Two-Term Differential Equations with Matrix Distributional Coefficients
We propose a regularization of the formal differential expression l(y) = iⁿy⁽ⁿ⁾(t) + q(t)y(t), t ∈ (a, b), of order n > 2 with matrix distribution q.
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irk-123456789-1656072020-02-15T01:27:34Z Two-Term Differential Equations with Matrix Distributional Coefficients Konstantinov, O.O. Статті We propose a regularization of the formal differential expression l(y) = iⁿy⁽ⁿ⁾(t) + q(t)y(t), t ∈ (a, b), of order n > 2 with matrix distribution q. Запропоновано регуляризацiю формального диференцiального виразу порядку n > 2 l(y) = iⁿy⁽ⁿ⁾(t) + q(t)y(t), t ∈ (a, b), з матричною узагальненою функцiєю q. 2015 Article Two-Term Differential Equations with Matrix Distributional Coefficients / O.O. Konstantinov // Український математичний журнал. — 2015. — Т. 67, № 5. — С. 625–634. — Бібліогр.: 15 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165607 517.9 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Konstantinov, O.O. Two-Term Differential Equations with Matrix Distributional Coefficients Український математичний журнал |
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We propose a regularization of the formal differential expression
l(y) = iⁿy⁽ⁿ⁾(t) + q(t)y(t), t ∈ (a, b), of order n > 2 with matrix distribution q. |
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Article |
| author |
Konstantinov, O.O. |
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Konstantinov, O.O. |
| author_sort |
Konstantinov, O.O. |
| title |
Two-Term Differential Equations with Matrix Distributional Coefficients |
| title_short |
Two-Term Differential Equations with Matrix Distributional Coefficients |
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Two-Term Differential Equations with Matrix Distributional Coefficients |
| title_fullStr |
Two-Term Differential Equations with Matrix Distributional Coefficients |
| title_full_unstemmed |
Two-Term Differential Equations with Matrix Distributional Coefficients |
| title_sort |
two-term differential equations with matrix distributional coefficients |
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Інститут математики НАН України |
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2015 |
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Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/165607 |
| citation_txt |
Two-Term Differential Equations with Matrix Distributional Coefficients / O.O. Konstantinov // Український математичний журнал. — 2015. — Т. 67, № 5. — С. 625–634. — Бібліогр.: 15 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT konstantinovoo twotermdifferentialequationswithmatrixdistributionalcoefficients |
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2025-07-14T19:12:04Z |
| last_indexed |
2025-07-14T19:12:04Z |
| _version_ |
1837650752550469632 |
| fulltext |
UDC 517.9
O. O. Konstantinov (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
TWO-TERM DIFFERENTIAL EQUATIONS
WITH MATRIX DISTRIBUTIONAL COEFFICIENTS
ДВОЧЛЕННI ДИФЕРЕНЦIАЛЬНI РIВНЯННЯ
З МАТРИЧНИМИ КОЕФIЦIЄНТАМИ-РОЗПОДIЛАМИ
We propose a regularization of the formal differential expression
l(y) = imy(m)(t) + q(t)y(t), t ∈ (a, b),
of order m > 2 with matrix distribution q. It is assumed that q = Q([m/2]), where Q = (Qi,j)
s
i,j=1 is a matrix function with
entries Qi,j ∈ L2[a, b] if m is even and Qi,j ∈ L1[a, b], otherwise. In the case of Hermitian matrix q, we describe self-
adjoint, maximal dissipative, and maximal accumulative extensions of the associated minimal operator and its generalized
resolvents.
Запропоновано регуляризацiю формального диференцiального виразу порядку m > 2
l(y) = imy(m)(t) + q(t)y(t), t ∈ (a, b),
з матричною узагальненою функцiєю q. Припускається, що q = Q([m/2]), де Q = (Qi,j)
s
i,j=1 — матрична функцiя
з елементами Qi,j ∈ L2[a, b] у випадку парного m i Qi,j ∈ L1[a, b] для непарного m. У випадку ермiтової
матрицi q описано самоспряженi максимальнi дисипативнi та максимальнi акумулятивнi розширення асоцiйованого
мiнiмального оператора та його узагальненi резольвенти.
1. Introduction. In [1] (see also [2]) it was proposed a regularization with the help of quasi–
derivatives of the two-term formal differential expression
l(y) = imy(m) + qy, m ≥ 3, (1)
with distributional potential q = Q([m/2]), where Q ∈ L2[a, b] if m is even and Q ∈ L1[a, b]
otherwise. Note that the case m = 2 was considered earlier in [3] which started a new development
in the theory of Schrödinger operators with distributional potentials. We mention here only papers
[4, 5] and references therein. In particular in [5] (see also [6]) spectral properties of Schrödinger
operators with matrix distributional potentials were studied.
