ly he se o
N£Ncomplex ma ices !such ha
(3.3) [!+C(¸)]R(¸)¡1[!+C(¸)]¤·j¸¡¸j¡2R(¸);
whe e C(¸)=B(¸; ¸)D(¸; ¸)¡1.
Simila ly, B1(¸)isalso hese o N£Ncomplex ma ices !such ha
(3.4)
1
X
k=0
(Q¤
k(¸)+!P¤
k(¸))(Qk(¸)+Pk(¸)!¤)·!¡!¤
¸¡¸:
Looking a (3.1) and (3.3) i is immedia e ha upon a linea ma ix ans o ma ion,
any o he se s Bn(¸)o B1(¸) is in a one o one co espondence wi h he se o N£N
complex ma ices Tsa is ying TT¤·I, which is a con ex se whose ex emal poin s a e
he ma ices e i ying TT¤=I, ha is, he uni a y ma ices ( his is a well-known esul
in ope a o heo y which can be p o ed o example wi h he aid o he singula alue
decomposi ion o ma ices). This implies ha hese se s Bn(¸)andB1(¸) a e con ex se s
whose ex emal poin s (Ex Bn(¸) and Ex B1(¸)) a e hose o which equali y is a ained
in (3.1) and (3.2) o (3.3) and (3.4) espec i ely.
By using o mulas (2.1), (2.2), (2.3) and (2.4) in (3.2) i is s aigh o wa d o see ha
an equi alen condi ion o ! o be an ex emal poin o Bn(¸) is ha he ma ix
(3.5) (!P¤
n(¸)+Q¤
n(¸))A¤
n(Pn¡1(¸)!¤+Qn¡1(¸))
is he mi ian.
I is clea ha o all n¸1weha eB1(¸)µBn+1(¸)µBn(¸). I is also clea ha
!belongs o he se o in e io poin s o Bn(¸)o B1(¸)(In Bn(¸)andIn B1(¸)) i a
s ic inequali y is a ained in (3.1) and (3.2) o (3.3) and (3.4) espec i ely.
We ha e he ollowing esul s:
Theo em 1. Le Vdeno e he se o solu ions o a comple ely inde e mina e ma ix
momen p oblem de¯ned by a na ix o measu es ºand le ¸2CnR.Thenweha e
B1(¸)=I(V)(¸):
RIESZ'S THEOREM FOR ORTHOGONAL MATRIX POLYNOMIALS 7
The key o p o e his heo em will be he inclusions
(3.6) In Bn(¸)µI(V2n¡2)(¸)µBn(¸):
The p oo s o hese inclusions p esen mo e di±cul ies han in he scala case. We will
p o e hen la e in Lemmas 1 o 7. Indeed, in he scala case he se I(V2n¡2)(¸)isgi en
by
I(V2n¡2)(¸)=Bn(¸)n½¡qn¡1(¸)
pn¡1(¸)¾:
The poin ¡qn¡1(¸)=pn¡1(¸) lies on he bo de o he ci cle Bn(¸). When amo es along
he eal axis, he quo ien
¡qn(¸)¡aqn¡1(¸)
pn(¸)¡apn¡1(¸)
desc ibes all he poin s o he ci cum e ence o he closed disk Bn(¸) excep o he limi
poin ¡qn¡1(¸)=pn¡1(¸). The well known quad a u e o mula (see [A,p. 20])gi es ha
e e y poin de¯ned by he o me quo ien o a2Rbelongs o I(V2n¡2)(¸). I is easy
o see ha ¡qn¡1(¸)=pn¡1(¸)=2I(V2n¡2)(¸), bu his is o no impo ance because aking
in o accoun ha I(V2n¡2)(¸) is a con ex se and he simple geome y o he ci cles Bn(¸)
i is immedia e o deduce ha In Bn(¸)µI(V2n¡2)(¸). This inclusion is no a all so
immedia e in he ma ix case. We p o e i in Lemmas 2 o 7 by means o new ideas.
P oo o Theo em 1
Suppose ¯ s ha !2B1(¸). B1(¸)iscon ainedinBn(¸) o all nand hus we can pu
!=!n, o all n,being!nin Bn(¸). Since he in e io se In Bn(¸)isdenseinBn(¸), we
can ¯nd ´nin In Bn(¸)such ha lim
n!1 k!n¡´nk=0. SinceIn Bn(¸)µI(V2n¡2)(¸),
he e exis s a ma ix o measu es ¾nin V2n¡2such ha ´n=I(¾n)(¸), o all n.Since he
se ¹¸µ:¿¹(R)·cgis aguely compac , whe e cis a posi i e cons an (see Lemma
3.8 in [DL2]) and ¾n(R)=S0 o n¸0, he e exis s ¾a ague accumula ion poin o
asubsequence(¾np)o (¾n). Like in he p oo o Lemma 3.10 o [DL2]weha e¾2V.
