ENTROPY FUNCTIONALS AND THEIR EXTREMAL VALUES FOR SOLVING THE STIELTJES MATRIX MOMENT PROBLEM

Entropy functionals and their extremal values have been studied by many authors (see, for example, [1], [2], [3], [5]). But similar functionals were not considered for solution interpolation problems in the matrix Stieltjes class. In this paper, entropy functionals over solutions of the Stieltjes matrix moment problem are defined and studied for the first time. Given integers m,n ≥ 1, we let C denote the linear space of columns of complex numbers x = col (


Introduction
Entropy functionals and their extremal values have been studied by many authors (see, for example, [1], [2], [3], [5]). But similar functionals were not considered for solution interpolation problems in the matrix Stieltjes class. In this paper, entropy functionals over solutions of the Stieltjes matrix moment problem are defined and studied for the first time.
Given integers m, n ≥ 1, we let C m denote the linear space of columns of complex numbers x = col x 1 x 2 . . . x m of size m equipped with the inner product (x, y) = m j=1x i y j . Let C m×n be the set of complex matrices with m rows and n columns. Denote by C m×m H the set of all Hermitian matrices. A Hermitian matrix A is called nonnegative if (x, Ax) ≥ 0 ∀x ∈ C m . By C m×m ≥ denote the set of nonnegative matrices. A nonnegative matrix A is called positive if (x, Ax) > 0 for any nonzero vector x ∈ C m . Let C m×m > be the set of positive matrices. By I m ∈ C m×m denote the identity matrix and by O m×n ∈ C m×n denote the zero matrix. We will often omit the subscripts of the identity matrix and the zero matrix if these subscripts are clear from the context. For Hermitian matrices A, B we write If the matrix A is invertible then by A − * denote the matrix A −1 * . If f (z) is a matrix function (MF) then by f * (z) denote the MF (f (z)) * . Let f (z) be an invertible MF. By f −1 (z) and f − * (z) denote MFs (f (z)) −1 and ( f (z)) −1 * respectively. By definition z = |z| exp(i arg z), −π < arg z ≤ π and √ z = |z| exp i arg z 2 . We will also write C + = {z ∈ C : Im z > 0}, C − = {z ∈ C : Im z < 0}, R + = {x ∈ R : x ≥ 0} and R − = {x ∈ R : x < 0}.
Denote by B the σ-algebra of Borel subsets of the real line R. A mapping σ : for any infinite sequence (A j ) ∞ j=1 of pairwise disjoint Borel subsets of R.
We first recall some facts about the Stieltjes matrix moment problem (see [6]- [10]). Let (s j ) 2n+1 j=0 be an arbitrary sequence of complex m × m matrices. We consider the following block matrices: Assume that the block matrices H 1 and H 2 satisfy the following conditions: In the Stieltjes matrix moment problem it is required to describe all matrix-valued nonnegative measures σ on the half-axis R + such that Let M + denote the set of all solutions σ to the Stieltjes matrix moment problem. Under the above assumptions it is known that M + = ∅. With each solution of the Stieltjes matrix moment problem we associate a MF as follows: By F + denote the set of associated MFs. It is obvious that associated MFs are holomorphic MFs in C \ R + . The Stieltjes inversion formula establishes a one-to-one correspondence between F + and M + . Let The pair of meromorphic m × m MF col (p(z) q(z)) in C \ R + is said to be Stieltjes if for this pair there exists a discrete the set of points D pq in C \ R + such that On the set of Stieltjes pairs, we introduce the equivalence ratio: the pairs col (p 1 (z) q 1 (z)) and col (p 2 (z) q 2 (z)) are said to be equivalent if there exists a MF Q(z) such that the MF Q(z), (Q(z)) −1 are both meromorphic in C \ R + and The set of equivalence classes of Stieltjes pairs will be denoted by S ∞ . A polynomial MF The formula establishes a bijective correspondence between F + and S ∞ . Substituting the Stieltjes pairs col (I O) and col (O I) in (1), we obtain extremal MFs Suppose that t 0 ∈ R − . The matrix interval [ s F (t 0 ), s K (t 0 ) ] is called the matrix Weyl interval (see [6]- [10]). We can prove that {s(x 0 ) : From (1) it follows that (s(z) − s * (z))/i ≥ O, z ∈ C + . Consequently, at almost all points t ≥ 0 exist nontangential limits s(t) = lim y→+0 s(t + iy). Let t 0 be a point belongs to R − . For any s belongs to F + the entropy functional I(s; t 0 ) is defined by the formula The main result in our paper is as follows.
Theorem 1. The entropy functional has an upper bound with equality if and only if Moreover, the matrixs(t 0 ) coincides with the center of the matrix Weyl interval

