An operator approach to indefinite Stieltjes moment problem
In the present paper we solve the indefinite Stieltjes moment problem MPkκ(s) within the M.G. Krein theory of u-resolvent matrices applied to a Pontryagin space symmetric operator A[0,N] generated by J[0,N]. The u-resolvent matrices of the operator A[0,N] are calculated in terms of generalized Stiel...
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Інститут прикладної математики і механіки НАН України
2017
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| Цитувати: | An operator approach to indefinite Stieltjes moment problem / V.A. Derkach, I.M. Kovalyov // Український математичний вісник. — 2017. — Т. 14, № 1. — С. 42-85. — Бібліогр.: 50 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859742461775052800 |
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| author | Derkach, V.A. Kovalyov, I.M. |
| author_facet | Derkach, V.A. Kovalyov, I.M. |
| citation_txt | An operator approach to indefinite Stieltjes moment problem / V.A. Derkach, I.M. Kovalyov // Український математичний вісник. — 2017. — Т. 14, № 1. — С. 42-85. — Бібліогр.: 50 назв. — англ. |
| collection | DSpace DC |
| container_title | Український математичний вісник |
| description | In the present paper we solve the indefinite Stieltjes moment problem MPkκ(s) within the M.G. Krein theory of u-resolvent matrices applied to a Pontryagin space symmetric operator A[0,N] generated by J[0,N]. The u-resolvent matrices of the operator A[0,N] are calculated in terms of generalized Stieltjes polynomials using the boundary triple’s technique. Criterions for the problem MPkκ(s) to be solvable and indeterminate are found. Explicit formulae for Pade approximants for generalized Stieltjes fraction in terms of generalized Stieltjes polynomials are also presented.
|
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Український математичний вiсник
Том 14 (2017), № 1, 42 – 85
An operator approach to indefinite
Stieltjes moment problem
Vladimir Derkach, Ivan Kovalyov
(Presented by M. M. Malamud)
Abstract. A function f meromorphic on C\R is said to be in the
generalized Nevanlinna class Nκ (κ ∈ Z+), if f is symmetric with respect
to R and the kernel Nω(z) := f(z)−f(ω)
z−ω has κ negative squares on C+.
The generalized Stieltjes class Nk
κ (κ, k ∈ Z+) is defined as the set of
functions f ∈ Nκ, such that zf ∈ Nk. The full indefinite Stieltjes
moment problem MP kκ (s) consists in the following: Given κ, k ∈ Z+,
and a sequence s = {si}∞i=0 of real numbers, describe the set of functions
f ∈ Nk
κ, which satisfy the asymptotic expansion
f(z) = −s0
z
− · · · − s2n
z2n+1
+ o
(
1
z2n+1
)
(z = −y ∈ R−, y↑∞)
for all n big enough. We associate to this expansion a special continued
fraction, so-called generalized Stieltjes fraction, a three-term difference
equations, generalized Stieltjes polynomials and a generalized Jacobi ma-
trix J[0,N ].
In the present paper we solve the indefinite Stieltjes moment problem
MP kκ (s) within the M.G. Krein theory of u-resolvent matrices applied to
a Pontryagin space symmetric operator A[0,N ] generated by J[0,N ]. The
u-resolvent matrices of the operator A[0,N ] are calculated in terms of
generalized Stieltjes polynomials using the boundary triple’s technique.
Criterions for the problem MP kκ (s) to be solvable and indeterminate are
found. Explicit formulae for Pade approximants for generalized Stieltjes
fraction in terms of generalized Stieltjes polynomials are also presented.
2010 MSC. Primary 30E05; Secondary 15B57, 46C20, 47A57.
Key words and phrases. Indefinite Stieltjes moment problem, Gener-
alized Stieltjes function, Generalized Stieltjes fraction, Boundary triple,
Weyl function, Resolvent matrix.
Received 30.03.2017
This work was supported by the grant of Volkswagen Foundation and by Ministry of
Education and Science of Ukraine (project numbers 0115U000556, 0115U000136)
ISSN 1810 – 3200. c⃝ Iнститут математики НАН України
V. Derkach, I. Kovalyov 43
Dedicated to E.R. Tsekanovskii on the occasion of his 80th Birthday
A function f meromorphic on C\R is said to be in the generalized
Nevanlinna class Nκ (κ ∈ Z+), if f is symmetric with respect to R and the
kernel Nω(z) :=
f(z)−f(ω)
z−ω has κ negative squares on C+. The generalized
Stieltjes class Nk
κ (κ, k ∈ Z+) is defined as the set of functions f ∈ Nκ,
such that zf ∈ Nk. The full indefinite Stieltjes moment problem MP kκ (s)
consists in the following: Given κ, k ∈ Z+, and a sequence s = {si}∞i=0
of real numbers, describe the set of functions f ∈ Nk
κ, which satisfy the
asymptotic expansion
f(z) = −s0
z
− · · · − s2n
z2n+1
+ o
(
1
z2n+1
)
(z = −y ∈ R−, y↑∞)
for all n big enough. We associate to this expansion a special continued
fraction, so-called generalized Stieltjes fraction, a three-term difference
equations, generalized Stieltjes polynomials and a generalized Jacobi ma-
trix J[0,N ].
In the present paper we solve the indefinite Stieltjes moment problem
MP kκ (s) within the M.G. Krein theory of u-resolvent matrices applied to
a Pontryagin space symmetric operator A[0,N ] generated by J[0,N ]. The
u-resolvent matrices of the operator A[0,N ] are calculated in terms of
generalized Stieltjes polynomials using the boundary triple’s technique.
Criterions for the problem MP kκ (s) to be solvable and indeterminate are
found. Explicit formulae for Pade approximants for generalized Stieltjes
fraction in terms of generalized Stieltjes polynomials are also presented.
1. Introduction
The classical Stieltjes moment problem solved in [48] consists in the
following: given a sequence of real numbers si (i ∈ Z+ := N∪ {0}) find a
positive measure σ with a support on R+, such that∫
R+
tidσ(t) = si (i ∈ Z+). (1.1)
It follows easily from (1.1) that the inequalities
Sn := (si+j)
n−1
i,j=0 ≥ 0, S+
n := (si+j+1)
n−1
i,j=0 ≥ 0 (n ∈ Z+) (1.2)
are necessary for solvability of the moment problem (1.1). If the matrices
Sn are nondegenerate for all n ∈ Z+, then the inequalities (1.2) are also
44 An operator approach to indefinite...
sufficient for solvability of the moment problem (1.4), see [1, Appendix].
Let
Dn := det Sn, D+
n := det S+
n (n ∈ Z+).
In the pioneering paper [48] by T. Stieltjes a continued fraction
1
−zm1 +
1
l1 + . . .
1
−zmn +
1
ln + . . .
(1.3)
mn :=
D2
n
D+
nD
+
n−1
, ln :=
(D+
n−1)
2
DnD
+
n
(n ∈ Z+).
was associated with the sequence of moments {si}∞i=0. The moment prob-
lem (1.1) is called determinate (indeterminate), if it has a unique (in-
finitely many) solutions. As was shown in [48] the moment problem (1.1)
is indeterminate, if and only if
M :=
∞∑
i=1
mi <∞, and L :=
∞∑
i=1
li <∞.
Although in [48] no mechanical interpretation for the fraction (1.3) was
given, it was shown in [28] that solutions of the problem (1.1) can be
interpreted as spectral functions of the so-called “Stieltjes strings”, i.e.
massless threads with countable sets of point masses. The truncated
Stieltjes moment problem, i.e. the problem (1.1) with a finite set of data
{si}2ni=0 was studied in [35, 41]. Matrix version of the Stieltjes moment
problem was studied in [24].
For every measure dσ on R+ the associated function
f(z) =
∫
R+
dσ(t)
t− z
z ∈ C\R+
belongs to the class N of functions holomorphic on C\R with nonnegative
imaginary part in C+ := {z : Im z > 0} and such that f(z) = f(z)
for z ∈ C+. Moreover, f belongs to the Stieltjes class S of functions
f ∈ N, which admit holomorphic and nonnegative continuation to R−.
By M.G. Krein’s criterion [33]
f ∈ S ⇐⇒ f ∈ N and zf ∈ N.
Notice, that by the Hamburger–Nevanlinna theorem [1] a measure σ is
a solution of the problem (1.1) if and only if the associated function f
V. Derkach, I. Kovalyov 45
satisfies the condition
f(z) = −s0
z
− s1
z2
− · · · − s2n
z2n+1
+ o
(
1
z2n+1
)
as z→̂∞ (1.4)
for every n ∈ N. The notation z→̂∞ means that z → ∞ nontangentially,
that is inside the sector ε < arg z < π − ε for some ε > 0.
Indefinite version of the class N was introduced in [36].
Definition 1.1. [36]A function f meromorphic on C\R with the set
of holomorphy hf is said to be in the generalized Nevanlinna class Nκ
(κ ∈ N), if the kernel Nω(z) :=
f(z)−f(ω)
z−ω has κ negative squares on C+,
i.e. if for every set of zj ∈ C+ ∩ hf (zi ̸= zj, i, j = 1, . . . , n) the form
n∑
i,j=1
f(zi)− f(zj)
zi − zj
ξiξj , ξj ∈ C
has at most κ and for some choice of zj (j = 1, . . . , n) exactly κ negative
squares and
f(z) = f(z) for all z ∈ C+ ∩ hf . (1.5)
The generalized Stieltjes class N+
κ was defined in [37] as the class of
functions f ∈ Nκ, such that zf ∈ N. Similarly, in [11, 12] the class Nk
κ
(κ, k ∈ N) was introduced as the set of functions f ∈ Nκ, such that
zf ∈ Nk, see also [22], where the class Nk
0 was studied. Clearly, N0
0 = S
and N0
κ = N+
κ . The classes Sk := Nk
0 were introduced in [19,22].
In the present paper we consider the following problems.
Truncated indefinite moment problem MPκ(s, ℓ). Given ℓ, κ ∈
Z+, and a finite sequence s = {si}ℓi=0 of real numbers, describe the set
Mκ(s, ℓ) of functions f ∈ Nκ, which satisfy the asymptotic expansion
f(z) = −s0
z
− · · · − sℓ
zℓ+1
+ o
(
1
zℓ+1
)
as z→̂∞. (1.6)
Truncated indefinite moment problem MP kκ (s, ℓ). Given ℓ, κ, k ∈
Z+, and a sequence s = {si}ℓi=0 of real numbers, describe the set Mk
κ(s, ℓ)
of functions f ∈ Nk
κ, which satisfy (1.6). A truncated moment problem is
called even or odd regarding to the oddness of the number ℓ+ 1 of given
moments.
Full indefinite moment problem MPκ(s). Given κ ∈ Z+, and an
infinite sequence s = {si}∞i=0, describe the set Mκ(s) of functions f ∈ Nκ,
which satisfy (1.6) for all ℓ ∈ N.
Full indefinite moment problem MP kκ (s). Given κ, k ∈ Z+, and an
infinite sequence s = {si}∞i=0, describe the set Mk
κ(s) := Mκ(s) ∩Nk
κ.
46 An operator approach to indefinite...
Indefinite moment problems MPκ(s) and MP 0
κ (s) in the classes Nκ
and N+
κ := N0
κ, respectively, were studied in [39, 40] by the methods
of extension theory of Pontryagin space symmetric operators developed
in [37, 38]. In particular, it was shown in [39] that the moment problem
MPκ(s) is solvable if the number ν−(Sn) of negative eigenvalues of Sn
does not exceed κ and S+
n > 0 for all n ∈ N. Further applications
of the operator approach to the moment problem MP kκ (s) were given
in [13]. A reproducing kernel approach to the moment problems MPκ(s)
was presented in [23]. A step-by-step algorithm of solving the moment
problems MPκ(s) was elaborated in [8, 9] and [2]. Applications of the
Schur algorithm to degenerate moment problem in the class Nκ were
given in [16].