We ha e ¾n(R)=¾(R) o n¸1, so by i ue o Theo em 3.1 o [DL2], (¾n)con e ges
weakly o ¾.Inpa icula
I(¾)(¸) = lim
p!1 I(¾np)(¸) = lim
p!1 ´np=lim
p!1 !np=!
andweha ep o ed ha B1(¸)µI(V)(¸).
Since I(V2n¡2)(¸)µBn(¸), o e e y n, he e e se inclusion is clea .
¥
Theo em 2. (Riesz's heo em o o hogonal ma ix polynomials) Le ¹be a posi i e
de¯ni e ma ix o measu es co esponding o a comple ely inde e mina e ma ix momen
p oblem. Then he ollowing condi ions a e equi alen :
(1) The e exis s ¸02CnRsuch ha I(¹)(¸0)is an ex emal poin (in he sense o
con exi y) o he se B1(¸0).
(2) Fo any ¸2CnR,I(¹)(¸)is an ex emal poin (in he sense o con exi y) o he
se B1(¸)
(3) Pis dense in L2(¹),equi alen ly(Pn( ))1
n=0 is an o hono mal basis o he Hilbe
space L2(¹).
8 P. LOPEZ-RODRIGUEZ
P oo o Theo em 2
(3) )(2) The polynomials a e dense in L2(¹) i o any unc ion in he space L2(¹)
we ha e equali y in Bessel's inequali y, which is equi alen o
(3.7)
1
X
k=0
( ;Pk)( ;Pk)¤=ZR
( )M( ) ¤( )d¿¹( ):
In pa icula , o ¸( )= I
¡¸2L
2(¹)weha e
( ¸;P
k)=ZR
I
¡¸d¹( )P¤
k( )
=ZR
d¹( )P¤
k( )¡P¤
k(¸)
( ¡¸)+ZR
d¹( )
¡¸P¤
k(¸)
=Q¤
k(¸)+I(¹)(¸)P¤
k(¸);
being I(¹)(¸)=ZR
d¹( )
¡¸ he S iel jes ans o m o ¹in he poin ¸,and
ZR
( )d¹( ) ¤( )=ZR
d¹( )
j ¡¸j2=I(¹)(¸)¡I(¹)(¸)¤
¸¡¸=ImI(¹)(¸)
Im¸;
so equali y in (3.7) is
1
X
k=0
(Q¤
k(¸)+!(¸)P¤
k(¸))(Qk(¸)+Pk(¸)!¤(¸)) = ImI(¹)(¸)
Im¸;
ha is I(¹)(¸)2Ex B1(¸).
(2) )(1) is ob ious.
(1) )(3) We suppose (1) holds and we claim n
¸0=I
( ¡¸0)n2 P, o n¸1. The
asse ion o n= 1 is he assump ion. We now p o e ha n+1
¸02 P unde he assump ion
n
¸02 P, so ha he claim is es ablished by induc ion. Fo gi en ²>0, he e exis s a
ma ix polynomial P2Psuch ha k n
¸0¡Pk2·²jIm¸0j. Di iding Pby (x¡¸0)Iwe
ge P(x)=(x¡¸0)Q(x)+A,wi hQano he polynomial o deg ee n¡1andAaN£N
complex ma ix. We ha e
k n+1
¸0¡A ¸0¡Qk2
2=
=¿ZRµI
( ¡¸0)n+1 ¡A
¡¸0
¡Q( )¶M( )µI
( ¡¸0)n+1 ¡A
¡¸0
¡Q( )¶¤
d¿¹( )
=¿ZR
1
j ¡¸0j2µI
( ¡¸0)n¡A¡Q( )( ¡¸0)¶M( )µI
( ¡¸0)n¡A¡Q( )( ¡¸0)¶¤
d¿¹( )
RIESZ'S THEOREM FOR ORTHOGONAL MATRIX POLYNOMIALS 9
·1
jIm¸0j2kI
( ¡¸0)n¡P( )k2
2·²2;
and since A
¡¸0
belongs o he closu e o Pwe deduce ha I
( ¡¸0)n+1 also does.
