Infinite Hankel Block Matrices, Extremal Problems

0.09 MB·

by  Lev Sakhnovich

Infinite Hankel Block Matrices, Extremal
9 Problems
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Lev Sakhnovich
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a
J

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Address: 735 Crawford ave., Brooklyn, 11223, New York, USA.

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E-mail address: [email protected]
P
S
. Abstract
h
t
a In this paper we use the matrix analogue of eigenvalue ρ2 to formulate and
m min
to solve the extremal Nehary problem. When ρ is a scalar, our approach
[ min
coincides with Adamjan-Arov-Krein approach.

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Mathematics Subject Classification (2000): Primary15A57;Secondary
v
0 47B10.
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Keywords. Matrix Nehary problem, minimal solution, matrix analogue of
2
1 eigenvalue, Adamjan-Arov-Krein approach.
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9 1 Introduction
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:
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In the paper we consider a matrix version of the extremal Nehary problem
i
X
[1],[4]. Our approach is based on the notion of a matrix analogue of the
r eigenvalue ρ2 . The notion of ρ2 was used in a number of the extremal
a min min
interpolation problems [2],[3],[7]. We note that ρ2 is a solution of a non-
min
linearmatrixinequalityoftheRiccatitype[2],[6],[7]. Ourapproachcoincides
with the Adamjan-Arov-Krein approach [1], when ρ2 is a scalar matrix.
min
Now we introduce the main definitions. Let H be a fixed separable Hilbert
space. By ℓ (H) we denote the Hilbert space of the sequences ξ = {ξ }∞,
2 k 1
where ξ ∈H and
k
∞
2 2
kξk = kξ k < ∞. (1.1)
k
Xk=1
1
The space of the bounded linear operators acting from ℓ (H ) into ℓ (H ) is
2 1 2 2
denoted by [ℓ (H ),ℓ (H )]. The Hankel operator Γ∈[ℓ (H ),ℓ (H )] has the
2 1 2 2 2 1 2 2
form
Γ = {γ }, 1≤j,k≤∞, γ ∈[H ,H ]. (1.2)
j+k−1 k 1 2
Let L [H ,H ] be the space of the measurable operator -valued functions
∞ 1 2
F(ξ)∈[H ,H ], |ξ| = 1 with the norm
1 2
kFk = esssupkFk < ∞, |ξ| = 1. (1.3)
∞
We shall say that an operator ρ∈[H,H] is strongly positive if there exists
such a number δ > 0 that
ρ > δI , (1.4)
H
where I is the identity operator in the space H. The relation
H
ρ≫0 (1.5)
means that the operator ρ is strongly positive. Further we use the following
version of the well-known theorem (see [1] and references there).
Theorem 1.1. Suppose given a sequence γ ∈[H ,H ], 1≤k < ∞ and a
k 1 2
strongly positive operator ρ∈[H ,H ]. In order for there to exist an operator
2 2
function F(ξ)∈L [H ,H ] such that
∞ 1 2
1
c (F) = ξkF(ξ)|dξ| = γ , k = 1,2,... (1.6)
k 2π Z k
|ξ|=1
and
F⋆(ξ)F(ξ)≤ρ2 (1.7)
it is necessary and sufficient that
Γ⋆Γ≤R2, (1.8)
where
R = diag{ρ,ρ,...}. (1.9)
(The integral in the right-hand side of (1.6) converges in the weak sense.)
Proof. Let us introduce the denotations
F (ξ) = F(ξ)ρ−1, γ = γ ρ−1. (1.10)
ρ k,ρ k
2
Relations (1.6) and (1.7) take the forms
1
ξkF (ξ)|dξ| = γ , k = 1,2,... (1.11)
2π Z ρ k,ρ
|ξ|=1
and
F⋆(ξ)F (ξ)≤I . (1.12)
ρ ρ H2
In case (1.11) and (1.12) the theorem is true (see [1]). Hence in case (1.6)
and (1.7) the theorem is true as well.
The aim of this work is the solution of the following extremal problem.
Problem 1.2.In the class of functions F(ξ)∈[H ,H ], |ξ| = 1 satisfying
1 2
condition (1.6) to find the function with the least deviation from the zero.
As a deviation measure we do not choose a number but a strictly positive
matrix ρ such that
min
F⋆(ξ)F(ξ)≤ρ2 . (1.13)
min
The case of the scalar matrix ρ was considered in the article [1]. The
min
transitionfromthescalar matrixρ tothegeneralcaseconsiderably widens
min