Denote by ν−(S) (ν+(S)) the number of negative (positive, resp.)
eigenvalues of the matrix S. Let H be the set of finite or infinite real
sequences s = {si}ℓi=0 and let Hκ,ℓ be the set of sequences s = {si}ℓi=0 ∈
H, such that
ν−(Sn) = κ (n = [ℓ/2] + 1). (1.7)
let Hk
κ,ℓ be the set of s = {si}ℓi=0 ∈ Hκ,ℓ, such that {si+1}ℓ−1
i=0 ∈ Hk,ℓ−1,
i.e.
ν−(S
+
n ) = k (n = [(ℓ+ 1)/2]). (1.8)
For an infinite sequence s = {si}∞i=0 one says s ∈ Hk
κ (or s ∈ Hk
κ) if (1.7)
(or (1.7) and (1.8)) is fulfilled for all n ∈ N.
A number nj ∈ N is called a normal index of the sequence s, if
detSnj ̸= 0. The ordered set of normal indices
n1 < n2 < · · · < nN
of s is denoted by N (s). A sequence s is called regular (see [17]), if
D+
nj = detS+
nj ̸= 0 for (1 ≤ j ≤ N).
As was shown in [9] there exists a sequence of monic polynomials
ai(z) = zℓi + a
(i)
ℓi−1z
ℓi−1 + . . .+ a
(i)
1 z + a
(i)
0
of degree ℓi = ni+1 − ni and real numbers bi ∈ R \ {0}, i ∈ N, such that
the convergents of the continued fraction
−b0
a0(z)−
b1
a1(z)− · · · − bn
an(z)− . . .
(1.9)
V. Derkach, I. Kovalyov 47
for sufficiently large n have the asymptotic expansion (2.1) for every ℓ ∈
N. This fact was known to L. Kronecker [42] and then it was reinvented
in [8]. The pairs (ai, bi) are called atoms, see [26] and the continued
fraction (1.9) is called the P–fraction, [45].
Consider the three-term recurrence relation (see [25])
biyi−1(z)− ai(z)yi(z) + yi+1(z) = 0, (1.10)
associated with the sequence of atoms (ai, bi), i ∈ N, and define polyno-
mials Pi(z) and Qi(z) of the first and second kind of the system (1.10)
subject to the initial conditions
P−1(z) ≡ 0, P0(z) ≡ 1, Q−1(z) ≡ −1, Q0(z) ≡ 0. (1.11)
Polynomials Pi(z) and Qi(z) are called Lanzcos polynomials of the first
and second kind. Moreover, the j-th convergent of the continued frac-
tion (1.9) takes the form (see [26, Section 8.3.7])
f [j](z) = −Qj(z)
Pj(z)
(1 ≤ j ≤ N).
As was shown in [9] the set Mκ(s, 2nN − 2) can be described in terms of
the Lanzcos polynomials of the first and second kind.
Odd indefinite Stieltjes moment problemsMP kκ (s, 2nN−2) for regular
sequences s were studied in [18]. For this problem one step of the Schur
algorithm is splited into two intermediate steps and this leads to the
expansion of f ∈ Mk
κ(s, 2nN − 2) into a generalized Stieltjes continued
fraction
f(z) =
1
−zm1(z) +
1
l1 + · · ·+ 1
−zmN (z) +
1
lN + τ(z)
, (1.12)
where mj are real polynomials, lj ∈ R\{0} and τ is a parameter function
from some generalized Stieltjes class Nk−kN
κ−κN , such that τ(z) = o(1) az
z→̂∞. Such continued fractions were studied in [17]. Associated to
the continued fraction (1.12) there is a system of difference equations
(see [50, Section 1]){
y2j − y2j−2 = ljy2j−1,
y2j+1 − y2j−1 = −zmj+1(z)y2j
(1.13)
48 An operator approach to indefinite...
Define the generalized Stieltjes polynomials P+
j (z) and Q+
j (z) of the first
and second kind as solutions of the system (1.13) subject to the initial
conditions
P+
−1(z) ≡ 0, P+
0 (z) ≡ 1, Q+
−1(z) ≡ 1, Q+
0 (z) ≡ 0.
The main result of [18] is the following
Theorem 1.2. Let s = {si}2nN−2
i=0 ∈ Hk,reg
κ,2nN−2, N (s) = {nj}Nj=1 be the
set of nomal indices, κ, k ∈ Z+, N ∈ N. Then:
(1) A nondegenerate odd moment problem MP kκ (s, 2nN−2) is solvable,
iff
κN := ν−(SnN ) ≤ κ and kN := ν−(S
+
nN−1) ≤ k.
(2) f ∈ Mk
κ(s, 2nN − 2) iff f admits the representation
f(z) =
Q+
2N−1(z)τ(z) +Q+
2N−2(z)
P+
2N−1(z)τ(z) + P+
2N−2(z)
, (1.14)
where τ(z) ∈ Nk−kN
κ−κN and 1
τ(z) = o(z) as z→̂∞.
In what follows for every 2 × 2 matrix W = (wij)
2
i,j=1 we associate
the linear-fractional transformation (LFT)
TW [τ ] :=
w11τ + w12
w21τ + w22
.
Denote by W+
[0,N−1](z) the coefficient matrix of the LFT (1.14)
W+
[0,N−1](z) =
(
Q+
2N−1(z) Q+
2N−2(z)
P+
2N−1(z) P+
2N−2(z)
)
.
Then the equality (1.14) can be rewritten in the form
f(z) = TW+
[0,N−1]
(z)[τ(z)]. (1.15)
The mvf W+
[0,N−1](z) admits the factorization
W+
[0,N−1](z) =M1(z)L1 . . . LN−1MN (z),
where the matrices Mj(z) and Lj are defined by
Mj(z) =
(
1 0
−zmj(z) 1
)
and Lj =
(
1 lj
0 1
)
j = 1, N. (1.16)
V. Derkach, I. Kovalyov 49
Similarly, the set of solutions of the moment problem MP kκ (s, 2nN − 1)
can be described via the LFT (1.15) with the the coefficient matrix
W++
[0,N−1](z) =
(
Q+
2N−1(z) Q+
2N (z)
P+
2N−1(z) P+
2N (z)
)
In the present paper we apply an operator approach to truncated and
full indefinite moment problems. For this purpose we associate with the
system of atoms {ai, bi}, i ∈ N, the so-called monic generalized Jacobi
matrix (GJM), see [9, 10]. This GJM J[0,N−1] generates a symmetric
operator A[0,N−1] with deficiency indices (1, 1) acting in an indefinite
inner product space H[0,N−1]. Then we invoke to the theory of boundary
triples developed in [20,30,32] and to the M.G. Krein theory of resolvent
matrices elaborated in [21, 34] and [15], see Sections 2.2 and 4.1 in the
present paper. We show that the matrices W+
[0,N−1](z) and W++
[0,N−1](z)
are u-resolvent matrices of the operator A[0,N−1] corresponding to some
boundary triples which are found explicitly in Section 4.2.
In the case of an infinite sequence s ∈ Hk,reg
κ , it is shown that the
coefficient matrix W+
[0,j](z) admits the factorization
W+
[0,j](z) = W++
[0,N−1](z)W
+
[N,j](z),
where W+
[N,j](z) is the resolvent matrix of some classical Stieltjes moment
problem. This fact allows to derive some facts for indefinite moment prob-
lem from the classical ones. In particular, it is shown that an indefinite
Stieltjes moment problem MP kκ (s) is solvable if and only if
ν−(Snj ) ≤ κ and ν−(S
+
nj ) ≤ k for all j ∈ N;
and MP kκ (s) is indeterminate if and only if
M :=
∞∑
j=1
mj(0) <∞ and L :=
∞∑
j=1
lj <∞. (1.17)
If (1.17) is in force, then the mvf’s W+
[0,j](z) are proved to converge to an
entire mvf W+
[0,∞](z) of order 1/2, which turns out to be an u-resolvent
matrix of minimal operator Amin generated by the GJM J[0,∞) in a Pon-
tryagin space H[0,∞). The LFT (1.15) generated by the mvf W+
[0,∞)(z)
provides a description of the set Mk
κ(s) in Theorem 5.2.
In Section 6 it is shown that the convergents of the continued frac-
tion (1.12) coincides with the diagonal and sub-diagonal Pade approxi-
mants of the corresponding formal power series.
50 An operator approach to indefinite...
2. Preliminaries
2.1. Generalized Nevanlinna functions
Recall that a function f meromorphic on C+ = {z : Im z > 0} is said
to be from the class Nκ (κ ∈ Z+), if the kernel Nω(z) has κ negative
squares on C+ ∩ hf and (1.5) holds. In particular, the class N := N0
consists of functions f holomorphic on C+, which map C+ into itself.
Notice that every real polynomial m(z) = mνz
ν + . . . + m1z + m0
of degree ν belongs to a class Nκ, where the index κ = κ−(m) can be
calculated by (see [37, Lemma 3.5])
κ−(m) =
{ [
ν+1
2
]
, if mν < 0 and ν is odd;[
ν
2
]
, otherwise.
Let s = {si}2ni=0 be a sequence of real numbers and let Sn := (si+j)
n−1
i,j=0
be a Hankel matrix of order n and denote Dn := det Sn (n ∈ Z+).
A function f ∈ Nκ is said to belong to the class Nκ,−ℓ if f admits
the following asymptotic expansion at ∞
f(z) ∼ −s0
z
− s1
z2
· · · − sℓ
zℓ+1
+ o
(
1
zℓ+1
)
as z→̂∞. (2.1)
The notation z→̂∞ means that z → ∞ nontangentially, that is inside
the sector ε < arg z < π − ε for some ε > 0. Let us also set
Nκ,−∞ :=
∩
ℓ≥0
Nκ,−ℓ.
Denote by ν−(S) (ν+(S)) the number of negative (positive, resp.)
eigenvalues of the matrix S. Let H be the set of finite real sequences
s = {si}ℓi=0 and let Hκ,ℓ be the set of sequences s = {si}ℓi=0 ∈ H, such
that
ν−(Sn) = κ (n = [ℓ/2] + 1).
The index ν−(Sn) for a Hankel matrix Sn can be calculated by the Frobe-
nius rule (see [27, Theorem X.24]). In particular, if all the determinants
Dn := detSn (n ∈ Z+) do not vanish, then ν−(Sn) coincides with the
number of sign alternations in the sequence
D0 := 1, D1, D2, . . . , Dn.
Proposition 2.1. [37] Let f ∈ Nκ, κ ∈ N. Then:
(1) f ∈ Nκ ⇐⇒ − 1
f ∈ Nκ.
V. Derkach, I. Kovalyov 51
(2) If f ∈ Nκ,−ℓ, then there exists κ′ ≤ κ, such that {si}ℓi=0 ∈ Hκ′,ℓ.
Denote by Hk
κ,ℓ the class of real sequences s = {si}ℓi=0 ∈ Hκ,ℓ, such
that {si+1}ℓ−1
i=0 ∈ Hk,ℓ−1, i.e. (1.8) holds. The following proposition is an
easy corollary of Proposition 2.1, see also [11].
Proposition 2.2. The following equivalences hold:
(1) f ∈ Nk
κ ⇐⇒ − 1
f ∈ N−k
κ .
(2) f ∈ Nk
κ ⇐⇒ zf(z) ∈ N−κ
k , in particular, f ∈ N+
κ ⇐⇒ zf(z) ∈
S−κ.
If, in addition, f ∈ Nk
κ has an asymptotic expansion (1.6) then {si}ℓi=0 ∈
Hk′
κ′,ℓ with κ
′ ≤ κ, k′ ≤ k.
2.2. Pontryagin spaces, symmetric operators, boundary
triples
Let H be a Hilbert space and let G be a selfadjoint operator in H such
that 0 ∈ ρ(G) and the total multiplicity of negative eigenvalues of G is
equal κ. The space H with the indefinite inner product
[f, g] := (Gf, g) f, g ∈ H
is called the Pontryagin space with negative index κ and is denoted by
(H, [·, ·]). A closed linear operator A in (H, [·, ·]) is called symmetric in
(H, [·, ·]), if
[Af, g] = [f,Ag] for all f, g ∈ dom(A).
A linear subspace T ⊂ H2 is called a linear relation T in H, see [4]. In
particular, the graph of the operator A in (H, [·, ·]) is a linear relation in H.