Now, i 2L
2(¹) is o hogonal o Pand we conside he S iel jes ans o m
I( ¹)(z)=ZR
( )
¡zd¹( );z2CnR
using ha n
¸0; n
¸02 P o n¸1, we see ha
I( ¹)(n)(z)=µ; o z=¸0; ¸0;and n¸0;
bu hen we ha e an analy ic unc ion I( ¹)inadomainDsuch ha I( ¹)(n)(z0)=µ
o n¸0andace ainz02D.SoI( ¹)isequal oµin D.Weconclude ha I( ¹)
is iden ically ze o in each o he wo hal planes CnR,and hen =µ,¹a.e., hence he
polynomials a e dense in L2(¹).
¥
We also ha e he ollowing Theo em:
Theo em 3. I ¹2V2n¡2is such ha I(¹)(¸)2Ex Bn(¸), henPn¡1=L2(¹)
P oo o Theo em 3
The hypo hesis means ha equali y is a ained in (3.2), ha is, he unc ion ¸( )= I
¡¸
can be app oxima ed by ma ix polynomials up o deg ee n¡1. Now he p oo ¯nishes
exac ly in he same way as he p oo o Theo em 2.
¥
4. P oo s o he inclusions.
In his las sec ion we s udy in de ail he se Bn(¸) and o he ela ed se s, wi h he
pu pose o p o ing he inclusions (3.6). We ema k ha hese inclusions a e alid wi hou
supposing he ma ix momen p oblem o be comple ely inde e mina e. We ¯ s p o e he
second inclusion o (3.6)
Lemma 1.
I(V2n¡2)(¸)µBn(¸)
P oo
Le 's suppose ¹is a measu e in V2n¡2. We know ha he ¯ s no hono mal ma ix
polynomials P0;:::;P
n¡1 o m an o hono mal sys em in he space L2(¹). A e he abo e
calcula ions, om Bessel's inequali y o he unc ion ¸( )= I
¡¸,wededuce ha
n¡1
X
k=0
(Q¤
k(¸)+I(¹)(¸)P¤
k(¸))(Qk(¸)+Pk(¸)I(¹)(¸)¤)·ImI(¹)(¸)
Im¸:
16 P. LOPEZ-RODRIGUEZ
Since ibelongs o Ke (!P¤
n¡1(¸)+Q¤
n¡1(¸)), and using ha Tis he mi ian, we ha e ha
o any m+1·i·Nand o any 1 ·j·m,
i(!P¤
n(¸)+Q¤
n(¸))A¤
nu¤
j= i(!P¤
n(¸)+Q¤
n(¸))A¤
n(Pn¡1(¸)!¤+Qn¡1(¸) ¤
j
= i(!P¤
n¡1(¸)+Q¤
n¡1(¸))An(Pn(¸)!¤+Qn(¸)) ¤
j=µ:
This means ha i(!P¤
n(¸)+Q¤
n(¸))A¤
nbelongs o Im(!P¤
n¡1(¸)+Q¤
n¡1(¸))?, and con-
sequen ly (4.9) holds i we p o e ha o any ec o uin Im(!P¤
n¡1(¸)+Q¤
n¡1(¸))?we
ha e
lim
p!1 u(AnPn(¸)Pn¡1(¸)¡1¡Hp)¡1P¤
n¡1(¸)¡1=µ:
Since P¤
n¡1(¸)isanin e iblema ixand um+1;:::;u
Ngis a basis o Im(!P¤
n¡1(¸)+
Q¤
n¡1(¸))?i is enough o p o e ha o m+1·i·Nwe ha e
lim
p!1 ui[AnPn(¸)Pn¡1(¸)¡1¡Hp]¡1=µ:
Obse e ha
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;
whe e ei=(0;:::;1;:::;0), being he 1 in he posi ion i.
The ma ix CAnPn(¸)Pn¡1(¸)¡1C¤¡Mpis o he o m
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 de e minan o his ma ix is a polynomial in he a iable po deg ee N¡m,whe eas
he p incipal mino s Ai;j o io jbigge han ma e polynomials in he a iable po
deg ee N¡m¡1. Fo his eason, any en y Ei;j o [CAnPn(¸)Pn¡1(¸)¡1C¤¡Mp]¡1
wi h io jbigge han m ends o 0 when p ends o in¯ni y, and consequen ly
lim
p!1 ei[CAnPn(¸)Pn¡1(¸)¡1C¤¡Mp]¡1=µ; o m+1·i·N
which p o es he esul .
¥
We ¯nish by p o ing ha he con ex hull o he se ¡n(¸)con ains hese In Bn(¸)o
in e io poin s o Bn(¸).

RIESZ'S THEOREM FOR ORTHOGONAL MATRIX POLYNOMIALS 17
Lemma 7.