the class of the problems having one and onlyone solution. This is important
both from the theoretical and the applied view points. We note that the ρ2
min
is an analogue of the eigenvalue of the operator Γ⋆Γ.
2 Extremal problem
In this section we consider a particular extremal problem. Namely, we try to
find ρ which satisfies the condition
min
Γ⋆Γ≤R2 , R = diag{ρ ,ρ ,...}. (2.1)
min min min min
In order to explain the notion of ρ we introduce the notations B =
min r
[γ ,γ ,...], B = col[γ ,γ ,...]. Then the matrix Γ has the following struc-
2 3 c 2 3
ture
γ B
Γ = 1 r , (2.2)
(cid:20) Bc Γ1 (cid:21)
where
Γ = {γ }, 1≤j,k < ∞, γ ∈[H ,H ]. (2.3)
1 j+k k 1 2
It means that
A A
Γ⋆Γ = 11 12 , (2.4)
(cid:20) A⋆12 A22 (cid:21)
3
where
A = γ⋆γ +B⋆B , A = γ⋆B +B⋆Γ, A = Γ⋆Γ +B⋆B . (2.5)
11 1 1 c c 12 1 r c 22 1 1 r r
Further we suppose that
R2 −A ≫0. (2.6)
22
Then relation (1.8) is equivalent to the relation
ρ2≥A +A (R2 −A )−1A⋆ . (2.7)
11 12 22 12
Definition 2.1. We shall call the strongly positive operator ρ∈[H ,H ] a
2 2
minimal solution of inequality (2.1) if the following requirements are fulfilled:
1.Inequality (2.6) is valid.
2.
ρ2 = A +A (R2 −A )−1A⋆ . (2.8)
min 11 12 min 22 12
It follows from (2.8) that ρ2 coincides with the solution of the non-linear
min
equation
q2 = A +A (Q2 −A )−1A⋆ , (2.9)
11 12 22 12
where q∈[H ,H ], Q = diag{q,q,...}. Let us note that a solution q2 of
2 2
equation (2.8) is an analogue of the eigenvalue of the operator Γ⋆Γ.
Now we are giving the method of constructing ρ . We apply the method
min
of successive approximation. We put
q2 = A , q2 = A +A (Q2 −A )−1A⋆ , (2.10)
0 11 n+1 11 12 n 22 12
where
Q = diag{q ,q ,...}, n≥0. (2.11)
n n n
Further we suppose that
Q2 −A ≫0. (2.12)
0 22
It follows from relations (2.10)-(2.12) that
q2≥q2, Q2≥Q2≫0, n≥0. (2.13)
n 0 n 0
Astheright-handside of(2.10)decreases withthegrowthofq2. thefollowing
n
assertions are true (see[7]).
Lemma 2.2.
1.The sequence q2,q2,... monotonically increases and has the strong limit q2.
0 2
4
2.The sequence q2,q2,... monotonically decreases and has the strong limit q2.
1 3
3.The inequality
q2≤q2 (2.14)
is true.
Corollary 2.3. If condition (2.12) is fulfilled and
q2 = q2 (2.15)
then
ρ2 = q2 = q2 (2.16)
min
A.Ran and M.Reurings [6] investigated equation (2.10) when A are finite
ij
order matrices. Slightly changing their argumentation we shall prove that
the corresponding results are true in our case as well.
Theorem 2.4. Let A be defined by relations (2.5) and let condition (2.13)
ij
be fulfilled. If the inequalities
A ≥0, A ≥0, A A⋆ ≫0 (2.17)
11 22 12 12
are valid, then equation (2.10) has one and only one strongly positive solution
q2 and
ρ2 = q2 = q2 = q2. (2.18)
min
Proof. In view of Lemma 2.2, we have the relations
q2 = A +A (Q2 −A )−1A⋆ , (2.19)
11 12 22 12
q2 = A +A (Q2 −A )−1A⋆ , (2.20)
11 12 22 12
where Q = diag{q,q,...}, Q = diag{q,q,...}. According to (2.14) the in-
equality
y = q2 −q2≥0 (2.21)
holds. The direct calculation gives
y = B⋆YB, Y = diag{y,y,...}, (2.22)
with
B = T(I +TYT)−1/2TA⋆ (2.23)
12
Here T = (Q2 −A )−1/2. Let us introduce the operator
22
0
P = diag{p,p,...}, p = q −A . (2.24)
11
5
From assumption (2.12) and relation (2.17) we deduce that
B⋆PB≪B⋆(Q2 −A )B = p. (2.25)
22
Relation (2.25) can be written in the form
B⋆B ≪I, where B = P1/2Bp−1/2. (2.26)
1 1 1
Formula (2.22) takes the form
y = B⋆Y B , y = p−1/2yp−1/2, Y = P−1/2YP−1/2. (2.27)
1 1 1 1 1 1
Inequality (2.26) imply that equation (2.27) has only trivial solution y = 0.
1
The theorem is proved.
We can omit the condition A A⋆ ≫0, when
12 12
dimH = m < ∞, k = 1,2. (2.28)
k
In this case the following assertion is true.
Theorem 2.5. Let A be defined by relations (2.5) and let conditions (2.12)
ij
and (2.28) be fulfilled. If the inequalities
A ≥0, A ≥0, (2.29)
11 22