The main purpose of this paper is the extension of the results of [1] to the case of matrix differential
operators of the form (1), acting in the Hilbert space L2([a, b],Cs) ≡ (L2([a, b])
s, s ∈ N. In the case
of formally self-adjoint quasidifferential expression we apply the boundary triple technique to give
the explicit description of the various classes of extensions of the corresponding minimal operator.
The paper is organized as follows. In Section 1 we recall basic definitions and known facts
concerning the matrix quasidifferential operators. Section 2 presents the regularization of the formal
differential expression (1) using the quasiderivatives. In Section 3 the boundary triplets for the
minimal symmetric operators are constructed and maximal dissipative, maximal accumulative and
self-adjoint extensions of these operators are explicitly described in terms of boundary conditions.
Section 4 deals with the formally self-adjoint quasidifferential operators with real-valued coefficients.
In this case we prove that every maximal dissipative (or accumulative) extension of the minimal
operator is self-adjoint and describe all such extensions. In Section 5 the extensions with separated
boundary conditions are considered. Section 6 deals with generalized resolvents of the minimal
operator.
c© O. O. KONSTANTINOV, 2015
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 5 625
626 O. O. KONSTANTINOV
2. Matrix quasidifferential expressions. In this section we recall some basic facts concerning
the matrix quasidifferential operators on a finite interval. For a more detailed discussion of quasi-
differential equations the reader is referred to [7, 8] in the scalar coefficients case and to [9, 10] for
general case with matrix coefficients.
Let m, s ∈ N and a finite (closed) interval [a, b] be given. For a given set T, Ms(T ) denotes the
set of (s×s)-matrices with entries in T. Denote by Zm,s([a, b]) the set of the (m×m)-matrix-valued
functions A with entries ak,j satisfy
1) ak,j ∈Ms(L1[a, b]), k, j = 1, 2, . . . ,m,
2) ak,j = 0, j ≥ k + 2, ak,k+1 is invertible a. e. on J for k = 1, 2, . . . ,m− 1.
Such matrices will be referred to as Shin – Zettl matrices of order m. Define inductively the associated
quasiderivatives of orders k ≤ m of a (vector) function y ∈ Dom (A) in the following way:
D[0]y := y,
D[k]y := a−1k,k+1(t)
(D[k−1]y)′ −
k∑
j=1
ak,j(t)D
[j−1]y
, k = 1, 2, . . . ,m− 1,
where am,m+1 := Is, the identity (s× s)-matrix, and the associated domain Dom (A) is defined by
Dom(A) :=
{
y
∣∣∣D[k]y ∈ AC([a, b],Cs), k = 0,m− 1
}
.
The above yields D[m]y ∈ L1([a, b],Cs). The quasidifferential expression l(y) of order m associated
with A is defined by
l(y) := imD[m]y. (2)
It gives rise to the associated maximal quasidifferential operator
Lmax : y 7→ l(y),
Dom(Lmax) =
{
y ∈ Dom(A)
∣∣∣D[m]y ∈ L2([a, b],Cs)
}
in the Hilbert space L2([a, b],Cs), and the associated minimal quasidifferential operator is defined as
the restriction of Lmax onto the set
Dom(Lmin) :=
{
y ∈ Dom(Lmax)
∣∣∣D[k]y(a) = D[k]y(b) = 0, k = 0,m− 1
}
.
If the matrix functions ak,s are sufficiently smooth, then all the brackets in the definition of the
quasiderivatives can be expanded, and we come to the usual ordinary differential operators.
Let us recall the definition of the formally adjoint quasidifferential expression l+(y). The formally
adjoint (also called the Lagrange adjoint) matrix A+ for A ∈ Zm,s([a, b]) is defined by
A+ := −Λ−1m ATΛm,
where AT is the conjugate transposed matrix to A and
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 5
TWO-TERM DIFFERENTIAL EQUATIONS WITH MATRIX DISTRIBUTIONAL COEFFICIENTS 627
Λm :=
0 0 . . . 0 −Is
0 0 . . . Is 0
...
...
...
...
...
0 (−1)m−1Is . . . 0 0
(−1)mIs 0 . . . 0 0
.
One can easily see that Λ−1m = (−1)m−1Λm. In the similar way one can define the Shin – Zettl
quasiderivatives associated with A+ which will be denoted by D{0}y,D{1}y, . . . , D{m}y, acting on
the domain
Dom(A+) :=
{
y
∣∣∣D{k}y ∈ AC([a, b],Cs), k = 0,m− 1
}
.
