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\headline{\rm \ifodd\pageno\centerline{\tensmc Proceedings of the IWOP'96}\hfil\rm\folio\else
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{  \eightrm  \copyright 1997 \,\,  M. Alfaro et al. (Eds.) \,\, Proceedings of the
 
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International \, Workshop on Orthogonal \, Polynomials 

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in Mathematical Physics. Legan\'es, 26-26 June, 1996.}  }
 
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\topmatter 
\title On orthogonal matrix polynomials \endtitle
\endtopmatter
\centerline{ {\bf  Antonio J. Duran} }

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\bigskip
%\topmatter
\title \leftline{ 1  Introduction  } 
\endtitle
%\endtopmatter 

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In this paper we deal with orthogonal matrix polynomials.
First of all, we establish some basic notations and results we need later.


A matrix polynomial $P$ is a matrix whose entries are polynomials,
or, equivalently, a combination $P(t)=A_0+A_1t+\cdots +A_nt^n$, where
$A_0, \cdots , A_n$ are numerical matrices (the size of all
the matrices which appear in this paper is $N\times N$).

A positive definite matrix of measures $W$ is a matrix whose entries
are Borel measures and such that for any Borel set $A\in \R $, the
numerical
matrix $W(A)$ is positive semidefinite.

A sequence of matrix polynomials $(P_n)_n$, \text
{dgr}$(P_n)=n$, is orthonormal with respect to a positive definite
matrix of measures $W$ if:
$$
\int P_n(t)dW(t)P_m^*(t)=\delta _{n,m} Id.
$$
Matrix orthonormality is equivalent to the following  three-term
matrix  recurrence relation
$$
tP_n(t)=A_{n+1}P_{n+1}(t)+B_nP_n(t)+A_n^*P_{n-1}(t),
$$
with initial condition $P_{-1}=\theta $ and $P_0$ a nonsingular numerical
matrix, and
where  the $B_n$'s are hermitian  and the $A_n$'s are nonsingular (see
[1, 4, 8]).
Using the polar decomposition of a matrix, the $A_n$'s can be taken
to be  positive definite, while the $QR$  decomposition of Francis
and Kublanovskaja allows us to take the $A_n$'s to be lower triangular
matrices.

We can associate to the sequence of orthogonal matrix polynomials
$(P_n)_n$
the $(2N+1)$-banded infinite Hermitian matrix which  is called the
$N$-Jacobi
matrix
$$
J=\pmatrix B_0&A_1&&&\\
A_1^*&B_1&A_2&&\\
&A_2^*&B_2&A_3&\\
&&\ddots &\ddots &\ddots \endpmatrix .
$$
As in the scalar case, the polynomials of the second kind are defined
as
$$
Q_n(z)=\int {P_n(z)-P_n(t)\over z-t}dW(t).
$$
The sequence $(Q_n)_n$ satisfies the same three-term recurrence 
relation  as that satisfied by $(P_n)_n$ but with  initial conditions
$Q_0=
\theta $,
$Q_1$ a nonsingular  numerical matrix.

The paper is divided in two parts. In the first one we show a close link
between scalar polynomials satisfying higher order recurrence relations
and orthogonal matrix polynomials. Using this relationship, we give a 
nontrivial interpretation of scalar orthogonality as matrix
orthogonality, and, as a consequence, we will see that orthogonal matrix
polynomials are going to be a useful tool to study some problems in the
classical theory of orthogonal polynomials with respect to a positive
measure.
In the second part, we will give a survey of some recent results on
orthogonal matrix polynomials in the real line:
properties of their zeros, quadrature formulae, asymptotic behaviour,
density results, etc.



\bigskip
\topmatter
\title \leftline{ {2} Scalar polynomials and matrix polynomials } \endtitle
\endtopmatter

\vskip -.3cm

According to this plan, the starting point for the first part will be
the well-known Favard's theorem which establishes that for
a sequence of polynomials $(p_n)_n$ (dgr$(p_n)=n$) the following
conditions
are equivalent:

\noindent
$\bullet $ $(p_n)_n$ is a sequence of orthonormal polynomials with respect
to 
a positive measure.

\noindent
$\bullet $ There are two sequences $(a_n)_n$ ($a_n\not =0$, $n\in \N$),
$(b_n)_n$
for which
$$
tp_n(t)=a_{n+1}p_{n+1}(t)+b_np_n(t)+a_np_{n-1}(t). \tag 1
$$
The formula (1) is known as the three-term recurrence relation.

