The Discrete Self-Adjoint Dirac Systems of General Type: Explicit Solutions of Direct and Inverse Problems, Asymptotics of Verblunsky-Type Coefficients and the Stability of Solving of the Inverse Problem
We consider discrete self-adjoint Dirac systems determined by the potentials (sequences) {Ck} such that the matrices Ck are positive definite and j-unitary, where j is a diagonal m × m matrix which has m1 entries 1 and m2 entries –1 (m1 +m2 = m) on the main diagonal. We construct systems with the ra...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Цитувати: | The Discrete Self-Adjoint Dirac Systems of General Type: Explicit Solutions of Direct and Inverse Problems, Asymptotics of Verblunsky-Type Coefficients and the Stability of Solving of the Inverse Problem / I. Roitberg, A. Sakhnovich // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 4. — С. 532-548. — Бібліогр.: 22 назв. — англ. |
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irk-123456789-1458852019-02-03T01:23:07Z The Discrete Self-Adjoint Dirac Systems of General Type: Explicit Solutions of Direct and Inverse Problems, Asymptotics of Verblunsky-Type Coefficients and the Stability of Solving of the Inverse Problem Roitberg, I. Sakhnovich, A. We consider discrete self-adjoint Dirac systems determined by the potentials (sequences) {Ck} such that the matrices Ck are positive definite and j-unitary, where j is a diagonal m × m matrix which has m1 entries 1 and m2 entries –1 (m1 +m2 = m) on the main diagonal. We construct systems with the rational Weyl functions and explicitly solve the inverse problem to recover systems from the contractive rational Weyl functions. Moreover, we study the stability of this procedure. The matrices Ck (in the potentials) are the so-called Halmos extensions of the Verblunsky-type coefficients ρk. We show that in the case of the contractive rational Weyl functions the coefficients ρk tend to zero and the matrices Ck tend to the identity matrix Im. Розглянуто дискретнi самоспряженi системи Дiрака, визначенi потенцiалами (послiдовностями) {Ck} так, що матрицi Ck є позитивновизначеними та j-унiтарними, де j = це дiагональна матриця розмiру m × m, що має на головнiй дiагоналi m1 та m2 елементiв, якi дорiвнюють вiдповiдно 1 та 1 (m1 + m2 = m). У роботi побудовано системи з рацiональними функцiями Вейля та точно розв’язано обернену задачу вiдновлення системи за стискальними рацiональними функцiями Вейля. Крiм цього, у роботi дослiджується стiйкiсть цiєї процедури. Матрицi Ck (з потенцiалiв) = це так званi розширення Халмоша коефiцiєнтiв ρk типу Верблюнського. У роботi доведено, що у випадку стискальної рацiональної функцiї Вейля коефiцiєнти ρk прямують до нуля, а матрицi Ck прямують до одиничної матрицi Im. 2018 Article The Discrete Self-Adjoint Dirac Systems of General Type: Explicit Solutions of Direct and Inverse Problems, Asymptotics of Verblunsky-Type Coefficients and the Stability of Solving of the Inverse Problem / I. Roitberg, A. Sakhnovich // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 4. — С. 532-548. — Бібліогр.: 22 назв. — англ. 1812-9471 DOI: https://doi.org/10.15407/mag14.04.532 Mathematics Subject Classification 2000: 34B20, 39A12, 39A30, 47A57 http://dspace.nbuv.gov.ua/handle/123456789/145885 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| description |
We consider discrete self-adjoint Dirac systems determined by the potentials (sequences) {Ck} such that the matrices Ck are positive definite and j-unitary, where j is a diagonal m × m matrix which has m1 entries 1 and m2 entries –1 (m1 +m2 = m) on the main diagonal. We construct systems with the rational Weyl functions and explicitly solve the inverse problem to recover systems from the contractive rational Weyl functions. Moreover, we study the stability of this procedure. The matrices Ck (in the potentials) are the so-called Halmos extensions of the Verblunsky-type coefficients ρk. We show that in the case of the contractive rational Weyl functions the coefficients ρk tend to zero and the matrices Ck tend to the identity matrix Im. |
| format |
Article |
| author |
Roitberg, I. Sakhnovich, A. |
| spellingShingle |
Roitberg, I. Sakhnovich, A. The Discrete Self-Adjoint Dirac Systems of General Type: Explicit Solutions of Direct and Inverse Problems, Asymptotics of Verblunsky-Type Coefficients and the Stability of Solving of the Inverse Problem Журнал математической физики, анализа, геометрии |
| author_facet |
Roitberg, I. Sakhnovich, A. |
| author_sort |
Roitberg, I. |
| title |
The Discrete Self-Adjoint Dirac Systems of General Type: Explicit Solutions of Direct and Inverse Problems, Asymptotics of Verblunsky-Type Coefficients and the Stability of Solving of the Inverse Problem |
| title_short |
The Discrete Self-Adjoint Dirac Systems of General Type: Explicit Solutions of Direct and Inverse Problems, Asymptotics of Verblunsky-Type Coefficients and the Stability of Solving of the Inverse Problem |
| title_full |
The Discrete Self-Adjoint Dirac Systems of General Type: Explicit Solutions of Direct and Inverse Problems, Asymptotics of Verblunsky-Type Coefficients and the Stability of Solving of the Inverse Problem |
| title_fullStr |
The Discrete Self-Adjoint Dirac Systems of General Type: Explicit Solutions of Direct and Inverse Problems, Asymptotics of Verblunsky-Type Coefficients and the Stability of Solving of the Inverse Problem |
| title_full_unstemmed |
The Discrete Self-Adjoint Dirac Systems of General Type: Explicit Solutions of Direct and Inverse Problems, Asymptotics of Verblunsky-Type Coefficients and the Stability of Solving of the Inverse Problem |
| title_sort |
discrete self-adjoint dirac systems of general type: explicit solutions of direct and inverse problems, asymptotics of verblunsky-type coefficients and the stability of solving of the inverse problem |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2018 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/145885 |
| citation_txt |
The Discrete Self-Adjoint Dirac Systems of General Type: Explicit Solutions of Direct and Inverse Problems, Asymptotics of Verblunsky-Type Coefficients and the Stability of Solving of the Inverse Problem / I. Roitberg, A. Sakhnovich // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 4. — С. 532-548. — Бібліогр.: 22 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
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| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2018, Vol. 14, No. 4, pp. 532–548
doi: https://doi.org/10.15407/mag14.04.532
The Discrete Self-Adjoint Dirac Systems of
General Type: Explicit Solutions of Direct
and Inverse Problems, Asymptotics of
Verblunsky-Type Coefficients and the
Stability of Solving of the Inverse Problem
Inna Roitberg and Alexander Sakhnovich
To V.A. Marchenko with admiration
We consider discrete self-adjoint Dirac systems determined by the poten-
tials (sequences) {Ck} such that the matrices Ck are positive definite and
j-unitary, where j is a diagonal m ×m matrix which has m1 entries 1 and
m2 entries −1 (m1 +m2 = m) on the main diagonal. We construct systems
with the rational Weyl functions and explicitly solve the inverse problem to
recover systems from the contractive rational Weyl functions. Moreover, we
study the stability of this procedure. The matrices Ck (in the potentials)
are the so-called Halmos extensions of the Verblunsky-type coefficients ρk.