Identifying the operator A with its graph we will consider the set of linear
operators as a subset of the set of linear relations in H. If the operator
A is non-densely defined in H, then its adjoint A[∗] can be defined as a
linear relation in H by the equality
A[∗] = {{g, g′} ∈ H2 : [Af, g] = [f, g′] for all f ∈ domA}.
An approach to extension theory of symmetric operators based on the
notion of “abstract boundary conditions”, was proposed by Calkin [7],
and later on it was developed independently in [30,32]. Recall the defini-
tion of the boundary triple from [32] (see also [20, 21, 46] for the present
notations).
52 An operator approach to indefinite...
Definition 2.3. A collection Π = {H,Γ0,Γ1} consisting of a Hilbert
space H and two linear mappings Γ0 and Γ1 from A[∗] to H, is said to be
a boundary triple for A[∗] if:
(i) the abstract Green’s identity
[f ′, g]− [f, g′] = Γ1f̂Γ0ĝ − Γ0f̂Γ1ĝ
holds for all f̂ =
(
f
f ′
)
, ĝ =
(
g
g′
)
∈ A[∗];
(ii) the mapping Γ :=
(
Γ0
Γ1
)
: A[∗] → C2 is surjective.
Associated with a boundary triple Π there are two self-adjoint exten-
sions of the operator A given by
A0 = ker Γ0 and A1 = ker Γ1.
Let Nz := ker(A[∗] − zI) and let us set
N̂z :=
{(
fz
zfz
)
, fz ∈ Nz
}
⊂ A[∗]. (2.2)
A symmetric operator A in (H, [·, ·]) is called simple, if
span {Nz : z ̸= z̄} = H. (2.3)
Definition 2.4. The abstract Weyl function of A, corresponding to the
boundary triple Π = {H,Γ0,Γ1} is defined by
M(z)Γ0f̂z = Γ1f̂z, f̂z ∈ N̂z, z ∈ ρ(A0),
where N̂z is defined by (2.2).
The notion of the Weyl function for a Hilbert space symmetric opera-
tor was introduced in [19–21,46] both for densely and nondensely defined
operators. The definition of the Weyl function for a nondensely defined
Pontryagin space symmetric operator was given in [15]. As was shown
in [15] the Weyl function M(z) of a symmetric operator A acting in a
Pontryagin space H with negative index κ, is well defined and belongs to
the class Nκ′ with κ′ ≤ κ.
A boundary triple Π = {H,Γ0,Γ1} allows to give a description of all
self-adjoint extensions of A, which are disjoint with A0 in the form
Ab = ker(Γ1 − bΓ0), b ∈ R.
V. Derkach, I. Kovalyov 53
The resolvent set ρ(Ab) of the linear relation Ab is defined as the set of
points z ∈ C, such that
ran(Ab − zI) = H and ker(Ab − zI) = {0}.
For a simple symmetric operator A the resolvent set of its extension Ab
is characterized by the following statement
Proposition 2.5. [15]Let A be a simple symmetric operator in (H, [·, ·]),
let Π = {H,Γ0,Γ1} be a boundary triple for A[∗] and z ∈ ρ(A0), b ∈ R.
Then
z ∈ ρ(Ab) ⇐⇒M(z)− b ̸= 0.
2.3. Monic generalized Jacobi matrix
Let a(z) = zℓ + al−1z
ℓ−1 + · · · + a0 be a monic real polynomial of
degree ℓ, and let Ea and Ca be ℓ× ℓ matrices
Ea =
a1 · · · aℓ−1 1
... ··· ···
aℓ−1 ···
1 0
, Ca =
0 1 0
...
. . . . . .
0 · · · 0 1
−a0 −a1 · · · −aℓ−1
.
(2.4)
It follows from the equalities
EaCa = C∗
aEa (2.5)
(see [29, Chapter 12]) that the matrix EaCa is symmetric in the standard
scalar product in Cℓ.
Let us associate with the system of atoms {ai, bi}, i ∈ N, the so-called
monic generalized Jacobi matrix (GJM) (see [9, 10])
J =
Ca0 D0
B1 Ca1 D1
B2 Ca2
. . .
. . . . . .
, (2.6)
where the diagonal entries are companion matrices associated with the
polynomials ai(z) (see [43]) and Di and Bi+1 are ℓi × ℓi+1 and ℓi+1 × ℓi
matrices, respectively, determined by
Di =
0 0 · · · 0
...
... · · ·
...
0 0 · · · 0
1 0 · · · 0
, Bi+1 =
0 0 · · · 0
...
... · · ·
...
0 0 · · · 0
bi+1 0 · · · 0
, i ∈ Z+. (2.7)
54 An operator approach to indefinite...
The matrix J defined by (2.6)– (2.7) is called a GJM associated with
the system of atoms {ai, bi}, i ∈ N.The shortened GJM J[i,j] is defined
by
J[i,j] =
Cai Di
Bi+1 Cai+1
. . .
. . . . . . Dj−1
Bj Caj
, i ≤ j and i, j ∈ Z+. (2.8)
Let Pi(z) andQi(z) be the Lanzcos polynomials of the first and second
kind determined by (1.10) and (1.11). The Lanzcos polynomials satisfy
the following generalized Liouville–Ostrogradskii formula
Qi+1(z)Pi(z)−Qi(z)Pi+1(z) = b̃i. (2.9)
Let H[0,N ] be the indefinite inner product space of sequences from
CnN+1 endowed with the indefinite inner product
[x, y][0,N ] = (G[0,N ]x, y), G[0,N ] = diag(̃b0E−1
0 , b̃1E
−1
1 , . . . , b̃NE
−1
N ).
where
b̃0 = b0 and b̃i = b0b1 . . . bi, i ∈ N.
It follows from the equalities (2.5) that the matrix G[0,N ]J
T
[0,N ] is self-
adjoint in the standard scalar product in CnN+1 ,
G[0,N ]J
T
[0,N ] = J[0,N ]G[0,N ] (2.10)
and hence the matrix JT[0,N ] generates a self-adjoint operator in H[0,N ].
Let us set
π[0,N ](z) = G−1
[0,N ]
π0(z)...
πN (z)
and πi(z) =
Pi(z)
zPi(z)
...
zli−1Pi(z)
, (i = 0, N).
(2.11)
Alongside with π[0,N ](z) let us define the vector-function
ξ[0,N ](z) = G−1
[0,N ]
ξ0(z)...
ξN (z)
and ξi(z) =
Qi(z)
zQi(z)
...
zli−1Qi(z)
, (i = 0, N).
Lemma 2.6. For every N ∈ N the following equalities hold
(J⊤[0,N ] − zInN+1)π[0,N ](z) = −(̃bN )
−1PN+1(z)enN , (2.12)
(J⊤[0,N ] − zInN+1)ξ[0,N ](z) = e0 − (̃bN )
−1QN+1(z)enN . (2.13)
V. Derkach, I. Kovalyov 55
2.4. A system of difference equations and generalized Stieltjes
polynomials
In the present paper we will consider so-called regular sequences s
from Hk
κ introduced in [17].
Definition 2.7. The sequence s ∈ Hk
κ is said to belong to the class Hk,reg
κ
if one of the following equivalent conditions holds
(1) Pi(0) ̸= 0 for every i ≤ N ;
(2) D+
ni−1 ̸= 0 for every i ≤ N ;
(3) D+
ni ̸= 0 for every i ≤ N .
For a sequence s from Hk,reg
κ the following theorem holds, see [17].
Theorem 2.8. [17] Let s ∈ Hk,reg
κ,ℓ . Then there exists a sequence of
polynomials mj(z) and numbers lj such that the 2j−th convergent
u2j
v2j
of
the continued fraction
1
−zm1(z) +
1
l1 + . . .
1
−zmj(z) +
1
lj + . . .
. (2.14)
coincides with the j−th convergent of the P−fraction (1.9) correspond-
ing to the sequence s. The parameters lj and mj(z) of the generalized
S−fraction (2.14) are connected with the parameters bj and aj(z) of the
P−fraction (1.9) by the equalities
b0 =
1
d1
, a0(z) =
1
d1
(
zm1(z)−
1
l1
)
, (2.15)
bj =
1
l2jdjdj+1
, aj(z) =
1
dj+1
(
zmj+1(z)−
(
1
lj
+
1
lj+1
))
, (2.16)
where dj is the leading coefficient of mj(z) (j = 1, . . . , N − 1).
The continued fraction (2.14) will be called a generalized S−fraction.
In the case when mj(z) ≡ mj are constant numbers it reduces to the
classical S-fraction (1.3) and the formulas (2.15), (2.16) are well known,
[48]. The parameters lj and mj(z) in (2.14) can be calculated recursively
56 An operator approach to indefinite...
by (2.15) and (2.16). Alternatively, mj and lj can be represented in terms
of the sequence s (see [17]):
mj(z) =
(−1)ν+1
D
(j−1)
ν
∣∣∣∣∣∣∣∣∣∣
0 . . . 0 s
(j−1)
ν−1 s
(j−1)
ν
... . . . . . .
...
s
(j−1)
ν−1 . . . . . . . . . s
(j−1)
2ν−2
1 z . . . zν−2 zν−1
∣∣∣∣∣∣∣∣∣∣
, (2.17)
where D(j)
ν := detS
(j)
ν , ν = nj − nj−1 and
lj = (−1)ν+1 D
(j−1)
ν(
D
(j−1)
ν
)+ (j = 1, . . . , N − 1).
Let us consider a system of difference equations associated with the
continued fraction (2.14){
y2j − y2j−2 = ljy2j−1,
y2j+1 − y2j−1 = −zmj+1(z)y2j
. (2.18)
If the j–th convergent of this continued fraction is denoted by uj
vj
, then
uj , vj can be found as solutions of the system (2.18) (see [50, Section 1])
subject to the following initial conditions
u−1 ≡ 1, u0 ≡ 0; v−1 ≡ 0, v0 ≡ 1. (2.19)
The first two convergents of the continued fraction (2.14) take the form
u1
v1
=
1
−zm1(z)
= TM1 [∞],
u2
v2
=
l1
−zl1m1(z) + 1
= TM1L1 [0].
Similarly, the (2j − 1)-th and (2j)-th convergents are given by
u2j−1
v2j−1
= TW2j−1 [∞],
u2j
v2j
= TW2j [0].
Definition 2.9. [18] Let s ∈ Hk,reg
κ,ℓ . Define polynomials P+
i (z), Q+
i (z)
by
P+
−1(z) ≡ 0, P+
0 (z) ≡ 1, Q+
−1(z) ≡ 1, Q+
0 (z) ≡ 0,
P+
2i−1(z) = − 1
b̃i−1
∣∣∣∣Pi(z) Pi−1(z)
Pi(0) Pi−1(0)
∣∣∣∣ and P+
2i (z) =
Pi(z)
Pi(0)
,
Q+
2i−1(z) =
1
b̃i−1
∣∣∣∣Qi(z) Qi−1(z)
Pi(0) Pi−1(0)
∣∣∣∣ and Q+
2i(z) = −Qi(z)
Pi(0)
.
(2.20)
The polynomials P+
i (z), Q+
i (z) are called Stieltjes polynomials.
V. Derkach, I. Kovalyov 57
As was noticed in [18] Stieltjes polynomials are solutions of the sys-
tem (2.18).
Proposition 2.10. Let s ∈ Hk,reg
κ,ℓ and let P+
i (z) and Q+
i (z) be the gen-
eralized Stieltjes polynomials defined by (2.20). Then solutions {ui}Ni=0
and {vi}Ni=0 of the system (2.18), (2.19) take the form
ui = Q+
i (z), vi = P+
i (z) (i = −1, 0, . . . , N).
Remark 2.11. The Stieltjes polynomials satisfy the following properties
P+
2i−1(0) = 0, P+
2i−2(0) = 1 and Q+
2i−1(0) = 1.
By Definition 2.9 and (2.9)
P+
2i−1(0) = − 1
b̃i−1
∣∣∣∣Pi(0) Pi−1(0)
Pi(0) Pi−1(0)
∣∣∣∣ = 0 and P+
2i−2(0) =
Pi(0)
Pi(0)
= 1.