In Bn(¸)µco(¡n(¸))
P oo
Since ¡n(¸)isdenseinEx Bn(¸), ha is we ha e ¡n(¸)=Ex Bn(¸), we deduce ha
co ¡n(¸)=co¡n(¸)=co(Ex Bn(¸)) = Bn(¸)
he las equali y by i ue o K ein-Millman's heo em. Now, applying well-known a gu-
men s o con exi y we ha e ha
In (co(¡n(¸)) = co ¡n(¸)=Bn(¸):
As we ha e p e iously men ioned, he e exis s an in e ible linea ope a o Lde¯ned on he
N£Ncomplex ma ices ans o ming Bn(¸) bijec i ely on o he se B= T:TT¤·Ig.
I is immedia e ha he image se L(¡n(¸)) is dense in Ex B= T:TT¤=Ig,and ha
In (co(L(¡n(¸))) = B. To p o e ha In Bn(¸)µco(¡n(¸)) i is enough o p o e ha
In Bµco(L(¡n(¸))).
Fo his, i he e exis s x2In BnIn (co(L(¡n(¸))), we can sepa a e xand In (co(L(¡n(¸)))
wi h a linea ope a o ¤ such ha ¤(x)=1and¤(z)·1 o any zin In (co(L(¡n(¸))).
Since In (co(L(¡n(¸))) = Bwe ha e ha ¤(z)·1 o any zin Band hus k¤k·1, bu
his is in con adic ion wi h ¤(x)=1becausexin an in e io poin o B.
¥
Acknowledgemen s
The au ho exp esses his g a i ude o P o esso An onio J. Du ¶an o p oposing he
p oblem and o help ul sugges ions o he ¯nal d a , and o P o esso Luis R. Piazza o
ui ul discussions abou ques ions o con exi y.
Re e ences
[A]. N. I. Akhieze , The classical momen p oblem and some ela ed ques ions in analysis, english
ansla ion, Oli e and Boyd, Edinbu gh, 1965.
[AN]. A.I. Ap eka e and E.M. Nikishin, The sca e ing p oblem o a disc e e S u m-Liou ille ope a o ,
Ma h. USSR-Sb. 49 (1984), 325-355.
[B]. Ju. M. Be ezanskii, Expansions in eigen unc ions o sel adjoin ope a o s, T ansla ions o Ma he-
ma ical Monog aphs, Ame ican Ma hema ical Socie y, P o idence, Rhode Island, 1968.
[D1]. A. J. Du ¶an, On o hogonal polynomials wi h espec o a posi i e de¯ni e ma ix o measu es,
Can. J. Ma h. 47 (1995), 88-112.
[D2]. A.J.Du ¶an, Ma ko 's Theo em o o hogonal ma ix polynomials,Can.J.Ma h.48 (6) (1996),
1180-1195.
[DL1]. A. J. Du ¶an and P. L¶opez-Rod ¶³guez, O hogonal ma ix polynomials: ze os and Blumen hal's
Theo em, J. App ox. Theo y 84 (1996), 96-118.
[DL2]. A. J. Du ¶an and P. L¶opez-Rod ¶³guez, The Lpspace o a posi i e de¯ni e ma ix o measu es and
densi y o ma ix polynomials in L1, J. App ox. Theo y 90 (1997), 299-318.
[DL3]. A. J. Du ¶an and P. L¶opez-Rod ¶³guez, Densi y ques ions o he unca ed ma ix momen p oblem,
Canadian J. o Ma h. 49 (4) (1997), 708-721.
18 P. LOPEZ-RODRIGUEZ
[K]. M. K ein, In¯ni e J-ma ices and a ma ix momen p oblem, Dokl.Akad. Nauk SSSR 69, n . 2
(1949), 125-128 ( ansla ion om ussian by Wal e Van Assche in a pe sonal no e).
[N]. Ne anlinna, R., Asymp o ische En wickelungen besch Äank e Funk ionen und das S iel jessche Mo-
men enp oblem,Ann.Acad.Sci.Fenn.A185(1922).
[Na]. H. Nagel, Ä
Ube die quad ie ba en He mi eschen Ma izen en s ehenden Ope a o en, Ma h. Ann.
112 109 (1936), 247-285.
[R]. M. Rosenbe g, The squa e-in eg abili y o ma ix- alued unc ions wi h espec o a non-nega i e
he mi ian measu e,DukeMa h.J.31 (1964), 291-298.
[Ri]. M. Riesz, Su lep oblemedesmomen se le h¶eo eme de Pa se al co espondan , Ac a Li . ac
Sci. (Szeged) 1(1922), 209-225.
Ped o L¶
opez Rod ¶
³guez, Depa amen o de An¶
alisis Ma em¶
a ico, Uni e sidad de Se illa,
Apdo. 1160. 41080-Se illa, Spain. E-mail: plo[email p o ec ed]