We show that in the case of the contractive rational Weyl functions the co-
efficients ρk tend to zero and the matrices Ck tend to the identity matrix
Im.
Key words: discrete self-adjoint Dirac system, Weyl function, inverse
problem, explicit solution, stability of solution of the inverse problem,
asymptotics of the potential, Verblunsky-type coefficient.
Mathematical Subject Classification 2010: 34B20, 39A12, 39A30, 47A57.
1. Introduction
The discrete self-adjoint Dirac systems of general type have the form
yk+1(z) = (Im + izjCk)yk(z) (k ∈ N0) , (1.1)
where N0 stands for the set of non-negative integers, Im is the m ×m identity
matrix, “i” is the imaginary unit (i2 = −1) and the m × m matrices {Ck} are
positive and j-unitary:
Ck > 0, CkjCk = j, j :=
[
Im1 0
0 −Im2
]
(m1 +m2 = m;m1,m2 6= 0). (1.2)
c© Inna Roitberg and Alexander Sakhnovich, 2018
https://doi.org/10.15407/mag14.04.532
The Discrete Self-Adjoint Dirac Systems of General Type. . . 533
First, we will consider (in Section 2) explicit solutions of the direct and inverse
problems for system (1.1), (1.2) in terms of the Weyl–Titchmarsh (or simply
Weyl) functions. Direct and inverse problems of general type for this system
were studied (in terms of the Weyl functions) in [5] and explicit solutions for
the case m1 = m2, in [4]. In Section 2 and Appendix, we complete the results
from [5] by adding the properties of the Weyl functions in the lower half-plane and
generalize the explicit results from [4] for the case where m1 does not necessarily
equal m2. We will often shorten our proofs in Section 2 and Appendix and refer
to more detailed proofs in [4, 5]. However, a complete procedure of explicitly
solving the inverse problem from Section 2 is missing in [4] (and so it is new for
m1 = m2 as well).
The case of explicit solutions of direct and inverse problems corresponds to
the rational Weyl functions. The results in Section 2 are based on our generalized
Bäcklund–Darboux (GBDT) approach, which was initiated by the seminal book
[14] by V.A. Marchenko. For various versions of Bäcklund–Darboux transforma-
tions and related commutation methods see, for instance, [1, 2, 7, 9, 11, 15, 17, 21]
and references therein.
Section 3 is dedicated to the asymptotics of the potentials (sequences) {Ck}
corresponding to the rational Weyl functions. For this purpose, we first derive
the asymptotics of the so-called [20] Verblunsky-type coefficients.
Finally, in Section 4, we study the stability of our method of explicit solving
of the inverse problem for system (1.1), (1.2), and these results are new even
for the cases m1 = m2 and m1 = m2 = 1. We note that various important
early results on the stability of solutions for inverse problems were obtained by
V.A. Marchenko (see, e.g., [13]).
In the paper, N denotes the set of natural numbers, R denotes the real axis,
C stands for the complex plane, and C+ (C−) stands for the open upper (lower)
half-plane. The spectrum of a square matrix A is denoted by σ(A).
2. GBDT and direct and inverse problems
1. The fundamental m×m solution {Wk} of (1.1) is normalized by
W0(z) = Im. (2.1)
For the case z ∈ C+, the definition of the Weyl function ϕ(z) of Dirac system
(1.1), (1.2) was given in [5] in terms of Wk(z). Below we define the Weyl function
in C−, which is somewhat more convenient for our purposes. Clearly, this Weyl
function has the properties similar to those in [5, Theorem 3.8].
Definition 2.1. The Weyl function of Dirac system (1.1) (which is given on
the semi-axis 0 ≤ k <∞ and satisfies (1.2)) is an m1 ×m2 matrix function ϕ(z)
in the lower half-plane such that the following inequalities hold:
∞∑
k=0
q(z)k
[
ϕ(z)∗ Im2
]
Wk(z)∗CkWk(z)
[
ϕ(z)
Im2
]
<∞ (z ∈ C−), (2.2)
q(z) := (1 + |z|2)−1. (2.3)
534 Inna Roitberg and Alexander Sakhnovich
The properties of the Weyl function are described in the theorem below, which
is proved in Appendix (using the standard Weyl disk procedure).
Theorem 2.2. There is a unique Weyl function of the discrete Dirac system
(1.1), which is given on the semi-axis 0 ≤ k <∞ and satisfies (1.2). This Weyl
function ϕ is analytic and contractive (i.e., ϕ∗ϕ ≤ Im2) on C−.
In the proof of Theorem 2.2, given in Appendix, we will need the inequalities
Ck ≥ j, (2.4)
which (together with the inequalities Ck ≥ −j) immediately follow from [5,
Proposition 2.2].