Q+
2i−1(0)=
1
b̃i−1
∣∣∣∣Qi(0) Qi−1(0)
Pi(0) Pi−1(0)
∣∣∣∣=Qi(0)Pi−1(0)−Qi−1(0)Pi(0)
b̃i−1
=1.
Lemma 2.12. Let s ∈ Hk,reg
κ,ℓ and let Pi(z) and Qi(z) be polynomials of
the first and second kind associated with the monic GJM J. Then:
(1) The distance li can be calculated by
li = −Qi(0)
Pi(0)
+
Qi−1(0)
Pi−1(0)
. (2.21)
(2) For every N ∈ N the following formula holds
−QN (0)
PN (0)
=
N∑
i=1
li. (2.22)
Proof. (1) Considering (2.18) at z = 0, we obtain Q+
2i(0) = liQ
+
2i−1(0) +
Q+
2i−2(0) and hence
liQ
+
2i−1(0) = Q+
2i(0)−Q+
2i−2(0).
By Definition 2.9 and by the generalized Liouville–Ostrogradskii formula
(2.9)
Q+
2i−1(0) =
1
b̃i−1
(Qi(0)Pi−1(0)−Qi−1(0)Pi(0) = 1.
This implies (2.21).
(2) Summing the equalities (2.21) for i = 1, . . . , N one obtains (2.22).
58 An operator approach to indefinite...
Lemma 2.13. Let s ∈ Hk,reg
κ,ℓ , let Pi(z) be the polynomials of the first
kind associated with the monic GJM, let P+
i (z) and Q+
i (z) be Stieltjes
polynomials of the first and second kind defined by (2.20), xi = l1+· · ·+li,
i ∈ N. Then
∞∑
i=0
di+1 =
∞∑
i=0
|P 2
i (0)|̃b−1
i and
∞∑
i=2
di+1x
2
i =
∞∑
i=2
|Q2
i (0)|̃b−1
i .
2.5. The class Uκ(J) and linear fractional transformations
Let κ ∈ N and let J be the 2× 2 signature matrix J =
(
0 −i
i 0
)
.
Definition 2.14. A 2×2 mvf W(z) = (wi,j(z))
2
i,j=1 that is meromorphic
in C+ is said to be in the class Uκ(J) of generalized J-inner mvf’s if:
(i) the kernel
KW
ω (z) =
J −W(z)JW(ω)∗
−i(z − ω̄)
has κ negative squares in H+
W × H+
W and
(ii) J −W(µ)JW(µ)∗ = 0 for a.e. µ ∈ R,
where H+
W denotes the domain of holomorphy of W in C+.
The set of meromorphic mvf’s which satisfy only the first assumption
(i) is denoted by Pκ(J). The class P(J) := P0(J) was introduced and
studied by M.S. Livsič [44] in connection with the theory of characteristic
functions of quasi-hermitian operators, see also [49], in the case of un-
bounded operators. A complete factorization theory for mvf’s from the
class P(J) was developed by V.P. Potapov [47]. The subclass of J-inner
mvf’s U(J) plays an improtant role in this theory. Notice that mon-
odromy matrices of canonical systems and resolvent matrices of many
interpolation problems belong to the class U(J), [5]. The definition and
some properties of the class Uκ(J) are contained in [3].
Consider the linear fractional transformation (LFT)
TW [τ ] = (w11τ(z) + w12)(w21τ(z) + w22)
−1
associated with the mvf W(z). The LFT associated with the product
W1W2 of two mvf’s W1(z) and W2(z) coincides with the composition
TW1 ◦ TW2 .
As is known, if W1 ∈ Uκ1(J) and W2 ∈ Uκ2(J) then W1W2 ∈ Uκ′(J),
where κ′ ≤ κ1 + κ2. In the following statement a partial case, when the
preceding inequality becomes equality, is considered, see [18].
V. Derkach, I. Kovalyov 59
Lemma 2.15. Let mi(z) be real polynomials and li ∈ R \ {0} (i =
1, . . . , j), let the mvf’s Mi(z), Li, W2j−1(z) and W2j(z) be defined by
(1.16) and
W2j−1(z) =M1(z)L1 . . . Lj−1Mj(z), W2j(z) =M1(z)L1 . . .Mj(z)Lj .
Then:
(1) Mi ∈ Uκ−(mi)(J) and for every meromorphic function τ , such that
τ(z)−1 = o(z) as z→̂∞ the following equivalence holds
τ ∈ Nk′
κ′ ⇐⇒ TMi [τ ] ∈ N
κ−(mi)+k
′
κ−(zmi)+κ′
, κ′, k′ ∈ N.
(2) Li ∈ U(J) and for every meromorphic function τ , such that τ(z) =
o(1) as z→̂∞ the following equivalence holds
τ ∈ Nk′
κ′ ⇐⇒ TLi [τ ] ∈ N
κ−(zli)+k
′
κ′ .
(3) W2j−1 ∈ Uκj (J), where
κj =
j∑
i=1
κ−(zmi(z)), kj =
j∑
i=1
κ−(zmi) +
j−1∑
i=1
κ−(zli) (2.23)
and for every meromorphic function τ , such that τ(z)−1 = o(z) as
z→̂∞ the following equivalence holds
τ ∈ Nk′
κ′ ⇐⇒ TW2j−1 [τ ] ∈ N
kj+k
′
κj+κ′
.
(4) W2j ∈ Uκj (J), where
κj =
j∑
i=1
κ−(mi), k+j =
j∑
i=1
κ−(zmi) +
j∑
i=1
κ−(zli). (2.24)
and for every meromorphic function τ , such that τ(z) = o(1) as
z→̂∞ the following equivalence holds
τ ∈ Nk′
κ′ ⇐⇒ TW2j [τ ] ∈ N
k+j +k′
κj+κ′
.
60 An operator approach to indefinite...
3. Boundary triples for the operator A[0,j]
3.1. General case
Let us fix j ∈ N and define the operator A[0,j] in the Pontryagin space
H[0,j] as the restriction of the operator JT[0,j] to the domain
domA[0,j] =
{
f ∈ H[0,j] : [f, enj ] = 0
}
(3.1)
As was shown in [10] the adjoint linear relation A[∗]
[0,j] of A[0,j] has the
following representation
A
[∗]
[0,j] =
{
f̂ =
[
f
JT[0,j]f + cfenj
]
:
f ∈ H[0,j]
cf ∈ C
}
. (3.2)
Mention some properties of the operator A[0,j].
Proposition 3.1. Let the operator A[0,j] be defined by (3.1). Then:
1. e0 is a generating vector for the operator A[0,j];
2. σp(A[0,j]) = ∅;
3. H = ran(A[0,j] − z)u span{e0} for all z ∈ C;
4. the operator A[0,j] is simple, see (2.3).
Proof. (1) It follows from (2.8), (2.4) and (3.1) that e0 ∈ dom(Ai[0,j]) for
all i = 0, nj+1 − 1 and
H[0,j] = span {ẽi : 0 ≤ i ≤ nj+1 − 1}, ẽi = Ai[0,j]e0. (3.3)
(2) To prove the second statement let us assume that z ∈ σp(A[0,j])
and A[0,j]f = zf . Decomposing the vector f by vectors enk from (3.3)
one obtains
f =
nj+1−1∑
i=0
ξkẽi.
If m is the largest k for which ξk ̸= 0 then the equality A[0,j]f = zf
implies
m∑
i=0
ξiẽi+1 =
m∑
i=0
zξiẽi
and hence one obtains ξm = 0. Therefore, ξk = 0 for all i ≤ nj+1 − 1.
(3) is implied by (3.3) and (2.6), since
(A[0,j] − z)ẽi = ẽi+1 − zẽi, i = 0, . . . , nj+1 − 2.
V. Derkach, I. Kovalyov 61
For vectors
f̂ =
[
f
JT[0,j]f + cfenj
]
and ĝ =
[
g
JT[0,j]g + cgenj
]
∈ A
[∗]
[0,j].
define Wronskian Wj [f̂ , ĝ] by
Wj [f̂ , ĝ] :=
∣∣∣∣cf cg
fj gj
∣∣∣∣ , fj := [f, enj ], gj := [g, enj ]. (3.4)
Proposition 3.2. Vectors
π̂[0,j](z) :=
[
π[0,j](z)
zπ[0,j](z)
]
, ξ̂[0,j](z) :=
[
ξ[0,j](z)
zξ[0,j](z) + e0
]
(z ∈ C)
belong to A
[∗]
[0,j] and admit the representations
π̂[0,j](z) =
[
π[0,j](z)
JT[0,j]π[0,j](z) + b̃−1
j Pj+1(z)enj
]
(3.5)
ξ̂[0,j](z) =
[
ξ[0,j](z)
JT[0,j]ξ[0,j](z) + b̃−1
j Qj+1(z)enj
]
. (3.6)
Wronskians Wj [f̂ , π̂[0,j](z)] and Wj [f̂ , ξ̂[0,j](z)] can be found by
Wj [f̂ , π̂[0,j](z)] :=
∣∣∣∣∣cf b̃−1
j Pj+1(z)
fj Pj(z)
∣∣∣∣∣ , Wj [f̂ , ξ̂[0,j](z)] :=
∣∣∣∣∣cf b̃−1
j Qj+1(z)
fj Qj(z)
∣∣∣∣∣ ,
(3.7)
and the generalized Liouville–Ostrogradskii formula (2.9) takes the form
Wj [ξ̂[0,j](z), π̂[0,j](z)] = 1, z ∈ C. (3.8)
Proof. The formula (3.5) is implied by (2.8), (2.10) and the equalities
JT[0,j]π[0,j](z) + b̃−1
j Pj+1(z)enj = G−1
[0,j]
J[0,j]
π0(z)...
πj(z)
+
0
...
Pj+1(z)
= zπ[0,j](z).
It follows from (3.5) that cg = b̃−1
j Pj+1(z) for ĝ = π̂[0,j](z). Inserting this
into (3.4) yields the first formula in (3.7). The proof of (3.6) and the
seecond formula in (3.7) is similar.
The formula (3.8) is implied by (2.9), (3.7), (3.5) and (3.6).
62 An operator approach to indefinite...
Proposition 3.3. Let f̂ =
[
f
f ′
]
, ĝ =
[
g
g′
]
∈ A
[∗]
[0,j]. Then
[f ′, g]− [f, g′] =Wj [f̂ , ĝ]. (3.9)
Proof. Let vectors f̂ , ĝ ∈ A
[∗]
[0,j] be of the form
f̂ =
[
f
f ′
]
=
[
f
JT[0,j]f + cfenj
]
and ĝ =
[
g
g′
]
=
[
g
JT[0,j]g + cgenj
]
.
Since the matrix JT[0,j] generates a self-adjoint operator in H[0,j], see (2.10),
one gets from (3.4)
[f ′, g]− [f.g′] =
[
JT[0,j]f+ cfenj , g
]
−
[
f, JT[0,j]g + cgenj
]
= cfgj − fjcg =Wj [f̂ , ĝ]. (3.10)
This completes the proof.
The following Christoffel–Darboux formulas are implied by (3.9).
Corollary 3.4. For all j ∈ Z+, z ∈ C and z0 ∈ R the following formulas
hold
Wj [ξ̂[0,j](z), ξ̂[0,j](z0)] = (z − z0)[ξ[0,j](z), ξ[0,j](z0)]H[0,j]
, (3.11)
Wj [ξ̂[0,j](z), π̂[0,j](z0)] = 1 + (z − z0)[ξ[0,j](z), π[0,j](z0)]H[0,j]
,
Wj [π̂[0,j](z), ξ̂[0,j](z0)] = −1 + (z − z0)[π[0,j](z), ξ[0,j](z0)]H[0,j]
,
Wj [π̂[0,j](z), π̂[0,j](z0)] = (z − z0)[π[0,j](z), π[0,j](z0)]H[0,j]
. (3.12)
Theorem 3.5. Let Pi(z) be the polynomials of the first kind associated
with the monic generalized Jacobi matrix J. Then:
(1) the boundary triple {C,Γ0,Γ1} of the linear relation A
[∗]
[0,j] can be
found by
Γ0f̂ = fj =
[
f, enj
]
and Γ1f̂ = cf ; (3.13)
(2) the defect subspace of the operator A[0,j] is given by
Nz(A[0,j]) = span π[0,j](z),
where π[0,N ](z) is given by (2.11);
V. Derkach, I. Kovalyov 63
(3) the Weyl function and the γ-field of the operator A[0,j] correspond-
ing to the boundary triple {C,Γ0,Γ1} are given by
M(z) =
Pj+1(z)
b̃jPj(z)
, γ(z) =
π[0,j](z)
Pj(z)
. (3.14)
Proof. (1) The Green’s formula for the triple {C,Γ0,Γ1} is derived from
the equality (3.10):
[f ′, g]− [f, g′] = cf
[
g, enj
]
− cg[f, enj ] = Γ1f̂Γ0ĝ − Γ0f̂Γ1ĝ. (3.15)
(2) It follows from (3.2) and (2.11) that π̂[0,j](z) ∈ A
[∗]
[0,j] and hence
the inclusion π[0,j](z) ∈ Nz(A[0,j]) holds.