In the late 80's the following problem, which can be considered
as an extension for higher order recurrence relation of Favard's theorem,
arises: {\sl To describe the canonical orthogonality for a sequence of 
polynomials
$(p_n)_n$, (dgr$(p_n)=n$) which satisfies
$$
t^Np_n(t)=c_{n,0}p_n(t)+\sum _{k=1}^N\left( \bar
c_{n,k}p_{n-k}(t)+c_{n+k,k}p_{n+k}(t)\right), \tag 2
$$
where $c_{n,0}$ is real, $c_{n,N}\not =0$, $n\in \N $, and $p_k=0$ for
$k<0$.}

Sequences of polynomials satisfying the previous $(2N+1)$-term recurrence
formula  appear in the context of the so-called discrete Sobolev inner
products,
i.e., inner products obtained by perturbing with a discrete
differential part the inner product defined by
a positive measure:
$$
<p,q>=\int _\R p(t)q(t)d\mu (t) + \sum
_{k,m=1}^nM_{k,m}p^{(k)}(x_k)q^{(m)}(x_m),
$$
where $\mu $ is a positive measure, $M_{k,m}\ge 0$ and $x_k$, $x_m$ are
real.

The solution of this problem is certainly unexpected, and roughly speaking
can be established as follows:
If $(p_n)_n$ satisfy (2) then the polynomials
$p_n(t)$ can be splitted up, attending to the congruence of the 
exponents of $t$ 
modulus
$N$, to get a sequence of matrix polynomials
$(P_n)_n$ which is orthonormal in the real line with respect to a 
positive definite matrix of measures. And conversely, given a sequence
$(P_n)_n$ of orthogonal matrix polynomials with respect to a positive
definite matrix of measures, a sequence of scalar polynomials,
satisfying the recurrence (2),
can be rebuilt from the polynomials in the rows of each $P_n$.

Besides the relationship this solution is showing, the result has an
important
consequence which we will explain later, but which it is worthy to point
out here:  this result provides a non-trivial interpretation of
scalar orthogonality (i.e. orthogonality with respect to a positive
measure)
as matrix orthogonality. This  non-trivial interpretation will be the key
to apply the matrix orthogonality  to solve
some problems in the  theory of orthogonal polynomials with respect
to a positive measure.

To establish the solution of the previous problem in full, we need to
introduce
the following operators which  will be used to split up the scalar
polynomials:
for a natural number $N$, the operators $R_{N,m}$,
$m=0,1,\cdots , N-1$ are defined in the linear space of polynomials
$\P $ as
$$
R_{N,m}(p)(t)=\sum _n {p^{(nN+m)}(0)\over (nN+m)!}t^n,
$$
that is, the operator $R_{N,m}$ takes from the polynomial $p$ those
powers of $t$ congruent with $m$ modulus $N$, then removes the common
factor $t^m$ and changes $t^N$ to $t$.

We can now establish the following theorem which is an extension
of Favard's theorem for higher order recurrence relation:

\proclaim {\bf Theorem 1} ([4])

If the polynomials $(p_n)_n$ (dgr $(p_n)=n$) satisfy a $(2N+1)$-term
recurrence formula then there exists a positive definite matrix of
measures $\mu =(\mu
_{m,m'})_{m,m'=0,\cdots , N-1}$ such that the polynomials
$(p_n)_n$ are orthonormal with respect to the inner product
$$
B(p,q)=\sum _{0\le m,m'\le N-1}\int _R R_{N,m}(p)R_{N,m'}(q)d\mu_{m,m'}.
$$
\endproclaim

Using this theorem the relationship between scalar polynomials satisfying
a higher order recurrence relation and matrix orthogonal polynomials
follows:

\proclaim {\bf Theorem 2} ([11])

Let $p_n(t)$ $(n=0,1,2,\cdots )$ (dgr$(p_n)=n$) be
  scalar polynomials satisfying
a
$(2N+1)$-term recurrence relation and let
$(P_n)_n$ be matrix polynomials given by 
$$        
 P_n(t)=
\pmatrix 
R_{N,0}(p_{nN})(t)      &\hdots &R_{N,N-1}(p_{nN})(t)    \\        
R_{N,0}(p_{nN+1})(t)    &\hdots &R_{N,N-1}(p_{nN+1})(t)  \\        
\vdots                  &\ddots &\vdots                  \\
R_{N,0}(p_{nN+N-1})(t)  &\hdots &R_{N,N-1}(p_{nN+N-1})(t)
\endpmatrix .     
$$
Then this sequence of matrix polynomials is orthonormal in the real line
with respect to a positive definite matrix of measures.