Another way to prove Theorem 2.2 and the uniqueness of the solution for the
inverse problem, which we will need further, is to consider the Dirac systems
ỹk+1(z) = (Im + iz j̃ C̃k)ỹk(z) (k ∈ N0) , (2.5)
j̃ := −JjJ∗ =
[
Im2 0
0 −Im1
]
, J :=
[
0 Im2
Im1 0
]
, C̃k := JCkJ
∗. (2.6)
Systems (2.5), (2.6) are dual to systems (1.1), (1.2), and it is immediate from
(1.2), (2.6) that the relations
J∗J = Im, C̃k > 0, C̃k j̃ C̃k = j̃ (2.7)
are valid. Hence, systems (2.5) are again self-adjoint Dirac systems. Similarly
to j̃ and C̃k, we use “tilde” in other notations (introduced for self-adjoint Dirac
systems), when it goes about systems (2.5). For instance, clearly we have m̃1 =
m2, m̃2 = m1. It is easy to see that the fundamental solution {W̃k(z)} of systems
(2.5) is connected with the fundamental solution {Wk(z)} of (1.1) by the equality
W̃k(z) = Wk(−z). (2.8)
Thus, according to (2.2) and (2.8), the function
ϕ̃(z) = ϕ(−z), (2.9)
where ϕ is the Weyl function of system (1.1), satisfies the inequalities
∞∑
k=0
q(z)k
[
Im2 ϕ̃(z)∗
]
W̃k(z)∗C̃kW̃k(z)
[
Im2
ϕ̃(z)
]
<∞ (z ∈ C+). (2.10)
Therefore, by virtue of [5, Definition 3.6], the matrix function ϕ̃(z) is the Weyl
function (on C+) of dual system (2.5). Moreover, we see that there is a one to
one correspondence (2.6), (2.9) between systems (1.1) and (2.5) and their Weyl
functions (on C− and C+, respectively). Hence, [5, Corollary 4.7] yields the
theorem below.
Theorem 2.3. Dirac system (1.1), (1.2) is uniquely recovered from its Weyl
function ϕ(z) (z ∈ C−) introduced by (2.2).
The Discrete Self-Adjoint Dirac Systems of General Type. . . 535
2. In order to consider the case of rational Weyl functions, we introduce
the generalized Bäcklund–Darboux transformation (GBDT) of discrete Dirac sys-
tems. Each GBDT of the initial discrete Dirac system is determined by a triple
{A,S0,Π0} of parameter matrices. Here, we take a trivial initial system and
choose n ∈ N (n > 0), two n× n parameter matrices A (detA 6= 0) and S0 > 0,
and an n×m parameter matrix Π0 such that
AS0 − S0A∗ = iΠ0jΠ
∗
0. (2.11)
Define recursively the sequences {Πk} and {Sk} (k > 0) by the relations
Πk+1 = Πk + iA−1Πkj, (2.12)
Sk+1 = Sk +A−1Sk(A∗)−1 +A−1ΠkΠ∗k(A∗)−1. (2.13)
From (2.11)–(2.13), the validity of the matrix identity
ASr − SrA∗ = iΠrjΠ
∗
r (r ≥ 0), (2.14)
follows by induction.
Definition 2.4. The triple {A,S0,Π0}, where detA 6= 0, S0 > 0 and (2.11)
holds, is called admissible.
In view of (2.13), for the admissible triple we have Sk > 0 (k ≥ 0). Thus, the
sequence
Ck := Im + Π∗kS
−1
k Πk −Π∗k+1S
−1
k+1Πk+1 (2.15)
is well-defined. We say that the sequence {Ck} is determined by the admissible
triple {A,S0,Π0}. We will need also the matrix function wA, which for each k ≥
0 is a so-called transfer matrix function in Lev Sakhnovich’s form [18,21,22] and
is defined by the relation
wA(k, λ) := Im − ijΠ∗kS
−1
k (A− λIn)−1Πk. (2.16)
Now, similarly to [4, 9], we obtain the theorem below.
Theorem 2.5. Let the triple {A,S0,Π0} be admissible and assume that the
recursions (2.12) and (2.13) are valid. Then the matrices Ck given by (2.15)
(i.e., determined by {A,S0,Π0}) are well-defined and satisfy (1.2). Moreover,
in this case the fundamental solution {Wk} of Dirac system (1.1) admits the
representation
Wk(z) = wA(k, −1/z) (Im + izj)k wA(0, −1/z)−1 (k ≥ 0), (2.17)
where wA is defined in (2.16).
Proof. Recall that since S0 > 0, relation (2.13) yields by induction that Sk >
0, and so the sequence {Ck} is well-defined.
536 Inna Roitberg and Alexander Sakhnovich
Next, formula (2.17) easily follows from the equality
wA(k + 1, λ)
(
Im −
i
λ
j
)
=
(
Im −
i
λ
jCk
)
wA(k, λ) (k ≥ 0), (2.18)
which is proved quite similarly to the proof of [4, (2.24)] (and so we omit this
proof here).
It remains to prove (1.2). The second equality in (1.2), that is, CkjCk = j,
follows from (2.18) and the equalities
wA(k, λ)jwA(k, λ)∗ = j, (2.19)
which are to be found in [22] (see also [21, (1.84)]). Indeed, we can easily check
that (
Im −
i
λ
j
)
j
(
Im +
i
λ
j
)
=
(
1 +
1
λ2
)
j, (2.20)
and formulas (2.18)–(2.20) imply that(
Im −
i
λ
jCk
)
j
(
Im +
i
λ
Ckj
)
=
(
1 +
1
λ2
)
j. (2.21)
Clearly, the second equality in (1.2) is immediate from (2.21).
Finally, the first equality in (1.2) is proved in the same way as [4, Proposi-
tion 3.1].