(3) Applying Γ0 and Γ1 to the defect vector f̂z := π̂[0,j](z), one obtains
Γ0f̂z = [π[0,j](z), enj ] = Pj(z) and Γ1f̂z = b̃−1
j Pj+1(z). (3.16)
This proves the formulas (3.14).
Remark 3.6. A similar construction of the boundary triple for “symmet-
ric” GJM’s was presented in [10]. Relations between the corresponding
objects for monic and “symmetric” GJM’s will be given in Section 5.2.
Theorem 3.7. Let the operator S[0,j] in the space H[0,j] be defined as the
restriction of the operator JT[0,j] to the domain
domS[0,j] =
{
f ∈ H[0,j] : [f, e0] = 0
}
. (3.17)
Then:
(1) the adjoint linear relation S
[∗]
[0,j] of S[0,j] has the following represen-
tation
S
[∗]
[0,j] =
{
f̂ =
[
f
JT[0,j]f + dfe0
]
:
f ∈ H[0,j],
df ∈ C
}
; (3.18)
(2) the boundary triple {C,Γ0,Γ1} for the linear relation S
[∗]
[0,j] can be
found by
Γ1f̂ = [f, e0] and Γ0f̂ = −df ;
(3) the defect subspace Nλ(S[0,j]) of the operator S[0,j] is spanned by
(JT[0,j] − z)−1e0 =
(
ξ[0,j] −
Qj+1(z)
Pj+1(z)
π[0,j]
)
;
64 An operator approach to indefinite...
(4) the Weyl function m[0,j](z) of the operator S[0,j] corresponding to
the boundary triple {C,Γ0,Γ1} is given by
m[0,j](z) = [(JT[0,j] − z)−1e0, e0] = −Qj+1(z)
Pj+1(z)
. (3.19)
Moreover
m[0,j](z) = −s0
z
− · · · −
s2nj+1−2
z2nj+1−1 + o
(
1
z2nj+1−1
)
as z→̂∞,
si =
[(
JT[0,j]
)i
e0, e0
]
.
(3.20)
Proof. (1) Assume that the linear operator S[0,j] in the space H[0,j] is
defined as the restriction of the operator JT[0,j] to the domain (3.17).
Consequently, the adjoint linear relation S
[∗]
[0,j] is given by (3.18).
(2) Suppose, f̂ = {f, JT[0,j]f + dfe0} and ĝ = {f, JT[0,j]g + dge0}, then
[f ′, g]− [f, g′] = [JT[0,j]f + dfe0, g]− [f, JT[0,j]g + dge0] =
= df [g, e0]− dg[f, e0] = Γ1f̂Γ0ĝ − Γ0f̂Γ1ĝ.
(3) Let us set
fz := ξ[0,j](z)−
Qj+1(z)
Pj+1(z)
π[0,j](z).
Then it follows from (2.12) and (2.13) that (JT[0,j] − z)fz = e0. Hence
fz = (JT[0,j] − z)−1e0 (3.21)
and
f̂z =
[
fz
zfz
]
=
[
fz
JT[0,j]fz − e0
]
. (3.22)
Therefore, fz ∈ Nz(S[0,j]).
(4) It follows from (3.13) and (3.22)
Γ0f̂z = 1 and Γ1f̂z = −Qj+1(z)
Pj+1(z)
. (3.23)
This proves the second formula in (3.19). The first formula is implied by
(3.21) and (3.23). Due to (3.19), we obtain
m[0,j](z) = − [e0, e0]
z
− · · · −
[(
JT[0,j]
)2nj+1−2
e0, e0
]
z2nj+1−1 + o
(
1
z2nj+1−1
)
as z→̂∞. Denote si =
[(
JT[0,j]
)i
e0, e0
]
, we get (3.20).
V. Derkach, I. Kovalyov 65
3.2. The case s ∈ Hk,reg
κ,ℓ
Definition 3.8. A symmetric operator A in a Pontryagin space (H, [·, ·])
is said to have k negative squares, if for every choice of fj ∈ domA the
form
n∑
i,j=0
[Afi, fj ]Hξiξ̄j
has at most k, and for some choice of fj ∈ domA exactly k negative
squares.
Let a symmetric operator A in a Pontryagin space (H, [·, ·]) have k
negative squares. Recall [14], that a boundary triple Π = {C,Γ+
0 ,Γ
+
1 } for
the linear relation A[∗] is said to be basic, if the Weyl function M(z) of
A corresponding to the boundary triple Π satisfies the conditions
lim
iy→0
M(iy) = ∞, lim
iy→∞
M+(iy) = 0. (3.24)
Proposition 3.9. Let s ∈ Hk,reg
κ,ℓ and let N (s) = {ni}N+1
i=1 . Then:
1. the operator A[0,N ] has k negative squares;
2. a boundary triple {C,Γ+
0 ,Γ
+
1 } for the linear relation A
[∗]
[0,N ] can be
chosen as follows
Γ+
1 f̂ =
1
PN (0)
[f, enN ] , Γ+
0 f̂ = −PN (0)cf + b̃−1
N PN+1(0)[f, enN ],
(3.25)
where cf is defined by the decomposition (3.2);
3. the corresponding Weyl function and the γ-field are given by
M+
[0,N ](z) =
P+
2N (z)
P+
2N+1(z)
, γ+(z) =
π[0,N ](z)
P+
2N+1(z)
; (3.26)
4. the boundary triple {C,Γ+
0 ,Γ
+
1 } for the linear relation A
[∗]
[0,N ] is ba-
sic.
Proof. (1) Arbitrary vector f ∈ dom(A[0,N ]) can be expanded by vectors
ei (0 ≤ i ≤ nN+1 − 2) from (3.3) as follows
f =
nN+1−2∑
i=0
ξiei, ξi ∈ C.
66 An operator approach to indefinite...
By (3.20)
[A[0,N ]ei, ej ]H[0,k]
= si+j+1, i, j = 0, . . . , nN+1 − 2. (3.27)
Since s ∈ Hk,reg
κ,ℓ one obtains from (3.27)
nN+1−2∑
i,j=0
[A[0,N ]ei, ej ]H[0,N ]
ξiξ̄j =
nN+1−2∑
i,j=0
si+j+1ξiξ̄j
and hence the operator A[0,N ] has k negative squares.
(2) Obviously the operators Γ+
0 , Γ
+
1 are connected with Γ0, Γ1 by
Γ+
1 f̂ =
1
β
Γ0 and Γ+
0 f̂ = β (αΓ0 − Γ1) ,
where β = PN (0) and α =
b̃−1
N PN+1(0)
PN (0) . Then, by [21, Proposition 1.7]{
C,Γ+
0 ,Γ
+
1
}
is a boundary triple of the linear relation A
[∗]
[0,N ]. Hence, the
corresponding Weyl function can be calculated by
M+(z) = 1
β2(α−M(z))
= 1
P 2
N (0)
(
PN+1(0)
b̃NPN (0)
− PN+1(z)
b̃NPN (z)
)
= −
PN (z)
PN (0)
b̃−1
N (PN+1(z)PN (0)− PN (z)PN+1(0))
=
P+
2N (z)
P+
2N+1(z)
.
(3) It follows from (3.16) that for f̂z = π̂[0,N ](z)
Γ+
0 f̂z =
1
b̃N
(PN+1(0)PN (z)− PN (0)PN+1(z)) = P+
2N+1(z),
Γ+
1 f̂z =
1
PN (0)
[π[0,N ](z), enN ] =
PN (z)
PN (0)
= P+
2N (z).
(3.28)
This proves the formulas (3.26).
(4) It follows from (3.26) that the Weyl function M+(z) satisfies the
conditions
lim
iy→0
M+(iy) = ∞, lim
iy→∞
M+(iy) = 0.
Therefore, by the above definition (3.24) the boundary triple {C,Γ+
0 ,Γ
+
1 }
for the linear relation A
[∗]
[0,N ] is basic.
Remark 3.10. Recall that in [14] the boundary triple {C,Γ0,Γ1} for
the linear relation A[∗] was called basic, if the extensions A0 = AK and
A1 = AF were the Krein and the Friedrichs extensions of A[0,N ], respec-
tively. We use here an equivalent definition to prevent introducing of
extra notations AF and AK (see [14, Proposition 3.1]).
V. Derkach, I. Kovalyov 67
4. Resolvent matrix
4.1. Review of the M.G. Krein theory of resolvent matrix
Let A be a symmetric operator in a Pontryagin space (H, [·, ·]) of
negative index κ0, let the defect indices of A be (1, 1), let a scale vector
u ∈ H be given and let à be a selfadjoint extension of A acting in a
Pontryagin space H̃(⊃ H) of negative index κ satisfying the minimality
condition
H̃ = span{u, (Ã− z)−1u : z ∈ ρ(Ã)}.
The function [(Ã− z)−1u, u]H is called the u-resolvent of the operator A
of index κ.
A description of u-resolutions of a symmetric operator was given in
[34, 38] in the framework of the theory of the u-resolvent matrix, which
will be briefly presented below. A point z ∈ C is called (see [34, 38]) a
u-regular point of the operator A if ran(A− z) is closed and
H = ran(A− z)u span u.
Denote by ρ(A, u) the set of u-regular points of the operator A. Define
two functionals P(z), Q(z) : H → C holomorphic in ρ(A, u) by the
formulas
f − (P(z)f)u ∈ ran(A− z), Q(z)f = [(A− z)−1(f − (P(z)f)u), u].
Define also two vector-valued functions P(z)[∗], Q(z)[∗] with values in H
by
[P(z)[∗], f ] := P(z)f, [Q(z)[∗], f ] := Q(z)f.
Direct verification shows that for all z ∈ ρ(A, u) we have
P̂(z)[∗] := {P(z)[∗], z̄P(z)[∗]} ∈ A[∗],
Q̂(z)[∗] := {Q(z)[∗], z̄Q(z)[∗] + u} ∈ A[∗].
(4.1)
A description of u-resolvents of a Hilbert space symmetric operator with
equal finite defect indices was obtained by M.G. Krein in [34], and for
densely defined operator in a Pontryagin space, by M.G. Krein and H.
Langer in [38]. An explicit formula for the resolvent matrix of a Hilbert
space symmetric operator in boundary triple’s notations was given in [20,
21]. For a nondensely defined symmetric operator in a Pontryagin space
with the defect indices (1, 1) such a formula and the description of u-
resolvents take the following form (see [15, Theorem 5.2]).