Conversely, if $P_n=(P_{n,m,j})^{N-1}_{m,j=0}$ is a sequence of
orthonormal
matrix
 polynomials, then the scalar polynomials defined by       
$$
p_{nN+m}(t)=\sum _{j=0}^{N-1}t^jP_{n,m,j}(t^N), \qquad
(n\in \N , 0\le m\le N-1 ),        
$$        
satisfy a  $(2N+1)$-term recurrence formula.

\endproclaim

We are now ready to show how to use Theorem 1 to give a non-trivial
interpretation of scalar orthogonality as matrix orthogonality. 
For the sake of simplicity, we take $N=2$. So,
let $(p_n)_n$ be scalar polynomials which are orthonormal with respect to
a
positive measure, or equivalently:
$$
tp_n(t)=a_{n+1}p_{n+1}(t)+b_np_n(t)+a_np_{n-1}(t). \tag 3
$$
Iterating (3) we get a five-term recurrence relation
$$
t^2p_n(t)=\alpha _{n+2}p_{n+2}(t)+\beta _{n+1}p_{n+1}(t)+
\gamma _np_n(t)+\beta _n p_{n-1}(t) +\alpha _np_{n-2}(t).
$$
So, there exists a $2\times 2$ positive definite matrix
of measures
$$
W=\pmatrix w_{11}& w_{12} \\ w_{21}&w_{22} \endpmatrix
$$
such that the polynomials $(p_n)_n$ are orthonormal with
respect to $W$ in the sense of Theorem 1, that is,
$$
\sum _{i,j=0}^1\int _\R R_{2,i}(p_n)R_{2,j}(p_m)dw_{i,j}=\delta _{n,m}.
$$
A natural question arises: is there some relationship between the
measure $\mu $ with respect to which the polynomials $(p_n)_n$ are
orthonormal and the positive definite matrix of measures $W$?

First of all,  it is not difficult to see that
each positive measure $\mu $ for which 
$$
\int p_np_md\mu =\delta
_{n,m}
$$
 gives a positive definite matrix of measures $W_\mu $
 with respect to
which $(p_n)_n$ are also orthonormal; this matrix of measures is defined
as follows:
$$
W_\mu =\pmatrix \psi (\mu )& \psi (t\mu ) \\ 
\psi (t\mu )&\psi (t^2\mu ) \endpmatrix , 
$$
where $\psi (\mu )$ is the image measure of $\mu $ under $\psi
(t)=t^2$. Indeed, the inner product defined by $W_\mu $ in the
space of polynomials  is just the same as that defined by $\mu $.

But, the key of the question is that other positive definite
 matrices of measures
$W$ can exist which are not of the form $W_\mu $. These matrices of
measures
may give (and in fact they do give) new information about the polynomials
$(p_n)_n$.


We illustrate this with two examples.

The first one is related to the asymptotic properties of the derivatives
of a sequence $(p_n)_n$ of scalar polynomials orthonormal with respect
to a positive measure. Indeed, it is very well-known the formula:
$$
{1\over \sum _n \vert p_n(x)\vert ^2}=\sup \{ \mu (\{ x\} ):
\text {$\mu $ is a weight for $(p_n)_n$ }
\}.
$$
However, similar formulas for the derivatives of the polynomials did
not seem to be known: to get them it is necessary to consider the matrix
orthogonality for the scalar polynomials $(p_n)_n$ as explain before;
then,
the following formula, where the matrix weights for $(p_n)_n$ appear
in an essential way, can be
proved 
$$
\split
{1\over \sum _n \vert p_n'(0)\vert ^2}=
\sup \{ &w _{22} (\{ 0\} )-\displaystyle {{\vert w _{12}\vert ^2(\{ 0\} )
\over w _{1,1}(\{ 0\} )}}:W=(w_{i,j})_{i,j=1}^2 \\
&\text {$W$ matrix weight for $(p_n)_n$ }
\}.
\endsplit
$$
Other formulas for higher derivatives evaluated at any real
number can be found in [4].