3. It is convenient to partition Π0 into the n×mi blocks ϑi and to partition
wA(0, λ) into the four blocks of the same orders as for j in (1.2):
Π0 = [ϑ1 ϑ2], wA(0, λ) =
[
a(λ) b(λ)
c(λ) d(λ)
]
. (2.22)
Theorem 2.6. Let a sequence {Ck} and so Dirac system (1.1), (1.2) be
determined by some admissible triple {A,S0,Π0}. Then the unique Weyl function
of this system is given by the formula
ϕ(z) = −izϑ∗1S
−1
0
(
In + zA×
)−1
ϑ2, A× = A+ iϑ2ϑ
∗
2S
−1
0 . (2.23)
Proof. Recall the definition (2.2) of the Weyl function ϕ(z), where q(z) =
(1 + |z|2)−1. First, let us show that the summation formula
r∑
k=0
q(z)kWk(z)∗CkWk(z) =
i
(
1 + |z|2
)
(z − z)
(
q(z)r+1Wr+1(z)
∗jWr+1(z)− j
)
(2.24)
is valid. Indeed, according to (1.1) and (1.2), we have
Wk+1(z)
∗jWk+1(z) = Wk(z)∗ (Im − izCkj) j (Im + izjCk)Wk(z)
= q(z)−1Wk(z)∗jWk(z) + i(z − z)Wk(z)∗CkWk(z),
The Discrete Self-Adjoint Dirac Systems of General Type. . . 537
that is,
q(z)kWk(z)∗CkWk(z)
=
iq(z)k−1
(z − z)
(q(z)Wk+1(z)
∗jWk+1(z)−Wk(z)∗jWk(z)) , (2.25)
and (2.24) is immediate from (2.25).
Next, we will need the inequality
wA
(
k,−1
z
)∗
jwA
(
k,−1
z
)
≤ j (z ∈ C−), (2.26)
which together with (2.19), follows from a more general formula (see, e.g., [21,
(1.88)]), of the form
wA (k, λ)∗ jwA (k, λ) = j − i(λ− λ)Π∗k(A∗ − λIn)−1S−1k (A− λIn)−1Πk. (2.27)
Formulas (2.17) and (2.26) yield (in C−) the inequality
Wr+1(z)
∗jWr+1(z)
≤
(
wA(0, −1/z)−1
)∗
(Im − izj)r+1 j (Im + izj)r+1wA(0, −1/z)−1. (2.28)
Setting
ϕ(z) = b(−1/z)d(−1/z)−1 (2.29)
and taking into account (2.22) and (2.29), we derive
(Im + izj)r+1wA(0, −1/z)−1
[
ϕ(z)
Im2
]
= (Im + izj)r+1
[
0
Im2
]
d(−1/z)−1
= (1− iz)r+1
[
0
Im2
]
d(−1/z)−1. (2.30)
It is immediate from (2.28) and (2.30) that
[
ϕ(z)∗ Im2
]
Wr+1(z)
∗jWr+1(z)
[
ϕ(z)
Im2
]
≤ 0 (z ∈ C−). (2.31)
For ϕ(z) given by (2.29), relations (2.24) and (2.31) imply that (2.2) holds, and
thus ϕ(z) is the Weyl function. (We did not discuss the singularities of d(−1/z)
and d(−1/z)−1, but ϕ(z) is analytic in C− because it is meromorphic and it is
the Weyl function.)
It remains to show that the right-hand sides of (2.23) and (2.29) coincide. By
virtue of (2.16) and (2.22), using the inversion formula from the system theory
(see, e.g., [21, Appendix B] and references therein), we obtain
b(λ)d(λ)−1 = −iϑ∗1S
−1
0 (A− λIn)−1ϑ2
(
Im2 + iϑ∗2S
−1
0 (A− λIn)−1ϑ2
)−1
538 Inna Roitberg and Alexander Sakhnovich
= −iϑ∗1S
−1
0 (A− λIn)−1ϑ2
(
Im2 − iϑ∗2S
−1
0 (A× − λIn)−1ϑ2
)
,
where A× = A+ iϑ2ϑ
∗
2S
−1
0 . Since iϑ2ϑ
∗
2S
−1
0 = A×−A = (A×−λIn)− (A−λIn),
we essentially simplify the right-hand side in the formula above:
b(λ)d(λ)−1 = −iϑ∗1S
−1
0 (A× − λIn)−1ϑ2. (2.32)
Hence, the right-hand sides of (2.23) and (2.29), indeed, coincide.
4. We note that the Weyl function ϕ(z) in (2.23) is rational and contractive
on C−. Moreover, ϕ(−1/z) is strictly proper rational and contractive. It is well
known (see, e.g., [10, 12]) that each strictly proper rational m1 × m2 matrix
function ψ(z) admits a representation (the so-called realization)
ψ(z) = C(zIn −A)−1B, (2.33)
where A is an n× n matrix, C is an m1 × n matrix and B is an n×m2 matrix.
Further in the text we will assume that (2.33) is a minimal realization, that is,
the value of n in (2.33) is minimal (among the corresponding values in different
realizations of ψ). The following proposition is immediate from [19, Lemma 3.1]
(and is based on several theorems from [12], for details, see [19]).
Proposition 2.7. Assume that a strictly proper rational m1 × m2 matrix
function ψ(z) is contractive on C− and that (2.33) is its minimal realization.
Then there is a unique Hermitian solution X of the Riccati equation
XBB∗X − i(A∗X −XA) + C∗C = 0 (2.34)
such that the relation
σ(A− iBB∗X) ⊂ (C+ ∪ R) (2.35)
holds. Moreover, this solution X is positive.
Next, we give an explicit procedure of solving the inverse problem to recover
Dirac system from its Weyl function.
Theorem 2.8. Let ϕ(z) be a rational m1 × m2 matrix function such that
ψ(z) = ϕ(−1/z) is a strictly proper rational matrix function, which is contractive
on R and has no poles on C−. Assume that (2.33) is a minimal realization of ψ
and that X > 0 is a solution of (2.34).