68 An operator approach to indefinite...
Theorem 4.1. Let A be a symmetric operator with defect indices (1, 1)
in a Pontryagin space H of negative index κ0, let Π = {C,Γ0,Γ1} be a
boundary triple for A[∗], let u ∈ H, let ρ(A, u) ̸= ∅ and let
W(z) =
(
−Γ0Q̂(z[∗] Γ0P̂(z)[∗]
−Γ1Q̂(z)[∗] Γ1P̂(z)[∗]
)∗
, z ∈ ρ(A, u). (4.2)
Then:
(1) W(z) = (wi,j(z))
2
i,j=1 is holomorphic on ρ(A, u), and the formula
[(Ã− z)−1u, u] =
w11(z)τ(z) + w12(z)
w21(z)τ(z) + w22(z)
, z ∈ ρ(A, u) ∩ ρ(Ã)
establishes a one-to-one correspondence between the set of all u-
resolvents of A of index κ and the set of all τ ∈ Nκ−κ0, such that
w21(z)τ(z) + w22(z) ̸≡ 0. (4.3)
(2) If, in addition, H0 := H[−] domA is not trivial and A0 = A u
{0} × H0 then à is an operator if and only if τ satisfies the Nevan-
linna condition
τ(iy) = o(y) as y → ∞. (4.4)
(3) If, in addition, the operator A has k0 negative squares and Π is
a basic boundary triple, then the formula (4.2) establishes a one-
to-one correspondence between the set of all u-resolutions of the
operator A of index κ, such that the extension à has k negative
squares and the set of τ ∈ Nk−k0
κ−κ0, such that (4.3) holds.
4.2. Resolvent matrix of the operator A[0,j]
Theorem 4.2. [10, Theorem 3.14] Let a boundary triple Π = {C,Γ0,Γ1}
for the operator A
[∗]
[0,j] be defined by (3.13) and let u = e0. Then the cor-
responding u−resolvent matrix of the operator A[0,j] takes the following
representation
W[0,j](z) =
(
−Qj(z) −b̃−1
j Qj+1(z)
Pj(z) b̃−1
j Pj+1(z)
)
. (4.5)
Proof. It follows from (3.5) and (3.6) that P̂[0,j](z)
[∗] = π̂[0,j](z) and
Q̂[0,j](z)
[∗] = ξ̂[0,j](z). By (3.13) we obtain
Γ1Q̂[0,j](z)
[∗] = b̃−1
j Qj+1(z) and Γ0Q̂[0,j](z)
[∗] = Qj(z),
Γ1P̂[0,j](z)
[∗] = b̃−1
j Pj+1(z) and Γ0P̂[0,j](z)
[∗] = Pj(z).
(4.6)
Substituting (4.6) in (4.2), we get (4.5). This completes the proof.
V. Derkach, I. Kovalyov 69
Proposition 4.3. Let a boundary triple {C,Γ0,Γ1} for the operator A
[∗]
[0,j]
be defined by (3.13) and let u = e0. Then the u-resolvent matrix W[0,j](z)
admits the following factorization
W[0,j](z) = W0(z)W1(z) . . .Wj(z), (4.7)
where the elementary matrices are defined by
W0(z) =
0 −1
1
a0(z)
b0
and Wi(z) =
0 −b̃−1
i−1
b̃i−1
ai(z)
bi
, i = 1, j.
Proof. Prove (4.7) by induction
(1) W[0,0](z) = W0(z), i.e. (4.7) holds.
(2) Inductive step. Let (4.7) hold for some i− 1, i.e.
W[0,i−1](z) = W0(z)W1(z) . . .Wi−1(z) =
(
−Qi−1(z) −b̃−1
i−1Qi(z)
Pi−1(z) b̃−1
i−1Pi(z)
)
.
Then
W[0,i−1](z)Wi(z) =
(
−Qi−1(z) −b̃−1
i−1Qi(z)
Pi−1(z) b̃−1
i−1Pi(z)
) 0 −b̃−1
i−1
b̃i−1
ai(z)
bi
=
(
−Qi(z) −b̃−1
i (−biQi−1(z) + ai(z)Qi(z))
Pi(z) b̃−1
i (−biPi−1(z) + ai(z)Pi(z))
)
= {by (1.10)}=
(
−Qi(z) −b̃−1
i Qi+1(z)
Pi(z) b̃−1
i Pi+1(z)
)
=W[0,i](z).
This proves (4.7).
Theorem 4.4. [10, Theorem 3.14] Let j ∈ N and let the Wronskians
Wj [f̂ , π̂[0,j](0)] and Wj [f̂ , ξ̂[0,j](0)] be defined by (3.7). Then:
(1) The formulas
Γ̃0f̂ =Wj [f̂ , π̂[0,j](0)], Γ̃1f̂ =Wj [f̂ , ξ̂[0,j](0)],
define a boundary triple Π̃ = {C, Γ̃0, Γ̃1} for the operator A
[∗]
[0,j].
(2) The u-resolvent matrix of A[0,j] corresponding to the boundary triple
Π̃ and the scale u = e0 has the form
W̃[0,j](z) =
[
−Wj [ξ̂[0,j](z), π̂[0,j](0)] −Wj [ξ̂[0,j](z), ξ̂[0,j](0)]
Wj [π̂[0,j](z), π̂[0,j](0)] Wj [π̂[0,j](z), ξ̂[0,j](0)]
]
.
(4.8)
70 An operator approach to indefinite...
(3) The Weyl function corresponding to the boundary triple Π̃ has the
form
M̃[0,j](z) =
Wj [π̂[0,j](z), ξ̂[0,j](0)]
Wj [π̂[0,j](z), π̂[0,j](0)]
.
Proof. The proof of (1) is based on the identities (3.15) and
Wj [f̂ , ξ̂[0,j](0)]Wj [ĝ, π̂[0,j](0)]
∗−Wj [f̂ , π̂[0,j](0)]Wj [ĝ, ξ̂[0,j](0)]
∗ =Wj [f̂ , ĝ].
Applying the operators Γ̃0 and Γ̃1 to
f̂ = Q̂[0,j](z)
[∗] = ξ̂[0,j](z) and ĝ = P̂[0,j](z)
[∗] = π̂[0,j](z)
one obtains the equalities
Γ̃0f̂ =Wj [ξ̂[0,j](z), π̂[0,j](0)], Γ̃1f̂ =Wj [ξ̂[0,j](z), ξ̂[0,j](0)].
Γ̃0ĝ =Wj [π̂[0,j](z), π̂[0,j](0)], Γ̃1ĝ =Wj [π̂[0,j](z), ξ̂[0,j](0)], (4.9)
In view of (4.2) this yields (4.8). The second part of the formula (4.8) is
implied by (3.11)–(3.12).
The statement (3) follows from (4.9).
Corollary 4.5. The formula (4.8) for the u-resolvent matrix of A[0,j]
corresponding to the boundary triple Π̃ can be rewritten as
W̃[0,j](z) =−I + z
[
−[ξ[0,j](z), π[0,j](0)][0,j] −[ξ[0,j](z), ξ[0,j](0)][0,j]
[π[0,j](z), π[0,j](0)][0,j] [π[0,j](z), ξ[0,j](0)][0,j]
]
.
(4.10)
The Weyl function M̃[0,j](z) corresponding to the boundary triple Π̃ is
equal to
M̃[0,j](z) =
Pj+1(z)Qj(0)− Pj(z)Qj+1(0)
Pj+1(z)Pj(0)− Pj(z)Pj+1(0)
and has the properties
lim
x↑0
M̃[0,j](x) = ∞, lim
x↓−∞
M̃[0,j](x) =
Qj(0)
Pj(0)
.
4.3. The case s ∈ Hk,reg
κ,2nj−2
Theorem 4.6. [10, Theorem 3.14] Let s = {si}
2nj−2
i=0 ∈ Hk,reg
κ,2nj−2, let
{C,Γ+
0 ,Γ
+
1 } be a boundary triple of the linear relation A
[∗]
[0,j]. Then:
V. Derkach, I. Kovalyov 71
(1) The corresponding u-resolvent matrix of the operator A[0,j] can be
found by
W+
[0,j](z) =
(
Q+
2j+1(z) Q+
2j(z)
P+
2j+1(z) P+
2j(z)
)
. (4.11)
(2) W+
[0,j] ∈ Uκj (J), where κj are calculated by (2.23).
Proof. It follows from (4.1) that P̂[0,j](z)
[∗] = π̂[0,j](z) and Q̂[0,j](z)
[∗] =
ξ̂[0,j](z). Calculating entries of W+
[0,j](z), we obtain
Γ+
0 Q̂[0,j](z)
[∗] = 1
b̃j
(Pj+1(0)Qj(z)− Pj(0)Qj+1(z)) = −Q+
2j+1(z),
Γ+
1 Q̂[0,j](z)
[∗] = 1
Pj(0)
[ξ[0,j](z), enj ] =
Qj(z)
Pj(0)
= −Q+
2j(z).
(4.12)
Inserting (4.12) and (3.28) in (4.2), one obtains (4.11).
In the following statement relations between the resolvent matrices
W̃[0,j](z) and W+
[0,j](z) is established.
Proposition 4.7. Let s = {si}
2nj−2
i=0 ∈ Hk,reg
κ,2nj−2. The resolvent matrices
W̃[0,j](z) and W+
[0,j](z) of the operator A[0,j] are related by
W+
[0,j](z) = W̃[0,j](z)V[0,j], where V[0,j] =
(
−1
Qj(0)
Pj(0)
0 −1
)
(4.13)
Proof. It follows from (4.8) and (3.7) that
W̃[0,j](z) =
−Qj+1(z)Pj(0)−Qj(z)Pj+1(0)
b̃j
−Qj+1(z)Qj(0)−Qj(z)Qj+1(0)
b̃j
Pj+1(z)Pj(0)−Pj(z)Pj+1(0)
b̃j
Pj+1(z)Qj(0)−Pj(z)Qj+1(0)
b̃j
and hence
W̃[0,j](z)V[0,j]=
Qj+1(z)Pj(0)−Qj(z)Pj+1(0)
b̃j
−Qj(z)(Qj+1(0)Pj(0)−Pj+1(0)Qj(0))
b̃jPj(0)
−Pj+1(z)Pj(0)−Pj(z)Pj+1(0)
b̃j
Pj(z)(Qj+1(0)Pj(0)−Pj+1(0)Qj(0))
b̃jPj(0)
Now (4.13) follows from the Liouville–Ostrogradskii formula (2.9).
Notice that the mvf W+
[0,j](z) coincides with the mvf W+
2j+1(z) intro-
duced in [18]. Recall that W+
2j+1(z) admits the following factorization
W+
2j−1(z) =M1(z)L1 . . . Lj−1Mj(z) (4.14)
72 An operator approach to indefinite...
where the matrices Mi(z), Li are defined by
Li =
(
1 li
0 1
)
i = 1, j − 1, Mi(z) =
(
1 0
−zmi(z) 1
)
, i = 1, j.
(4.15)
and the polynomials mi and the numbers li are defined by (2.17).
In the following proposition we introduce one more resolvent matrix
which will be used as a frame for the description of solutions of even
moment problem.
Proposition 4.8. Let s ∈ Hk,reg
κ,ℓ , N (s) = {ni}ji=1 and let a boundary
triple Π++ = {C,Γ++
0 ,Γ++
1 } for the linear relation A
[∗]
[0,j] be defined by
Γ++
1 f̂ = Γ+
1 f̂ + ljΓ
+
0 f̂ , Γ++
0 f̂ = Γ+
0 f̂ ,
where the boundary triple Π+ = {C,Γ+
0 ,Γ
+
1 } is given by (3.2). Then
(1) the corresponding u-resolvent matrix W++
[0,j](z) is given by
W++
[0,j](z) =
(
Q+
2j−1(z) Q+
2j(z)
P+
2j−1(z) P+
2j(z)
)
. (4.16)
(2) W++
[0,j] ∈ Uκj (J), where κj are calculated by (2.24).
The above two propositions allow to formulate the following factor-
ization result for the resolvent matrix W+
[0,j](z).
Theorem 4.9. Let s = {si}∞i=0 ∈ Hk,reg
κ and let N ∈ N be big enough so
that the equalities
ν−(SnN ) = ν−(Snj ) and ν−(S
+
nN
) = ν−(S
+
nj )
hold for all j > N , j ∈ N. Let J̃[N,j] be a Jacobi matrix
J̃[N,j] = J[N,j] + diag
(
1
mN lN−1
, 0, . . . , 0
)
.