The second example can be found in [11], and it consists in using the
matrix interpretation of scalar orthogonality to get a new and very simple
proof of Krein's theorem regarding orthogonal polynomials
with a spectrum with finitely many accumulation points.


These applications of matrix orthogonality to solve some problems of
scalar
orthogonality give a  new and strong motivation to do
 a systematic study of
 matrix orthogonality in the real line. 
The first steps
towards this systematic study
can be found in [5], [6], [7], [8], [9] and [10],
although some sparse results
have been obtained during the last fifty years; for
instance, Krein proved some results for
matrix moment problem (using operator theory) ([19], [20]); more
recently,
in the 80's, matrix orthogonality in the real  line has appeared in
connection
with  scattering and prediction theories ([14], [17]), and the analogous to
Favard's theorem has been established ([2]); other related results
can be found in [1], [12], [13], [18], [21], [22] or [23].
 

\bigskip
\topmatter
\title \leftline{ {3} Some results on  orthogonal matrix polynomials} \endtitle
\endtopmatter

\vskip -.3cm

The second part of this paper will
be devoted to show some recent results on  orthogonal matrix polynomials
which can be considered as the starting point for the systematic study
we mentioned above.

The first results  we are going to
consider are related
to the properties of zeros.

 As usual, we say that $a$ is a zero of the matrix
polynomial $P(t)$ if $a$ is a zero of det($P(t)$), i.e., if $P(a)$ 
is a singular matrix or, equivalently, $0$ is an eigenvalue of $P(a)$. 

Given a sequence of orthogonal matrix polynomials $(P_n)_n$, we have
the following properties for the zeros of the polynomial $P_n$:
\roster
\item"$\bullet $" They can have multiplicity bigger than one but always
bounded by $N$, (let us remind that $N$ is the size of the matrices
we are considering).
\item"$\bullet $" The zeros are real.
\item"$\bullet $" There also exists a separation property
between the zeros of $P_n$ and the zeros of $P_{n+1}$ but weaker than
in the scalar case.
\endroster

These properties can be easily deduced from the important fact that 
the zeros of $P_n$ are just the eigenvalues of the
truncated $N$-Jacobi matrix of size $nN$ (of course, with the same
multiplicity) associated to the polynomials $(P_n)_n$ (see [8]). 

Although these properties are quite natural, they are not complete 
in the following sense:
in the scalar case, $a$ is a zero of $p(t)$ if
$p(a)=0$, but in the matrix case, $a$ is a zero of $P(t)$ does
not imply $P(a)=\theta $, $a$ is a zero of $P(t)$ is equivalent to
say that $0$ is an eigenvalue of $P(a)$, but then, something
must be said about the eigenvectors of $P(a)$ associated to
$0$. The following property, which we have  called {\sl the fundamental
property of the zeros} (the reasons for this name will appear clear
when we  have shown the important consequences
of this property), gives an answer to this question
(see [5]):


\roster
\item"{}" If $a$ is a zero of $P_n$ of multiplicity $p$, then
the rank of $P_n(a)$ is just $N-p$ and the matrices
$\left( \text {Adj}\big( P_n(t)
\big) \right) ^{(p-1)}(a)$ and $P^*_{n-1}(a)A_n$ define two isomorphisms 
between the space of right eigenvectors and the space of left eigenvectors
of $P_n(a)$ associated to $0$. The inverse of
$P^*_{n-1}(a)A_n$ is given by the matrix 
$Q_n(a)$ 
($A_n$ is the matrix which appears in the three-term recurrence relation
for $(P_n)_n$, and $Q_n$ is the $n$-th polynomial of
the second kind).
\endroster

This property for the zeros of orthogonal matrix polynomials
is analogous to the fact that they are real and simple in the scalar case.
Indeed,  since in the scalar case the zeros are real and simple,
one can expand the ratio $\displaystyle {{p(z)\over p_n(z)}}$
(where $p$ is a polynomial of degree at most $n$)
into simple fractions:
$$
{p(z)\over p_n(z)}=\sum _{i=1}^n {b_i\over z-a_i};
$$
as a consequence, we can write
 $\displaystyle {{p(z)\over p_n(z)}}$ as the Hilbert transform
of the discrete measure 
$\displaystyle {\mu _n=\sum _{i=1}^n b_i\delta _{a_i}}$; we  can now
deduce properties of the measure $\mu _n$ from the ratio $\displaystyle
{{p(z)\over p_n(z)}}$, and viceversa.
In the matrix case, since the zeros can have multiplicity bigger than one,
one does not expect to be able to expand    the ratio
$P_n^{-1}(z)P(z)$ (or $P(z)P_n^{-1}(z)$)  in simple fractions
(and then to be able to write this ratio as the Hilbert
transform of a matrix of discrete   measures) as in the
scalar case. However, from the fundamental property it follows that
although the multiplicity of the zeros of $P_n$ can be  bigger
than one the  decomposition
$$
P(z) P_n^{-1}(z)=\sum _{k=1}^m{C_k\over z-x_k}, \quad \text {dgr $(P)\le
n-1$}
$$
is always possible.