Then ϕ(z) is the Weyl function of the Dirac system (1.1), (1.2), the potential
{Ck} of which is determined by the admissible triple
A = A− iBB∗X, S0 = X−1, ϑ1 = iX−1C∗, ϑ2 = B, Π0 = [ϑ1 ϑ2]. (2.36)
Proof. Since ψ(z) is contractive on R and has no poles on C−, it is contractive
on C−. Thus, according to Proposition 2.7, a positive definite solution X of (2.34)
The Discrete Self-Adjoint Dirac Systems of General Type. . . 539
exists. In view of (2.36), by choosing X > 0, we have S0 > 0. Moreover, relations
(2.34) and (2.35) yield the equality
ϑ2ϑ
∗
2 + i
((
A+ iϑ2ϑ
∗
2S
−1
0
)
S0 − S0
(
A+ iϑ2ϑ
∗
2S
−1
0
)∗)
+ ϑ1ϑ
∗
1 = 0, (2.37)
which is equivalent to (2.11). Hence the triple {A,S0,Π0} is admissible.
It remains to show that for the Weyl function ϕ(z) of the Dirac system (deter-
mined by this triple), the function ψ(z) = ϕ(−1/z) coincides with ψ(z) admitting
realization (2.33). Taking into account Theorem 2.6 and equalities (2.36), we see
that ψ(z) determined by our triple has the form
ψ(z) = iϑ∗1S
−1
0 (zIn −A)−1ϑ2 = C(zIn −A)−1B, (2.38)
and the right-hand sides of (2.33) and (2.38), indeed, coincide.
3. Verblunsky-type coefficients and asymptotics of the poten-
tials
Recall that the matrices Ck from the potential (sequence) {Ck} are positive
definite and j-unitary (i.e., they satisfy (1.2)). According to [5, Proposition 2.4],
it means that they admit the representations
Ck = DkHk, Dk := diag
{(
Im1 − ρkρ∗k
)− 1
2 ,
(
Im2 − ρ∗kρk
)− 1
2
}
, (3.1)
Hk :=
[
Im1 ρk
ρ∗k Im2
]
(ρ∗kρk < Im2). (3.2)
Here, the m1 × m2 matrices ρk are the so-called Verblunsky-type coefficients,
which were studied in detail in [20]. It is well known (see, e.g., [3]) that DkHk =
HkDk. Clearly, ρ∗kρk < Im2 yields ρkρ
∗
k < Im1 and vice versa.
In this section, we show that
lim
k→∞
[
Im1 0
]
Ck
[
Im1
0
]
= Im1 , (3.3)
and so ρk → 0 and Ck → Im. More precisely, we prove the following statement.
Theorem 3.1. Let the triple {A,S0,Π0} be admissible and assume that −i 6∈
σ(A). Then, for the potential {Ck} (of the Dirac system (1.1)) determined by
this triple, the asymptotic relations
lim
k→∞
ρk = 0, lim
k→∞
Ck = Im (3.4)
are valid.
Proof. Consider the equality
Sk+1 −
(
In + iA−1
)
Sk
(
In − i(A∗)−1
)
= Sk+1 − Sk −A−1Sk(A∗)−1 + iA−1(ASk − SkA∗)(A∗)−1. (3.5)
540 Inna Roitberg and Alexander Sakhnovich
Using (2.13) and (2.14), we rewrite (3.5):
Sk+1 −
(
In + iA−1
)
Sk
(
In − i(A∗)−1
)
= A−1Πk(Im − j)Π∗k(A∗)−1. (3.6)
Now, we partition Πk and, taking into account (2.12) and (2.22), write it down
in the form
Πk =
[(
In + iA−1
)k
ϑ1
(
In − iA−1
)k
ϑ2
]
. (3.7)
In view of (3.6) and (3.7), setting
Rr :=
(
In + iA−1
)−r
Sr
(
In − i(A∗)−1
)−r
, (3.8)
we have
Rk+1 −Rk = 2
(
In + iA−1
)−k−1
A−1
(
In − iA−1
)k
ϑ2
× ϑ∗2
((
In − iA−1
)k)∗ (
A−1
)∗ ((
In + iA−1
)−k−1)∗ ≥ 0. (3.9)
Since R0 = S0 > 0, relations (3.9) imply that there is a limit
lim
k→∞
R−1k = κR ≥ 0. (3.10)
On the other hand, from (3.7) and (3.8), we derive
[
Im1 0
]
Π∗kS
−1
k Πk
[
Im1
0
]
= ϑ∗1R
−1
k ϑ1, (3.11)
and so (3.10) yields
lim
k→∞
[
Im1 0
]
Π∗kS
−1
k Πk
[
Im1
0
]
= ϑ∗1κRϑ1. (3.12)
The definition (2.15) of Ck and the existence of the limit in (3.12) show that (3.3)
holds. It is easy to see that the first equality in (3.4) follows from (3.1)–(3.3).
Finally, the second equality in (3.4) is immediate from (3.1), (3.2) and the first
equality in (3.4).
Remark 3.2. According to Theorems 2.3, 2.6, 2.8 and Proposition 2.7, given
a potential {Ck} determined by some admissible triple we can recover another
admissible triple {A,S0,Π0}, which determines the same sequence {Ck} and has
additional property σ(A) ⊂ (C+ ∪ R). Namely, we construct first the Weyl
function using the initial triple and the procedure from Theorem 2.6. Next, we
recover another admissible triple {A,S0,Π0} such that σ(A) ⊂ (C+ ∪ R) in the
process of solving the inverse problem.
Thus, we assume σ(A) ⊂ (C+ ∪ R) without loss of generality, and so the
condition −i 6∈ σ(A) in Theorem 3.1 can be omitted.
The Discrete Self-Adjoint Dirac Systems of General Type. . . 541
We note that in the case of {Ck} determined by some admissible triple,
Verblunsky-type coefficients can be expressed explicitly. Indeed, in view of (3.1)
and (3.2), we have
ρk =
([
Im1 0
]
Ck
[
Im1
0
])−1 [
Im1 0
]
Ck
[
0
Im2
]
. (3.13)
Hence, taking into account (2.15) and (3.11), we derive
ρk =
(
Im1 + ϑ∗1R
−1
k ϑ1 − ϑ∗1R−1k+1ϑ1
)−1
×
[
Im1 0
]
(Π∗kS
−1
k Πk −Π∗k+1S
−1
k+1Πk+1)
[
0
Im2
]
. (3.14)
4. Stability of the procedure of solving the inverse problem
It is easy to see that the procedure (given in Theorem 2.8) to recover system
(1.1), (1.2) consists of two steps. The first step is the construction of X > 0 and
the second step is the construction of the potential {Ck} using this X.