are valid, then equation (2.10) has one and only one strongly positive solution
q2 and
ρ2 = q2 = q2 = q2. (2.30)
min
Proof. Let us consider the maps
F(q2) = A +A (Q2 −A )−1A⋆ , (2.31)
11 12 22 12
G(q2) = I +U(Q2 −D)−1U⋆, (2.32)
m
where
U = q−1A Q−1, D = Q−1A Q−1. (2.33)
0 12 0 0 22 0
Fixed points q2 and q2 of maps G and F respectively are connected by the
F G
relation
q = q−1q q−1. (2.34)
G 0 F 0
In view of (2.5) and (2.33) the matrix U has the form U = [u ,u ...], where
1 2
u are m×m matrices. Let d = dimkerU⋆. The relation x∈kerU⋆, x∈Cm is
k
true if and only if
u⋆x = 0, k = 1,2,... (2.35)
k
6
We shall use the decomposition
((kerU⋆)⊥) (kerU⋆). (2.36)
M
With respect to this decomposition the matrices u⋆ and q2 have the forms
k
u⋆ 0 q2 0
u⋆ = 1,k , q2 = 1 , (2.37)
k (cid:20) u⋆ 0 (cid:21) (cid:20) 0 I (cid:21)
2,k d
where u⋆ , u⋆ and q2 are matrices of order (m−d)×(m−d), d×(m−d)
1,k 2,k 1
and (m−d)×(m−d) respectively. We note that
q2 ≥I . (2.38)
11 m−d
Changing the decomposition of the space ℓ (H ) we can represent U⋆, D and
2 2
Q2 in the forms
U⋆ 0 d d Q2 0
U⋆ = 1 , D = 11 12 , Q2 = 11 , (2.39)
(cid:20) U2⋆ 0 (cid:21) (cid:20) d21 d22 (cid:21) (cid:20) 0 I (cid:21)
where U⋆ = col[u⋆ u⋆ ,...], (p = 1,2) and Q2 = diag{q2 ,q2 ,...}. By
p p,1 p,2 11 11 11
direct calculation we deduce that
(Q2 −D)−1 = Tdiag{Q2 −d −d (I −d )−1d⋆ ,I −d }−1T⋆, (2.40)
11 11 11 22 12 22
where
I 0
T = . (2.41)
(cid:20) (I −d22)−1d⋆12 I (cid:21)
Using formulas (2.39)-(2.41) we reduce the map G(q2) to the form
G (q2 ) = Aˆ +Aˆ (Q2 −Aˆ )−1Aˆ⋆ , (2.42)
1 11 11 12 11 22 12
where Aˆ = I +u ((I −d )−1u⋆, Aˆ = u +u (I −d )−1d⋆ ,
11 m−d 2 22 2 12 1 2 22 12
Aˆ = d +d (I −d )−1d⋆ . Relations (2.18), (2.33) and (2.38) imply that
22 11 12 22 12
D≪I, Q2 ≥I. (2.43)
11
Hence according to (2.43) the map G (q2 ) satisfies condition (2.12). By
1 11
repeating the described reduction method we obtain the following result:
either Aˆ⋆ = 0 or kerAˆ⋆ = 0 It is obvious that the theorem is true if Aˆ⋆ = 0.
12 12 12
If kerAˆ⋆ = 0, then the (m−d)×(m−d) matrix Aˆ Aˆ⋆ is positive, i.e. this
12 12 12
7
matrix is strongly positive. Now the assertion of the theorem follows directly
from Theorem 2.4.
Proposition 2.5. Let conditions of either Theorem 2.3 or of Theorem 2.4
be fulfilled.Then there exists one and only one operator function F(ξ) which
satisfies conditions (1.6) and (1.13).
Proof. The formulated assertion is true when
ρ = αI , α = kΓ⋆Γk. (2.44)
min H2
Using formulas (1.10) we reduce the general case to (2.44).The proposition
is proved.
Remark 2.6.The method of constructing the corresponding operator func-
tion is given in paper [1] for case (2.44). Using this method we can construct
the operator function F (ξ) and then F(ξ).
ρmin
Remark 2.7. Condition(2.44)isvalidinafewcases. Byourapproach(min-
imalρ) we obtaintheuniqueness ofthesolutionforabroadclassofproblems.
References
[1] V.M. Adamjan, D.Z. Arov, and M.G. Krein, Infinite Hankel Block Matri-
ces and Related Extension Problems, Amer. Math. Soc. Transl. (2), 111
(1978), 133-156
[2] B. Fritzsche, B. Kirstein, and L.A. Sakhnovich, Extremal Classical Inter-
polation Problems (matrix case), Linear Algebra and Appl. 430 (2009),
762-781.
[3] J.W. Helton and L.A. Sakhnovich, Extremal Problems of Interpolation
Theory, Rocky Mountain J. Math. 35 (2005), 819-841
[4] Z. Nehari, On Bounded Linear Forms, Ann. of Math. (2), 65 (1957),
153-162
[5] V.V.Peller and N.J. Joung, Superoptimal Analytic Approximation of Ma-
trix Functions, J. Funct. Anal. 120 (1994), 300-343
[6] A.C.M. Ran and M.C.B. Reurings, A Nonlinear Matrix Equation Con-
nected to Interpolation Theory, Linear Algebra and Appl. 379 (2004),
289-302
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[7] L.A.Sakhnovich, InterpolationTheoryanditsApplications,Kluwer, Dor-
drecht, 1997
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