The fundamental property is also the key to establish
the  following quadrature formulae, where an explicit expression for
the weights is given (see [5]; for other quadrature formulas
lacking an explicit expression for the weights see [8], [22])

\proclaim {\bf Quadrature formulae for  orthogonal matrix polynomials}
Let $(P_n)_n$ be a sequence of orthonormal polynomials
with respect to the positive definite matrix of measures $W$, and let
$x_{n,k}$,
$k=1,\cdots, m$, be the different zeros of
 $P_n$ ordered in increasing size and taking into account their
multiplicities  and let
$$
\Gamma _{n,k}={1\over (\text {det}(P_n(t))^{(l_k)}(x_{n,k})}\big( \text
{Adj}
(P_n(t))\big) ^{(l_k-1)}(x_{n,k})Q_n(x_{n,k}), 
$$
where $l_k$ is the multiplicity of $x_{n,k}$. Then
\roster
\item"(a)" If the degree of $P$ is less than $2n-1$ then
$$
\int P(t)d\mu (t)=\sum _{k=1}^m P(x_{n,k})\Gamma _{n,k}.
$$
\item"(b)"  $\Gamma _{n,k}$ are positive semidefinite matrices of rank
$l_k$,
$k=1,\cdots, m$.
\endroster

\endproclaim


We can now use this quadrature formulae to extend to the matrix case
one of the main asymptotic result in the scalar case: the Markov's theorem 
(see [5]).



\proclaim {\bf Markov's theorem for orthogonal matrix polynomials}
Let $(P_n)_n$ orthogonal matrix polynomials with respect to a positive
definite
and determinate matrix of measures $W$ (as in the scalar case,
determinate means that no other matrix of measures has the same moments
as those of $W$), then
$$
\lim _n P_n^{-1}(z)Q_n(z)=\int  {dW(t)\over z-t}\quad \text {for $z\in \C
\setminus \Gamma ,$}
$$
where $\Gamma $ is certain set of real numbers containing the
support of $W$, and the convergence
is uniform in compact sets.
\endproclaim

We remark that in the matrix version of Markov's theorem, the matrices
$P_n^{-1}$ and $Q_n$ must be multiplied in the order $P_n^{-1}Q_n$,
otherwise the result could be false.

The quadrature formula can also be used  to obtain some ratio
asymptotic results for orthogonal matrix polynomials (see [6]).


\proclaim {\bf Ratio asymptotics for orthogonal matrix polynomials}
Let $(P_n)_n$ be orthonormal matrix polynomials with respect
to the  positive definite matrix of measures $W$, satisfying
$$
\lim _nA_n=A, \quad \lim _n B_n=B,
$$
then
$$
\lim _n P_{n-1}(z)P_n^{-1}(z)=\left( \int {dW_{A,B}\over z-t}\right)
A\quad 
z\in \C \setminus \Gamma,
$$
where $\Gamma $ is certain bounded set of real numbers containing the
support of $W$ and the
convergence is uniform in compact sets. 
$W_{A,B}$  is the matrix of measures with respect to which the matrix
 polynomials $(R_n)_n$ satisfying the three-term recurrence relation
$$
tR_n(t)=AR_{n+1}(t)+BR_n(t)+AR_{n-1}(t)
$$
are orthonormal.
\endproclaim

As for the matrix version of Markov's theorem, the matrix polynomials
$P_n^{-1}$ and $P_{n-1}$ must be multiplied in
the order  $P_{n-1}P_n^{-1}$, otherwise the result could be false.