We start with the matrix function ϕ(z) such that ψ(z) = ϕ(−1/z) is a strictly
proper rational m1 × m2 matrix function which is contractive on C−. More
precisely, we start with a minimal realization (2.33) of ψ (or, equivalently, with
the triple {A,B, C}) and consider the stability of the recovery of X > 0 satisfying
additional condition (2.35). The existence and uniqueness of X > 0 satisfying
(2.35) follow from Proposition 2.7.
Definition 4.1. By Gn, we denote the class of triples
{
Ã, B̃, C̃
}
which deter-
mine minimal realizations ψ̃(z) = C̃
(
zIn−Ã
)−1B̃ of the m1×m2 matrix functions
ψ̃(z) contractive on C−.
The recovery of X > 0 satisfying (2.34), (2.35) from the minimal realization
(2.33) of ψ(z) (where
{
A,B, C
}
∈ Gn) is called stable if for any ε > 0 there is δ >
0 such that for each
{
Ã, B̃, C̃
}
, satisfying the conditions{
Ã, B̃, C̃
}
∈ Gn,
∥∥A− Ã∥∥+
∥∥B − B̃∥∥+
∥∥C − C̃∥∥ < δ, (4.1)
there is a solution X̃ = X̃∗ of the equation
X̃B̃B̃∗X̃ − i
(
Ã∗X̃ − X̃Ã
)
+ C̃∗C̃ = 0 (4.2)
in the neighbourhood
∥∥X − X̃∥∥ < ε of X.
The stability of the recovery of X follows (similarly to the case of the contin-
uous Dirac system) from [19, Theorem 3.3] based on [16, Theorem 4.4]. Namely,
applying [19, Theorem 3.3] to the triples {−A,B,−C} and
{
− Ã, B̃,−C̃
}
, we get
our next statement.
Proposition 4.2. The recovery of X > 0, satisfying (2.34), (2.35), from the
minimal realization (2.33) (with {A,B, C} ∈ Gn) is stable.
542 Inna Roitberg and Alexander Sakhnovich
Remark 4.3. Note that (according to [16, Theorem 4.4]) we may consider a
wider than Gn class of perturbed triples
{
Ã, B̃, C̃
}
, that is, such perturbed triples
that (4.2) has a Hermitian solution X̃ = X̃∗.
Recall that given the triple {A,B, C} and X > 0, we construct the matrices
A, Sk, Rk, . . . For the matrices constructed in a similar way in the case of the
triple
{
Ã, B̃, C̃
}
and of X̃ > 0 satisfying
X̃B̃B̃∗X̃ − i
(
Ã∗X̃ − X̃Ã
)
+ C̃∗C̃ = 0, (4.3)
we use notations with “tilde”: Ã, S̃k, R̃k, . . .
The stability of the second step of solving the inverse problem one can prove
under the additional condition κR = 0 or, equivalently,
lim
k→∞
Rk = +∞, (4.4)
which means that all the eigenvalues of Rk tend to infinity. Unlike the skew-self-
adjoint case [6], equality (4.4) is not fulfilled automatically.
A sufficient condition of stability can also be expressed in terms of matrices
Qr introduced by the relations
Qr :=
(
In − iA−1
)−r
Sr
(
In + i(A∗)−1
)−r
. (4.5)
Clearly, we assume in (4.5) that i 6∈ σ(A). Similarly to equality (3.6), from (2.13)
and (2.14), we have
Sk+1 −
(
In − iA−1
)
Sk
(
In + i(A∗)−1
)
= A−1Πk(Im + j)Π∗k(A∗)−1. (4.6)
Hence, taking into account (3.7) (in analogy with relation (3.9) for Rr), we derive
Qk+1 −Qk = 2
(
In − iA−1
)−k−1
A−1
(
In + iA−1
)k
ϑ1
× ϑ∗1
((
In + iA−1
)k)∗ (
A−1
)∗ ((
In − iA−1
)−k−1)∗ ≥ 0. (4.7)
Since Q0 = S0 > 0, relations (4.7) imply that there is a limit
lim
k→∞
Q−1k = κQ ≥ 0. (4.8)
Moreover, (3.7) and (4.5) yield
lim
k→∞
[
0 Im1
]
Π∗kS
−1
k Πk
[
0 Im1
]
= ϑ∗2κQϑ2. (4.9)
Formula (4.9) implies that
lim
k→∞
[
0 Im2
]
Ck
[
0 Im2
]
= Im2 , (4.10)
which gives another way of proving Theorem 3.1. The cases where (4.4) or the
equality
lim
k→∞
Qk = +∞ (4.11)
The Discrete Self-Adjoint Dirac Systems of General Type. . . 543
holds are considered in the stability theorem below. (Recall that the sequence
{Rk} is given by (3.8) or, equivalently, by (3.9) together with (2.36) and R0 =
S0.) In Proposition 4.5 at the end of this section we present a wide class where
(4.11) is valid.
Theorem 4.4. Consider the procedure (from Theorem 2.8) of the unique
recovery of the potential {Ck} of the discrete self-adjoint Dirac system (1.1),
(1.2) from a minimal realization (2.33), where ψ(z) = ϕ(−1/z) and ϕ(z) is the
Weyl function of the system (1.1), (1.2). Assume that X in this procedure is
chosen such that (2.35) holds (which is always possible). Assume also that either
the sequence {Rk} satisfies (4.4) or i 6∈ σ(A) and the sequence {Qk} satisfies
(4.11).
Then this procedure of the recovery of the potential {Ck} is stable in the class
of triples from Gn.
Proof. The recovery of X > 0 satisfying (2.34), (2.35) is possible according
to Proposition 2.7 and stable according to Proposition 4.2.