Then the resolvent matrix W+
[0,j](z) of A[0,j] admits the factorization
W+
[0,j](z) = W++
[0,N−1](z)W
+
[N,j](z), (4.17)
where W++
[0,N−1](z) is the resolvent matrix of the operator A[0,N−1] of the
form (4.16) corresponding to the boundary triple Π++
[0,N−1], and W+
[N,j](z)
is the resolvent matrix of the operator
Ã[N,j] = J̃[N,j]|dom(Ã[N,j])
, dom(Ã[N,j]) = {f ∈ H[N,j] : [f, enj ] = 0}
corresponding to the boundary triple Π+
[N,j] of the form (3.25).
V. Derkach, I. Kovalyov 73
Proof. By Proposition 4.8 the resolvent matrix of the operator Ã[N,j]
corresponding to the boundary triple Π+
[N,j] admits the factorization
W+
[N,j](z) = M̃N+1L̃N+1 . . . M̃j+1,
where M̃i and L̃i are matrices generated by numbers m̃i, l̃i defined
by (1.16). By [17, Theorem 4.1]
m̃i = mi, (i = N + 1, . . . , j + 1), l̃i = li, (i = N + 2, . . . , j),
and the equalities
1
m̃N+1 l̃N+1
= −ãN (0) = −aN (0)−
1
mN+1lN
=
1
mN+1lN+1
imply that l̃N+1 = lN+1. Therefore,
W+
[N,j](z) =MN+1LN+1 . . .Mj+1. (4.18)
It follows from (4.14) thatW+
[0,j](z) admits the factorization (4.17), where
W++
[0,j−1](z) =M1L1 . . .MjLj , W++
[0,N−1](z) =M1L1 . . .MNLN ,
(4.19)
The statement is implied now by (4.18) and (4.19).
4.4. Truncated indefinite moment problem
Consider the truncated indefinite moment problems MPκ(s, ℓ) and
MP kκ (s, ℓ). The moment problem MPκ(s, ℓ) is called even or odd regard-
ing to the oddness of the number ℓ+1 of given moments. The odd moment
problem MP kκ (s, 2n−2) is called nondegenerate, if Dn=detSn ̸= 0. The
moment problem MPκ(s, ℓ) was studied in [13,23,39,40]. Recall the fol-
lowing description of the set Mκ(s, 2nN − 2) from [10, Proposition 3.31].
Proposition 4.10. Let s = {si}2nN−2
i=0 ∈ Hκ,2nN−2 and let N (s) =
{ni}Ni=1 be the set of normal indices of s and let W[0,N−1] = (wij)
2
i,j=1.
The problem MPκ(s, 2nN −2) is solvable if and only if κN := ν−(SnN ) ≤
κ and the formula
f(z) =
w11(z)τ(z) + w12(z)
w21(z)τ(z) + w22(z)
, (4.20)
establishes a one-to-one correspondence between the class
Mκ(s, 2nN − 2) and the set of functions τ ∈ Nκ−κN , which satisfy
the Nevanlinna condition (4.4).
74 An operator approach to indefinite...
The operator approach to MPκ(s, 2nN − 2) in [39] is based on the
formula
f(z) = [(Ã− zI)−1e0, e0][0,N−1] (4.21)
which describes the set Mκ(s, 2nN − 2) when à ranges over the set of
all single-valued self-adjoint extensions of A[0,N−1]. In fact, in [39] a full
moment problem is considered. The description (4.20) is based on this
formula and on Theorem 4.1. Another proof of this formula was given
in [8, 9] and [10] by applications of the Schur algorithm.
The problem MP kκ (s, 2nN ) was posed and solved by the methods of
extension theory in [13, Theorem 4.14].
Theorem 4.11. Let s = {si}2nN−2
i=0 ∈ Hk,reg
κ,2nN−2, n = nN and let the
matrix Sn = (si+j)
n−1
i,j=0 be nondegenerate and κN := ν−(Sn) ≤ κ. Then
the problem MPκ(s, 2n− 2) is solvable and the formula
F (z) = [(Ã− z)−1u, u][0,N−1]
establishes a one-to-one correspondence between the set of all solutions of
the truncated indefinite moment problem MPκ(s, 2nN − 2) and the set of
all u-resolvents of the symmetric operator A[0,N−1] of index κ generated
by single-valued self-adjoint extensions Ã.
If, in addition, the matrix S+
nN−1 = (si+j+1)
nN−2
i,j=0 is nondegenerate
and kN := ν−(S
+
nN−1) ≤ k, then the problem MP kκ (s, 2nN − 2) is solv-
able and the formula (4.21) establishes a one-to-one correspondence be-
tween the set of all solutions of the truncated indefinite moment problem
MP kκ (s, 2nN−2) and the set of all u-resolvents of the symmetric operator
A[0,N−1] of index κ, such that the extension à has k negative squares.
Proof of Theorem 1.2. Since s = {si}2nN−2
i=0 ∈ Hk,reg
κ,2nN−2, the matrix
S+
n−1 = (si+j+1)
n−2
i,j=0 is nondegenerate and kN := ν−(S
+
n−1) ≤ k by Theo-
rem 4.11 the problemMP kκ (s, 2nN−2) is solvable and the formula (4.21)
establishes a one-to-one correspondence between the set of all solutions
of the truncated indefinite moment problem MP kκ (s, 2nN − 2) and the
set of all u-resolvents of the symmetric operator A[0,N−1] of index κ−κN ,
such that the extension à has k negative squares.
Consider now the boundary triple Π+
[0,N−1] = {C,Γ+
0 ,Γ
+
1 } for the
operator A[0,N−1]. By Proposition 3.9 the boundary triple Π+
[0,N−1] is
basic. Therefore, by Theorem 4.11 the set of all u-resolvents in (4.21) is
described by the formula
[(Ã− z)−1u, u][0,N−1] = TW+
[0,N−1]
[τ(z)],
V. Derkach, I. Kovalyov 75
where τ ∈ Nk−kN
κ−κN . Moreover, by Theorem 4.1 an u-resolvent [(Ã−
z)−1u, u][0,N−1] is generated by a single-valued self-adjoint extensions Ã
if and only if τ(z) = o(1) as z→̂∞. This completes the proof of Theo-
rem 1.2.
In the following proposition we recall a description of solutions of the
even indefinite Stieltjes moment problem presented in [18].
Theorem 4.12. Let s ∈ Hk,reg
κ,2nN−1, N (s) = {nj}Nj=1 and let the u-
resolvent matrix W++
[0,N−1](z) be given by (4.16). Then:
(1) A nondegenerate even moment problem MP kκ (s, 2nN − 1) is solv-
able, iff
κN := ν−(SnN ) ≤ κ and kN := ν−(S
+
nN
) ≤ k.
In this case the set Mk
κ(s, 2nN − 1) is parametrized by the formula
f(z) =
Q+
2N−1(z)τ(z) +Q+
2N (z)
P+
2N−1(z)τ(z) + P+
2N (z)
,
where τ(z) ∈ Nk−kN
κ−κN and τ(z) = o(1) as z→̂∞.
(2) The matrix W++
[0,N−1](z) admits the following factorization
W++
[0,N−1](z) =M1(z)L1 . . .MN−1(z)LN−1,
where the matrices Mi(z), Li are defined by (4.15).
5. The full indefinite moment problem MP k
κ (s)
5.1. Description of solutions
Let us associate with the infinite sequence s = {si}∞i=0 an indefinite
inner product space H[0,∞) of infinite sequences endowed with the inner
product
[x, y][0,∞) = (G[0,∞)x, y), G[0,∞) = diag(̃b0E−1
0 , b̃1E
−1
1 , . . .).
If s ∈ Hκ, then the space H[0,∞) is a Pontryagin space (see [6]), with
max{ind−H[0,∞), ind+H[0,∞)} = κ.
Associated to the matrix JT[0,∞) there is a minimal operator Amin, defined
as the closure of the operator JT[0,∞) restricted originally to the set of
76 An operator approach to indefinite...
finite sequences. The operator Amin turns out to be symmetric in the
space H[0,∞).
Alongside with the minimal operator we consider also the maximal
operator Amax, defined as the restriction of JT[0,∞) to the domain
dom(Amax) := {x ∈ H[0,∞) : J
T
[0,∞)x ∈ H[0,∞)}.
As is known [10,39] the operator Amin is self-adjoint in H[0,∞), if and only
if Amin = Amax. One can easily see that Amax = A
[∗]
min. It follows from the
equalities (2.12) that the defect subspace Nz(Amin) := ker(Amax − zI) is
either a one dimensional space generated by π(z) or is trivial. Combining
this observation with Lemma 2.13 one obtains the following
Proposition 5.1. Let s = {si}∞i=0 ∈ Hκ. Then the following statements
are equivalent:
(1) The moment problem MPκ(s) is indeterminate.
(2) The deficiency indices of the operator Amin are equal to one.
(3) The “moment of inertia” of the string is finite
∞∑
j=1
x2jmj(0) <∞, where xj = l1 + · · ·+ lj .
The results of [39] on indefinite moment problem can be reformulated
in notations connected with a monic GJM J as follows.
Theorem 5.2. Let ν−(Sn) = κ1 ≤ κ for all n ∈ N, n ≥ nN , and let the
moment problem MPκ(s) be indeterminate. Then:
(1) For all f ∈ H[0,∞) there exist finite limits
W∞[f, π(λ0)] = lim
j→∞
Wj [f, π(λ0)],
W∞[f, ξ(λ0)] = lim
j→∞
Wj [f, ξ(λ0)].
(2) A boundary triple for Amax can be defined by the equalities
Γ̃0f̂ =W∞[f̂ , π̂(0)], Γ̃1f̂ =W∞[f̂ , ξ̂(0)]. (5.1)
(3) The corresponding resolvent matrix takes the form
W̃[0,∞)(λ) =
(
−W∞[ξ̂(λ), π̂(0)] −W∞[ξ̂(λ), ξ̂(0)]
W∞[π̂(λ), π̂(0)] W∞[π̂(λ), ξ̂(0)]
)
.
The mvf W̃[0,∞)(z) = (w̃ij(z))
2
i,j=1 is entire of minimal exponential
type.
V. Derkach, I. Kovalyov 77
(4) The Weyl function of A, corresponding to the boundary triple (5.1),
takes the form
M̃[0,∞](z) = −W∞[π̂(z), π̂(0)]
W∞[π̂(z), ξ̂(0)]
.
(5) The formula
f(z) =
w̃11(z)τ(z) + w̃12(z)
w̃21(z)τ(z) + w̃22(z)
establishes a one-to-one correspondence between the class Mκ(s)
and the set of functions τ ∈ Nκ−κ1.
Assume now that s = {sj}∞j=0 ∈ Hk1
κ1 and let N be big enough, so
that
ν−(Sj) = κ1 = ν−(SnN ), ν−(S
(1)
j ) = k1 = ν−(S
(1)
nN
) for all j ≥ nN .
Then in view of Theorem 4.9 the indefinite Stieltjes moment problem
Mk
κ(s) can be reduced to a classical Stieltjes moment problem M0
0(s
(N)),
where the induced sequence s(N) can be calculated recursively as in [18,
Theorems 3.3, 3.5]. Alternatively, the sequence s(N) can be found as the
sequence of coefficients of the series expansion −
∞∑
i=0
s
(N)
i
zi+1 corresponding
to the continued fraction
1
−zmN+1 +
1
lN+1 +
1
−zmN+2 +
1
. . .
.
Notice that the sequence s(N) belongs to the class H0
0.
Theorem 5.3. Let s = {si}∞i=0 ∈ Hk0
κ0 for some κ0, k0 ∈ N. Then the
moment problem MP kκ (s) with κ, k ∈ N is solvable if and only if
κ0 ≤ κ, and k0 ≤ k;
and the moment problem MP kκ (s) is indeterminate if and only if
∞∑
j=1
mj(0) <∞ and
∞∑
j=1
lj <∞. (5.2)
If (5.2) holds then:
78 An operator approach to indefinite...
(1) The sequence of resolvent matrices W+
[0,n](z) converges to an entire
mvf W+
[0,∞)(z) = (w+
ij(z))
2
i,j=1 of minimal exponential type.