A partial extension of Blumenthal theorem has been
also proved for this matrix Nevai class,
i.e., the class of orthogonal matrix  polynomials for which 
$$
\lim _nA_n=A, \quad \lim _n B_n=B.
$$
In this case, we assume the stronger condition that the diagonals of the
$N$-Jacobi matrix associated to the polynomials $(P_n)_n$ are convergent
sequences (actually, these sequences are the same as in
the $(2N+1)$-term recurrence relation associated to the polynomials
$(P_n)_n$ in
the sense explained in the first part of this paper).
We write $c_k$ for the limit of
the $k$-th updiagonal, $k=0, 1, \cdots , N$ (since the $N$-Jacobi matrix
is  a
$(2N+1)$-banded hermitian matrix these $N$ updiagonals have all the
information).  Then, the support of the orthogonalizing matrix of measures
for
$(P_n)_n$ is a compact interval plus, possibly, two sequences of real
numbers
outside this interval which tend to its endpoints. This interval is given
by
$$
[c_0+\inf _{x\in [-\pi, \pi ]}t(x),c_0+\sup _{x\in [-\pi, \pi ]}t(x)],
$$
where $t(x)$ is the trigonometric polynomial
$$
t(x)=\left( \sum _{k=1}^N2\Re (c_k)T_k(\cos x)\right) -
\left( \sum _{k=1}^N2\Im (c_k)\sin xU_{k-1}(\cos (x))\right) 
=2\Re \left( \sum _{k=1}^Nc_ke^{ikx}\right) ,
$$
and $(T_k)_k, (U_k)_k$ are the Chebyshev polynomials of the first and
second
kind, respectively (see [8]).

The property (b) of the weights in the quadrature formulae is the key
to solve the following density problem for the truncated moment problem:
let $(P_n)_n$ be a sequence of orthogonal matrix polynomials
with respect to the positive
definite matrix of measures
$W$; we write
$$
\split 
V_n=\{ \nu : &
\text {$\nu $ is a positive definite matrix of measures for which} \\
&\int t^kd\nu (t)=\int t^kdW(t), k=0, \cdots , n 
\} .
\endsplit
$$
then 
 \proclaim {\bf Theorem} ([9])
If $\nu \in V_{2n-2}$ the following conditions are equivalent



\roster
\item $\nu $ is an extremal point, in the sense of convexity,
of
$V_{n-1}$.
\item The space of matrix polynomials of degree less than
$n$ is dense in
$L^2(\nu )$.
\item There exists a matrix $A$, with $A_nA=A^*A_n^*$, such that
$$
\nu =\sum _{i=1}^m G_i\delta _{x_i},
$$
where $x_i$ are the different zeros of
$P_n(t)-AP_{n-1}(t)$ and
$G_i$ are the matrices which appear in the partial fraction decomposition 
$$
(P_n(z)-AP_{n-1}(z))^{-1}(Q_n(z)-AQ_{n-1}(z))=\sum _{i=1}^m{G_i\over
z-x_i}.
$$
\endroster

\endproclaim


This theorem  have important divergences with its scalar version. Indeed,
in the scalar case, the points of the support of the extremal
measures support the maximum mass in $V_{2n-2}$ which at the point $x$ is
$$
{1\over \sum _{k=0}^{n-1}p^2_k(x)} .
$$
This property is no longer true for the matrix  case: the matrices in
$V_{2n-2}$ extremal in $V_{n-1}$ may not support the maximum mass
in $V_{2n-2}$. This happens exactly when the mass is a nonsingular matrix.
In such a case the polynomial $P_n(z)-AP_{n-1}(z)$ has 
 a zero of  multiplicity $N$  (the biggest possible)
at the point $x$; in that case $A$ is given by $A=P_n(x)P^{-1}_{n-1}(x)$
(see [9]).

The following result is also related to density questions, in this case
for the space $L^1(W)$, $W$ a positive definite matrix of measures (see
[10]
for the definition of the $L^p$ spaces for
a positive definite matrix of measures). The result is a partial extension
to the matrix case of the well-known Naimark's theorem which characterizes
 those measures among the convex set of measures having the same moments 
such that the linear space of polynomials is dense in $L^1$: just the
extremal measures (in the sense of convexity). 