Now, in order to show that the recovery of {Ck} is stable under condition (4.4),
we choose some small ε̂ > 0 and such a large N > 0 and a small neighbourhood of
{A,B, C} that
∥∥R−1k
∥∥ < ε̂ and
∥∥R̃−1k
∥∥ < 2ε̂ for X > 0 satisfying (2.34), (2.35), for
k > N , and for the matrices X̃ > 0 satisfying (4.3) (where the triples
{
Ã, B̃, C̃
}
∈
Gn belong to the mentioned above neighbourhood of {A,B, C} and X̃ are those
solutions of (4.3) which belong to the neighbourhood of X). Here, we use the
fact that the sequence
{
R̃k
}
is monotonically increasing and if R̃r0 is sufficiently
large, then R̃r (r > r0) is sufficiently large as well.
In view of (2.15) and (3.11), we see that for sufficiently small ε̂ the matrices[
Im1 0
]
Ck
[
Im1
0
]
,
[
Im1 0
]
C̃k
[
Im1
0
]
(4.12)
are sufficiently close to Im1 . This, in turn, means that (in view of (3.1) and (3.2))
the matrices ρk, ρ̃k are sufficiently small, and thus Ck and C̃k are sufficiently close
to Im. Therefore, for any ε > 0, we may choose ε̂ such that∥∥Ck − C̃k
∥∥ < ε for all k > N(ε̂).
Moreover, for any ε > 0, we may choose a neighbourhood ofX and of {A,B, C}
such that for
{
Ã, B̃, C̃
}
from this neighbourhood the inequalities∥∥Ck − C̃k
∥∥ < ε (0 ≤ k ≤ N(ε̂))
are valid as well. Thus, the recovery of {Ck} is stable, indeed.
The stability of the recovery of {Ck} under condition (4.11) is proved in a
similar way.
Now, consider the case where A is similar to a diagonal matrix D (A is
diagonalisable):
A = UDU−1. (4.13)
544 Inna Roitberg and Alexander Sakhnovich
Relations (2.35), (2.36) and (4.13) yield σ(D) ∈ (C+ ∪ R) or, equivalently,
i(D∗ −D) ≥ 0. (4.14)
Proposition 4.5. Let the sequence {Qk} be given by (4.5), where A and {Sk}
are constructed with the use of the procedure from Theorem 4.4, A is diagonaliz-
able (i.e., representation (4.13) holds) and i 6∈ σ(A). Then (4.11) is valid.
Proof. According to (4.7), we have
Qk+n −Qk = 2(A− iIn)−n−k(A+ iIn)kF (A∗ − iIn)k(A∗ + iIn)−n−k, (4.15)
F :=
n∑
`=1
(A− iIn)n−`(A+ iIn)`−1ϑ1ϑ
∗
1(A
∗ − iIn)`−1(A∗ + iIn)n−`, (4.16)
where F does not depend on k. Let us show that F is strictly positive, that is,
F > 0. Indeed, it is easy to see (more details are given in the similar part of the
proof of [6, Proposition 4.10]) that
Span
n⋃
`=1
(A− iIn)n−`(A+ iIn)`−1ϑ1 = Span
n⋃
`=1
A`−1ϑ1,
and we have only to prove that the pair {A, ϑ1} is controllable.
Since realization (2.33) is minimal, the pair {A∗, C∗} is controllable. In view
of (2.36), the controllability of the pair
{
X−1A∗X,ϑ1
}
follows from the control-
lability of {A∗, C∗}. Hence, the equality
X−1A∗X = A− iϑ1ϑ
∗
1X (4.17)
(which we derive below) implies that the pair {A, ϑ1} is controllable as well.
Finally, using (2.36), we rewrite (2.11) in the form
AX−1 −X−1A∗ = i(ϑ1ϑ
∗
1 − ϑ2ϑ∗2).
This yields in turn that X−1A∗X = A + iBB∗X − iϑ1ϑ
∗
1X. Applying now the
first equality from (2.36), we obtain (4.17). Thus {A, ϑ1} is controllable and the
inequality F > 0 is proved.
Next, we show that
(D − iIn)−1(D + iIn)
(
(D − iIn)−1(D + iIn)
)∗ ≥ In. (4.18)
Inequality (4.18) is equivalent to the inequality
(D + iIn)(D∗ − iIn) ≥ (D − iIn)(D∗ + iIn),
which follows from (4.14).
Now, formula (4.15), representation (4.13) and inequalities F > 0 and (4.18)
imply that
Qk+n −Qk ≥ εIn (4.19)
for some ε > 0, which does not depend on k. The asymptotics (4.11) is immediate
from (4.19).
The Discrete Self-Adjoint Dirac Systems of General Type. . . 545
Appendix. Proof of Theorem 2.2
Proof. It is easy to see that
(Im + izjCk)∗j (Im + izjCk) =
(
1 + z2
)
j, (A.20)
and so both (Im + izjCk) and Wr(z) =
∏r−1
k=0(Im + izjCk) are invertible for z 6=
±i. Now let us consider the sets Nr of the linear fractional transformations
ϕr(z,P) =
[
Im1 0
]
Wr(z)
−1P(z)
([
0 Im2
]
Wr(z)
−1P(z)
)−1
, (A.21)
where P(z) are nonsingular m ×m2 matrix functions with property-j. That is,
P(z) are meromorphic on C− matrix functions such that the inequalities
P(z)∗P(z) > 0, P(z)∗jP(z) ≤ 0 (A.22)
hold for all the points in C− (excluding, possibly, discrete sets of points). The
sets Nr are well-defined because the inequality
det
([
0 Im2
]
Wr(z)
−1P(z)
)
6= 0 (A.23)
follows from (A.22). Indeed, since relations (1.2) and (2.4) yield
(Im + izjCk)∗j(Im + izjCk) =
(
1 + |z|2
)
j + i(z − z)Ck ≥ q̃(z)j, (A.24)
q̃(z) := 1 + |z|2 + i(z − z) > 0, (A.25)
we have
Wr(z)
∗jWr(z) ≥ q̃(z)rj, i.e.,
(
Wr(z)
−1)∗ jWr(z)
−1 ≤ q̃(z)−rj. (A.26)
Thus, the inequalities
P(z)∗
(
Wr(z)
−1)∗ jWr(z)
−1P(z) ≤ 0,
[
0 Im2
]
j
[
0
Im2
]
< 0 (A.27)
are valid, and (A.23) is immediate from [21, Proposition 1.43].