Entropy functionals for the Hamburger matrix moment problem
We first recall some facts about the Hamburger matrix moment problem (see, for example, [4], [5], [12], [13]). Let (w j ) 2n j=0 be an arbitrary sequence of complex m × m matrices. We consider the following block matrices Assume that the block matrix H is positive. In the Hamburger matrix moment problem it is required to describe all the matrix-valued nonnegative measures τ such that Let M denote the set of solutions to problem (3). Under the above assumptions, M = ∅.
With each matrix measure τ ∈ M we associate a MF as follows: By F denote the set of associated MFs f . It is obvious that associated MFs are holomorphic MFs in C + . The Stieltjes inversion formula establishes a one-to-one correspondence between F and M.
A pair of meromorphic m × m MF col (p(w) q(w)) in C + is said to be Nevanlinna if for this pair there exists the discrete set of points D pq in C + such that On the set of Nevanlinna pairs, we introduce the equivalence ratio: the pairs col (p 1 (w) q 1 (w)) and col (p 2 (w) q 2 (w)) are said to be equivalent if there exists a MF Q(w) such that the MF Q(w), (Q(w)) −1 are both meromorphic in C + and The set of equivalence classes of Nevanlinna pairs will be denoted by R ∞ . A polynomial MF is called the resolvent matrix the Hamburger moment problem.
The formula establishes a bijective correspondence between F and R ∞ . By S denote the set of m × m MFs S(w) which are analytic and contractive (i.e., It is well known that the formula establishes a bijective correspondence between S and R ∞ . Here MFs Q(z), (Q(z)) −1 are both meromorphic in C + . By definition, put Combining (6), (7), (8), and (9), we get From (4) follows that (f (w) − f * (w))/i ≥ O, w ∈ C + . Consequently, at almost all points x ∈ R exist nontangential limits f (x) = lim y→+0 f (x + iy). Let w 0 = x 0 + iy 0 be a point belongs to C + . For any f belongs to F the entropy functional I(f ; w 0 ) is defined by the formula The following theorem was proved in [5].
Theorem 2. The entropy functional has an upper bound with equality if and only if In other words,

Entropy functionals for the symmetric Hamburger moment problem
Let (w j ) 4n+2 j=0 be a sequence of complex m × m matrices such that Corresponding moment problem (3) is said to be the symmetric Hamburger matrix moment problem if w 2j+1 = O m×m , j = 0, . . . , 2n. The symmetric moment problem we will study in these chapter. If we replace w 2j , j = 0, . . . , 2n + 1 by s j , we obtain a sequence Given the sequence (12) of m × m matrices, we construct the following block matrices: It is easily verified that the block matrices defined above satisfy the main identity By P denote block matrix These matrix has exactly one entry of 1 in each row and each column and 0s elsewhere, i.e., P is a permutation matrix. In particular, the matrix P is orthogonal P ′ P = P P ′ = I.
It is easy to see that Hear by A(w) denote some polynomial m × m(n + 1) MF.
It follows from (13), (16) that This implies that be a resolvent matrix for the Hamburger symmetric matrix moment problem that corresponds to sequence (12). Then Proof. Using (15), (16), and (17), we get and (19) are proved by analogy. This completes the proof of theorem 3. Lemma 1. Let U (z) be a resolvent matrix for the Hamburger symmetric matrix moment problem that corresponds to sequence (12). Then Proof. We have Using (14), we get Thus we have A 11 (y) = O m×m . The equalities A 12 (y) = A 21 (y) = A 22 (y) = O m×m are proved in a similar way. Finally, we obtain U (iy)J π U * (iy) − J π = O 2m×2m . It follows (see, for example, [1]) that U * (iy)J π U (iy) − J π = O 2m×2m . Lemma 1 is proved.
If some Nevanlinna pair is symmetric then all equivalent Nevanlinna pairs is also symmetric. The corresponding equivalence class of Nevanlinna pairs is called symmetric. The set of equivalence classes of Nevanlinna symmetric pairs will be denoted byR ∞ .
The set of Nevanlinna symmetric MF will be denoted byR.
Consider the symmetric Hamburger moment problem. By definition, put

Theorem 4. Let
be a resolvent matrix for the Hamburger symmetric matrix moment problem that corresponds to sequence (12). Then the formula establishes a bijective correspondence betweenF andR ∞ .
Theorem 5. The entropy functional has an upper bound for all f ∈F with equality if and only if Proof. Using (11), (19), we get (22). Next we have Here we used the formula (20). Thus Using (23), Theorem 4, and Theorem 2, we get Theorem 5.

Entropy functionals for the Stieltjes moment problem
Lemma 2. The formula establishes a bijective correspondence betweenR ∞ and S ∞ . Moreover, the formula establishes a bijective correspondence betweenF and F + .
Consequently 1 w f (w) Lemma 2 is proved.
Proof of Theorem 1. The proof is divided into steps.
Step 1. If f (w) and s(z) are as in Lemma 2 and Proof of step 1. We havẽ Substituting x 2 for t in the last integral, we get Indeed, we get this result if substituting √ t for x in the last integral and using the formula (see [11], p. 546) ln a, a > 0.
Step 2. There is an inequality Proof of step 2. Using Theorem 5 and (25), we get From [6], we get the following formula: This yields that It immediately follows (26).