\proclaim {\bf Theorem} ([10])
Let $W $ be a definite positive matrix of measures with finite moments,
then
\roster
\item The set $$
\split 
V_n=\{ \nu : &
\text {$\nu $ is a positive definite matrix of measures for which} \\
&\int t^kd\nu (t)=\int t^kdW(t), k\in \N 
\} 
\endsplit
$$
is a compact and convex set.
\item For $\nu \in V$, let $M(t)$  be the matrix of  Radon-Nikodym
derivatives of
the elements of
$\nu $ with respect to the trace measure. Let 
$$
0\le \lambda _1(t)\le \cdots \le \lambda _N(t)
$$
be the eigenvalues of $M(t)$. If there exists a constant
$c>0$, such that
 $\lambda _1(t)\ge c$ then the space of matrix polynomials
is dense in $L^1(\nu )$ if and only if $\nu $ is an extremal point, in the
convexity sense, of $V$.
\endroster

\endproclaim

The existence of extremal points in the convex set $V$ follows from the
compactness of $V$.

The hypothesis in (2) is superfluous for the scalar case: 
in that case the trace of $W$ coincides with $W$ and then the matrix of
the Radon-Nikodym derivatives is $M(t)=1$, hence  the eigenvalues of
$M(t)$ are reduced in this case  to $1$. In the matrix case, 
the eigenvalues of $M(t)$ have a more complicated structure, and to extend
Naimark's theorem to the matrix case we need, as an additional hypothesis,
that
the eigenvalues of $M(t)$ do not approach to $0$.


Some general results 
concerning orthogonal matrix polynomials and differential equations 
have recently been obtained (see [15, 16] for some examples of orthogonal
matrix polynomials satisfying differential equations).  In [7] the
following
problem has been studied: consider the second order differential
operators:
$$
l_{2,L}=A_2D''+A_1D'+A_0D^0,
$$
and
$$
l_{2,R}=D''A_2+D'A_1+D^0A_0,
$$
where $A_2, A_1, A_0$ are certain matrix functions. The problem is:
for which  matrices of measures $W$ the operator $l_2$ is symmetric
 with respect to the inner product defined by $W$?

The noncommutativity of the matrix product introduces a strong difference
between the left and right hand side. Indeed,
for $l_{2,R}$ the problem is equivalent to  the fact that
the orthonormal matrix polynomials  $(P_n)_n$  with respect to $W$ satisfy
the differential equation
$$
l_{2,R}(P_n)=\Gamma _nP_n,
$$
for certain hermitian matrices $\Gamma _n$. But this is not the case
for $l_{2,L}$.

However, there is a classification result for the left hand side case:

\proclaim {\bf Theorem} ([7])
The matrices of measures whose matrix inner product has a symmetric
left hand side second order differential operator  are of the
form
$D^*XD$, where $D$ is a nonsingular numerical matrix 
and $X=\nu I$, where
$\nu $ is a classic measure, that is,
the Jacobi, Laguerre or Hermite weights up to a linear change of variable.
\endproclaim

\bigskip
\topmatter
\title \leftline{ References } \endtitle
\endtopmatter

\vskip -.3cm



\item{[1]}
{\eightrm  D. Alpay,  I. Gohberg: }
{ \sl On orthogonal matrix-valued polynomials and applications.}
{\tensmc Operator Theory: Advances and Applications} {\bf 34} (1988), 79--135.

\item{[2]}
{\eightrm  A.I. Aptekarev, E.M. Nikishin:}
{ \sl The scattering problem for a discrete Sturm-Liouville operator.}
{\tensmc  Mat. USRR Sb.} {\bf 49} (1984), 325--355.


\item {[3]}
{\eightrm  A.J. Duran:}
{\sl A generalization of Favard's theorem for polynomials
satisfying a recurrence relation.}
{\tensmc J.  Approx. Th.} {\bf 74} (1993) 83-104.

\item {[4]}
{\eightrm  A.J. Duran:}
{\sl On orthogonal polynomials with respect to a positive
definite matrix of measures.}
{\tensmc Canad. J.  Math.} {\bf 47} (1995) 88-112.

\item {[5]}
{\eightrm  A.J. Duran:}
{\sl Markov's theorem for orthogonal matrix polynomials.}
{\tensmc Canad. J.  Math. } (to appear).

\item {[6]}
{\eightrm  A.J. Duran:}
{\sl Ratio asymptotics for orthogonal matrix polynomials.}
{\tensmc Preprint.}

\item {[7]}
{\eightrm  A.J. Duran:}
{\sl Matrix inner product having a matrix symmetric second order
differential operator:}
{\tensmc Rocky Mount. J. Math.} (to appear).