In view of (A.21), we have
ϕr+1(z,P) =
[
Im1 0
]
Wr(z)
−1P̃(z)
([
0 Im2
]
Wr(z)
−1P̃(z)
)−1
, (A.28)
where
P̃(z) = (Im + izjCr)
−1P(z). (A.29)
Relations (A.24), (A.25) and (A.29) imply that
P̃(z)∗jP̃(z) ≤ 0. (A.30)
Compare (A.21), (A.22) with (A.28), (A.30) to see that the sets (Weyl disks) Nr
are embedded:
Nr+1 ⊆ Nr. (A.31)
546 Inna Roitberg and Alexander Sakhnovich
Clearly, formulas (A.28)–(A.30) remain valid when we put there r = 0. For that
case, we partition P̃ and (in view of (2.1)) rewrite (A.28) in the form
ϕ1(z,P) = P̃1(z)P̃2(z)−1, P̃ =:
[
P̃1
P̃2
]
, (A.32)
where (according to (A.23) with r = 1) we have det P̃2(z) 6= 0. It follows from
(A.30) and (A.32) that the functions from N1 are contractive. Hence, (A.31)
implies that all the functions ϕr(z,P) given by (A.21) are analytic and contractive
in C−.
Next, using Montel’s theorem and arguments from Step 1 in the proof of [5,
Theorem 3.8], one can easily show that there is an analytic and contractive in
C− matrix function ϕ∞(z) such that
ϕ∞ ∈
⋂
r≥1
Nr. (A.33)
(We note the functions
[
Im1
ϕ
]
in the proof of [5, Theorem 3.8] should be substi-
tuted by
[
ϕ
Im2
]
for our case of the Weyl functions in C−.) Taking into account
(A.21) and (A.33), we write the representations[
ϕ∞(z)
Im2
]
= Wr+1(z)P(z, r + 1) (r ≥ 0), (A.34)
where P(z, r + 1) are nonsingular with property-j. Using the summation formula
(2.24) and representation (A.34), we derive[
ϕ∗∞ Im2
] r∑
k=0
q(z)kWk(z)∗CkWk(z)
[
ϕ∞
Im2
]
≤
i
(
1 + |z|2
)
(z − z)
Im2 . (A.35)
Compare (A.35) with Definition 2.1 of the Weyl function in order to see that
ϕ∞ is a Weyl function of (1.1), (1.2). Moreover, this Weyl function is analytic
and contractive in C−. It remains to show that the Weyl function is unique.
First notice that (2.25) yields
q(z)Wk+1(z)
∗jWk+1(z) ≥Wk(z)∗jWk(z) (k ≥ 0). (A.36)
Thus, we have q(z)k+1Wk+1(z)
∗jWk+1(z) ≥ j, and so (2.4) implies that[
Im1 0
] r∑
k=0
q(z)kWk(z)∗CkWk(z)
[
Im1
0
]
≥ (r + 1)Im1 . (A.37)
Therefore, there is an m1-dimensional subspace of vectors g ∈ Cm such that
∞∑
k=0
g∗q(z)kWk(z)∗CkWk(z)g =∞. (A.38)
Further proof of the uniqueness of the values, which the Weyl function may take
at any fixed z ∈ C−, is easy and coincides with the arguments used in [5, Theorem
3.8].
The Discrete Self-Adjoint Dirac Systems of General Type. . . 547
Acknowledgment. The research of Alexander Sakhnovich was supported
by the Austrian Science Fund (FWF) under Grant No. P29177.
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548 Inna Roitberg and Alexander Sakhnovich
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Received February 8, 2018.
Inna Roitberg,
University of Leipzig, 10 Augustusplatz, Leipzig, 04109, Germany,
E-mail: innaroitberg@gmail.com
Alexander Sakhnovich,
Universität Wien, Fakultät für Mathematik, Oskar-Morgenstern-Platz 1, A-1090 Vienna,
Austria,
E-mail: oleksandr.sakhnovych@univie.ac.at
Дискретнi самоспряженi системи Дiрака
загального типу: явнi розв’язки прямої i оберненої
задач, асимптотики коефiцiєнтiв типу
Верблюнського та стiйкiсть розв’язання
оберненої задачi
Inna Roitberg and Alexander Sakhnovich
Розглянуто дискретнi самоспряженi системи Дiрака, визначенi по-
тенцiалами (послiдовностями) {Ck} так, що матрицi Ck є позитивно-
визначеними та j-унiтарними, де j — це дiагональна матриця розмiру
m ×m, що має на головнiй дiагоналi m1 та m2 елементiв, якi дорiвню-
ють вiдповiдно 1 та −1 (m1 + m2 = m). У роботi побудовано системи з
рацiональними функцiями Вейля та точно розв’язано обернену задачу
вiдновлення системи за стискальними рацiональними функцiями Вейля.
Крiм цього, у роботi дослiджується стiйкiсть цiєї процедури. Матрицi
Ck (з потенцiалiв) — це так званi розширення Халмоша коефiцiєнтiв ρk
типу Верблюнського. У роботi доведено, що у випадку стискальної ра-
цiональної функцiї Вейля коефiцiєнти ρk прямують до нуля, а матрицi
Ck прямують до одиничної матрицi Im.
Ключовi слова: дискретна самоспряжена система Дiрака, функцiя
Вейля, обернена задача, явний розв’язок, стiйкiсть розв’язання оберне-
ної задачi, асимптотики потенцiалу, коефiцiєнт типу Верблюнського.
mailto:innaroitberg@gmail.com
mailto:oleksandr.sakhnovych@univie.ac.at
Introduction
GBDT and direct and inverse problems
Verblunsky-type coefficients and asymptotics of the potentials
Stability of the procedure of solving the inverse problem
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