(2) The mvf W+
[0,∞)(z) is the resolvent matrix of the operator Amax
corresponding to the boundary triple
Γ+
0 f̂ = −W∞[f̂ , π̂(0)], Γ+
1 f̂ = −W∞[f̂ , ξ̂(0)]− LW∞[f̂ , π̂(0)];
(5.3)
(3) The formula
f(z) =
w+
11(z)τ(z) + w+
12(z)
w+
21(z)τ(z) + w+
22(z)
establishes a one-to-one correspondence between the class Mk
κ(s)
and the set of functions τ ∈ Nk−k1
κ−κ1 .
Proof. 1. Verification of criterion (5.2): It follows from Proposition 4.7
that there is a one-to-one correspondence between solutions f of the
problem Mk0
κ0(s) and solutions φ of the problem M0
0(s
(N)) given by LFT
f(z) = TW++
[0,N ]
(z)[φ(z)]. (5.4)
Therefore, the indefinite moment problemMk0
κ0(s) is indeterminate if and
only if the classical Stieltjes problem M0
0(s
(N)) is indeterminate. As is
known, see [31, Appendix II.13] the problem M0
0(s
(N)) is indeterminate
if and only if
∞∑
j=N+1
mj <∞ and
∞∑
j=N+1
lj <∞,
i.e. if (5.2) holds. Notice that for the classical Stieltjes moment problem
M0
0(s
(N)) the corresponding massesm
(N)
j and lengthes l
(N)
j are constants,
which coincide with mj+N and lj+N , respectively.
Now it remains to notice that the set of solutions f of the prob-
lem Mk
κ(s) and the set of solutions φ of the problem Mk−k0
κ−κ0(s
(N)) are
also connected by the LFT (5.4) and since s(N) ∈ H0
0 the problem
Mk−k0
κ−κ0(s
(N)) is indeterminate if and only if the problem M0
0(s
(N)) is
indeterminate (see [14]), which leads again to the condition (5.2).
2. Verification of (1): The convergence of the sequence of the re-
solvent matrices W+
[0,n](z) is implied by Theorem 5.2 and the formula
(4.13)
W+
[0,n](z) = W̃[0,n](z)
(
−1 Qn(0)
Pn(0)
0 −1
)
,
V. Derkach, I. Kovalyov 79
where Qn(0)
Pn(0)
= −L := −
n∑
j=0
lj in view of (2.22). In view of the limit mvf
W+
[0,∞](z) is connected with W̃[0,∞](z) by the equality
W+
[0,∞](z) = W̃[0,∞](z)
(
−1 −L
0 −1
)
, (5.5)
3. Verification of (2): The formulas (5.3) for the triple {C,Γ+
0 ,Γ
+
1 }
can be rewritten as
Γ+
0 f̂ = −Γ̃0f̂ , Γ+
1 f̂ = −Γ̃1f̂ − LΓ̃0f̂ . (5.6)
By Theorem 5.2 this implies that {C,Γ+
0 ,Γ
+
1 } is a boundary triple for
Amax. Moreover, it follows from (5.6) that the resolvent matrix of Amin
corresponding to the boundary triple {C,Γ+
0 ,Γ
+
1 } is connected with the
resolvent matrix W̃[0,∞](z) by the equality (5.5) and hence it coincides
with W+
[0,∞](z).
4. Verification of (3): The last statement is implied by the formula
(5.4) and the description of solutions of the problem Mk−k0
κ−κ0(s
(N)) given
in [14].
5.2. Padé approximants
Definition 5.4. The [L/M ] Pade approximant for a formal power series
−
∞∑
j=0
sj
zj+1
(5.7)
is a ratio f [n/k](z) = A[n/k](1/z)
B[n/k](1/z)
of polynomials A[n/k], B[n/k] of formal
degree n, k, respectively, such that B[n/k](0) ̸= 0 and
f [n/k](z) +
∞∑
j=0
sj
zj+1
= O
(
1
zn+k+1
)
as z→̂∞.
Explicit formula for diagonal Pade approximants was found in [9].
Here we give another proof of this formula.
Proposition 5.5. Let s = {si}∞i=0 ∈ Hk,reg
κ . Then the [n/n] Pade ap-
proximant for a formal power series (5.7) exists if n ∈ N (s) and
f [nj/nj ](z) = −Qj(z)
Pj(z)
, j ∈ N.
80 An operator approach to indefinite...
Proof. It follows from (2.20) and Proposition 4.8 that the function
−Qj(z)
Pj(z)
=
Q+
2j(z)
P+
2j(z)
= TW++
[0,j]
(z)[0] ∈ M(s, 2nj − 1)
belongs to M(s, 2nj−1). Therefore, the function −Qj(z)
Pj(z)
has the asymp-
totic
−Qj(z)
Pj(z)
= −s0
z
− · · · −
s2nj−1
z2nj
+O
(
1
z2nj+1
)
as z→̂∞.
Next, A(z) := znjQj(
1
z ), B(z) := znjPj(
1
z ) are polynomials of formal
degree nj and B(0) = 1. By Definition 5.4 the function −Qj(z)
Pj(z)
is the
[nj/nj ] Pade approximant for the formal power series (5.7).
The following formula for sub-diagonal Pade approximants in terms
of generalized Stieltjes polynomials can be proved similarly.
Proposition 5.6. Let s = {si}∞i=0 ∈ Hk,reg
κ . Then the [nj/nj − 1] Pade
approximants for the formal power series (5.7) exists and has the form
f [nj/nj−1](z) =
Q+
2j−1(z)
P+
2j−1(z)
, j ∈ N.
Appendix.
Relations between monic and symmetric GJM’s
The exposition of all results in [10] is based on so-called symmetric
GJM’s. To make a connection between [10] and the present paper we
present in this appendix some formulas which relate symmetric and monic
GJM’s and the corresponding polynomials of the first and second type.
Recall that the symmetric GJM H associated with the sequence of
atoms (ai, bi) (i ∈ Z+) was defined in [10] by the formulas
H =
CTa0 θ1B̂1
B̂1 CTa1 θ2B̂2
B̂2 CTa2
. . .
. . . . . .
, where B̂i =
0 0 · · ·
√
|bi|
...
... · · ·
...
0 0 · · · 0
0 0 · · · 0
,
(A.1)
V. Derkach, I. Kovalyov 81
θi = sign(bi) and the entries Cai are defined by (2.4), i ∈ N. The short-
ened symmetric GJM H[0,j] is defined by
H[0,j] =
CTa0 θ1B̂1
B̂1 CTa1
. . .
. . . . . . θjB̂j
B̂j CTaj
.
Lemma A.1. Let J be a monic GJM, corresponding to a sequence of
atoms (ai, bi), i ∈ Z+, and let the matrix Ψ be defined by
Ψ = diag(Iℓ0 ,
√
|b1|Iℓ1 ,
√
|b1b2|Iℓ2 , . . .). (A.2)
Then the monic and symmetric GJM’s J and H are connected by
HT := Ψ−1JΨ.
Similarly, the matrices J[0,j] and H[0,j] are connected by the formula
HT
[0,j] := Ψ−1
[0,j]J[0,j]Ψ[0,j],
Ψ[0,j] = diag(Iℓ0 ,
√
|b1|Iℓ1 , . . . ,
√
|b1 . . . bj |Iℓj ).
Proof. The equality (A.1) is obtained from (2.6) and (A.2) by direct
computations.
Remark A.2. Let Ĥ[0,j] be the indefinite inner product space of se-
quences from Cnj+1 endowed with the indefinite inner product
[x, y][0,j] = (Ĝ[0,j]x, y), Ĝ[0,j] = diag(θ0E0, θ1E1, . . . , θjEj).
where
Eai =
a
(i)
1 · · · a
(i)
ℓi−1 1
... ··· ···
a
(i)
ℓi−1 ···
1 0
i = 0, j.
It follows from the equalities (see [29, Chapter 12])
EaCa = C∗
aEa
that the matrix Ĝ[0,j]H[0,j] is self-adjoint in the standard scalar product
in Cnj+1 , and hence the matrix H[0,j] generates a self-adjoint operator in
Ĥ[0,j].
82 An operator approach to indefinite...
The polynomials P̂j(z) and Q̂j(z) of the first and second kind of H
were defined in [10] as solutions of the following recurrent relations
θj
√
|bj |yj−1(z)− aj(z)yj(z) +
√
|bj+1|yj+1(z) = 0 (b0 = ε0) (A.3)
subject to the initial conditions
P̂−1(z) ≡ 0, P̂0(z) ≡ 1, Q̂−1(z) ≡ −1, Q̂0(z) ≡ 0,
Lemma A.3. Let J be the monic GJM and let Pj(z) and Qj(z) be the
Lanzcos polynomials of the first and second kind of J . Let H be the
symmetric GJM. Then the polynomials of the first and second kind of H
can be found by
P̂0(z) = P0(z) and P̂j(z) =
1√
|b1...bj |
Pj(z),
Q̂0(z) = Q0(z) and Q̂j(z) =
1√
|b1...bj |
Qj(z).
(A.4)
Proof. Substituting (A.4) into (A.3) one obtains in view of (1.10) for
j = 0
−a0(z)P̂0(z) +
√
|b1|P̂1(z) = −a0(z)P0 + P1(z) = 0,
and for arbitrary j ∈ N the left part of (A.3) takes the form
bjPj−1 − aj(z)Pj(z) + Pj+1(z)√
|b1 . . . bj |
= 0.
The equalities (A.4) for the polynomials Q̂j(z) are proved similarly.
Remark A.4. The Liouville–Ostrogradskii formula (2.9) for polynomi-
als P̂j and Q̂j takes the form
θ0 . . . θj
√
|bj+1|
(
Q̂j+1(z)P̂j(z)− Q̂j(z)P̂j+1(z)
)
= 1.
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Contact information
Vladimir Derkach Dragomanov National
Pedagogical University,
Kyiv, Ukraine
Donetsk National University,
Vinnytsya, Ukraine
E-Mail: derkach.v@gmail.com
Ivan Kovalyov Dragomanov National
Pedagogical University,
Kyiv, Ukraine
E-Mail: i.m.kovalyov@gmail.com
|
| id | nasplib_isofts_kiev_ua-123456789-169313 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1810-3200 |
| language | English |
| last_indexed | 2025-12-01T18:28:39Z |
| publishDate | 2017 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Derkach, V.A. Kovalyov, I.M. 2020-06-10T13:35:31Z 2020-06-10T13:35:31Z 2017 An operator approach to indefinite Stieltjes moment problem / V.A. Derkach, I.M. Kovalyov // Український математичний вісник. — 2017. — Т. 14, № 1. — С. 42-85. — Бібліогр.: 50 назв. — англ. 1810-3200 2010 MSC. Primary 30E05; Secondary 15B57, 46C20, 47A57. https://nasplib.isofts.kiev.ua/handle/123456789/169313 In the present paper we solve the indefinite Stieltjes moment problem MPkκ(s) within the M.G. Krein theory of u-resolvent matrices applied to a Pontryagin space symmetric operator A[0,N] generated by J[0,N]. The u-resolvent matrices of the operator A[0,N] are calculated in terms of generalized Stieltjes polynomials using the boundary triple’s technique. Criterions for the problem MPkκ(s) to be solvable and indeterminate are found. Explicit formulae for Pade approximants for generalized Stieltjes fraction in terms of generalized Stieltjes polynomials are also presented. en Інститут прикладної математики і механіки НАН України Український математичний вісник An operator approach to indefinite Stieltjes moment problem Article published earlier |
| spellingShingle | An operator approach to indefinite Stieltjes moment problem Derkach, V.A. Kovalyov, I.M. |
| title | An operator approach to indefinite Stieltjes moment problem |
| title_full | An operator approach to indefinite Stieltjes moment problem |
| title_fullStr | An operator approach to indefinite Stieltjes moment problem |
| title_full_unstemmed | An operator approach to indefinite Stieltjes moment problem |
| title_short | An operator approach to indefinite Stieltjes moment problem |
| title_sort | operator approach to indefinite stieltjes moment problem |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/169313 |
| work_keys_str_mv | AT derkachva anoperatorapproachtoindefinitestieltjesmomentproblem AT kovalyovim anoperatorapproachtoindefinitestieltjesmomentproblem AT derkachva operatorapproachtoindefinitestieltjesmomentproblem AT kovalyovim operatorapproachtoindefinitestieltjesmomentproblem |