\item {[8]}
{\eightrm   A.J. Duran and P. Lopez-Rodriguez:}
{\sl Orthogonal matrix polynomials: zeros and Blumenthal's theorem.}
{\tensmc J. Approx. Th.} {\bf 84} (1996) 96-118.


\item {[9]}
{\eightrm   A.J. Duran and P. Lopez-Rodriguez:}
{\sl Density questions for the truncated matrix moment problem.}
{\tensmc Canad. J.  Math. }(to appear).


\item {[10]}
{\eightrm   A.J. Duran and P. Lopez-Rodriguez:}
{\sl The $L^p$ space of a positive definite matrix of measures and density
of matrix polynomials in $L^1$.}
{\tensmc J. Approx. Th.}  (to appear).

\item {[11]}
{\eightrm   A.J. Duran and W. Van Assche:}
{\sl Orthogonal matrix polynomials  and
higher order recurrence relation.}
{\tensmc Linear Algebra and Appl.} {\bf 219} (1995) 261-280.

\item{[12]}
{\eightrm  H. Dym:}
{ \sl On hermitian block Hankel matrices, matrix polynomials, the Hamburger
moment problem, interpolation and maximum entropy.}
{\tensmc Int. Eq. and Operator Th.} {\bf 12} (1989), 757--811.


\item{[13]}
{\eightrm  P.A. Fuhrmann:}
{ \sl Orthogonal matrix polynomials and System Theory.}
{\tensmc Rend. Sem. Mat. Univ. Politec. Torino } (1988), 68--124.

\item{[14]}
{\eightrm  J.S. Geronimo:}
{ \sl Scattering theory and matrix orthogonal polynomials on the real line.}
{\tensmc Circuits Systems Signal Process.} {\bf 1} (1982), 471--495.


\item{[15]}
{\eightrm  L. Jodar,  R. Company, E. Navarro:}
{ \sl Laguerre matrix polynomials and systems of second order differential
equations.}
{\tensmc Applied Numer. Math.}  {\bf 15} (1994),   129--141.

\item{[16]}
{\eightrm  L. Jodar, M. Legua:}
 {\sl Solving second order matrix differential equations with regular
singular points avoiding the increase of the problem dimension}
{\tensmc Applied Math. Comput.} {\bf 53} (1993),   191--206.


\item{[17]}
{\eightrm  T. Kailath, B. Porat:}
 {\sl State-space generators for orthogonal polynomials}
{\tensmc Prediction theory and harmonic analysis,} Editors (1983), 131-163.

\item{[18]}
\eightrm  I.V. Kovalishina:
 \sl Analytic theory of a class of interpolation problems
\tensmc Math. USRR Izvestiya {\bf 22} (1984), 419--463.

\item{[19]}
{\eightrm  M.G. Krein:}
{ \sl Fundamental aspect of the representation theory of Hermitian
operators with deficiency index (m,m)} 
{\tensmc Amer. Math. Soc. Transl. }{\bf 97} (1970), 75--143.

\item{[20]}
{\eightrm  M.G. Krein:}
{ \sl Infinite $J$-matrices and a matrix moment problem.}
{\tensmc Dokl. Akad. Nauk SSSR} {\bf 69} (1949), 125--128 (translated by
W. Van Assche)


\item{[21]}
{\eightrm  M. Shmoish:}
{ \sl On generalized spectral functions, the parametrization of block
Hankeland block Jacobi matrices.}
{\tensmc Linar Algebra and Appl.} {\bf 202} (1994), 91--128.


\item{[22]}
{\eightrm  A. Sinap,  W. Van Assche:}
{ \sl Polynomial interpolation and Gaussian quadrature for matrix value
functions. }{ \tensmc Linear Algebra and Appl.} {\bf 207} (1994), 71--114.

\item{[23]}
{\eightrm  D. Zhani:}
{ \sl Probleme des moments matriciels sur la droite: construction d'une
famille de solutions et question d'unicite.}
{\tensmc Publications du Departament de Mathematiques, Universite
Claude-Bernard-Lyon I Nouvelle serie} {\bf 21} (1984),
1--84.


\vskip 1cm 



\noindent
Antonio Duran 

\noindent 
Departamento de Analisis Matematico, Universidad de Sevilla,  

\noindent 
Apdo. 1160, 41080-Sevilla, Spain. 

\noindent
{\eightrm E-mail}: {\tt duran\@ cica.es}
\end