The formally adjoint quasidifferential expression is defined as l+(y) := imD{m}y. Denote the
associated maximal and minimal operators by L+
max and L+
min respectively. The following results are
proved in [9] (see also [10]).
Lemma 1 (Green’s formula). For any y ∈ Dom(Lmax), z ∈ Dom(L+
max) there holds
b∫
a
(
D[m]y · z − y ·D{m}z
)
dt =
m∑
k=1
(−1)k−1D[m−k]y ·D{k−1}z
∣∣∣t=bt=a .
Lemma 2. For any (α0, α1, . . . , αm−1), (β0, β1, . . . , βm−1) ∈ Cms there exists a function y ∈
∈ Dom(Lmax) such that
D[k]y(a) = αk, D[k]y(b) = βk, k = 0, 1, . . . ,m− 1.
Theorem 1. The operators Lmin, L
+
min, Lmax, L
+
max are closed and densely defined in
L2 ([a, b],Cs) , and satisfy
L∗min = L+
max, L∗max = L+
min.
If l(y) = l+(y), then the operator Lmin = L+
min is symmetric with the deficiency indices (ms,ms) ,
and
L∗min = Lmax, L∗max = Lmin.
3. Regularizations by quasiderivatives. Consider the formal matrix differential expression
l(y) = imy(m)(t) + q(t)y(t), m ≥ 2,
assuming that
q = Q(n), n =
[m
2
]
,
Q ∈
Ms(L2[a, b]), m = 2n,
Ms(L1[a, b]), m = 2n+ 1,
(3)
where the derivatives of Q are understood in the sense of distributions. Introduce the quasiderivatives
as follows:
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 5
628 O. O. KONSTANTINOV
D[r]y = y(r), r = 0, 1, . . . ,m− n− 1,
D[m−n+k]y = (D[m−n+k−1]y)′ + i−m(−1)k
(
n
k
)
QD[k]y, k = 0, 1, . . . , n− 1,
D[m]y =
(D[m−1]y)′ + i−m(−1)n
(
n
n
)
QD[n]y, m = 2n+ 1,
(D[m−1]y)′ +QD[n]y + (−1)n+1Q2y, m = 2n,
(4)
where
(
k
j
)
are the binomial coefficients. It is easy to verify that for sufficiently smooth matrix
functions Q we have l(y) = imD[m]y. The Shin – Zettl matrix corresponding to (4) has the form
A :=
0 Is 0 . . . 0 . . . 0 0
0 0 Is . . . 0 . . . 0 0
...
...
...
...
...
...
...
...
−i−m
(
n
0
)
Q 0 0 . . . 0 . . . 0 0
0 i−m
(
n
1
)
Q 0 . . . 0 . . . 0 0
...
...
...
...
...
...
...
...
0 0 0 . . . 0 . . . 0 Is
(−1)
m
2 δ2n,mQ
2 0 0 . . . i−m(−1)n+1
(
n
n
)
Q . . . 0 0
,
(5)
where δij is the Kronecker symbol. Note that under assumptions (3), all the coefficients of the Shin –
Zettl matrix (5) are integrable matrix functions. The regularization of the initial formal differential
expression (1) is defined by (2) and generates the corresponding quasidifferential operators Lmin and
Lmax.
Remark 1. For m − n ≤ r < m the quasiderivatives D[r] depend on the choice of the an-
tiderivative Q of order n of the (matrix) distribution q which is defined up to a a polynomial of
order ≤ n − 1. Nevertheless in the sense of distributions imD[m]y = l[y] does not depend on this
polynomial. Morever, it is easy to see that the corresponding maximal and minimal operators also do
not depend on the choice of antiderivative (cf. [2]).
In the case s = 1,m = 2 the above regularization was proposed in [3]. The extension to arbitrary
even m was announced in [11]. The case of general m ≥ 3 was considered in [1, 2]. Here we extend
this approach on the arbitrary s ≥ 1.
4. Extensions of symmetric quasidifferential operators. Throughout the rest of the paper
we assume that the matrix distribution q is Hermitian (due to Remark 1 one can suppose that Q is
Hermitian). It follows from Theorem 1 that the minimal quasidifferential operator Lmin is symmetric
with deficiency indices (ms,ms). Therefore it is interesting to describe various classes of extensions
of Lmin in L2([a, b],Cs). For this purpose we will exploit the theory of boundary triplets [12].
Definition 1. Let T be a closed densely defined symmetric operator in a Hilbert space H with
equal (finite or infinite) deficiency indices. The triplet (H,Γ1,Γ2) , where H is an auxiliary Hilbert
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 5
TWO-TERM DIFFERENTIAL EQUATIONS WITH MATRIX DISTRIBUTIONAL COEFFICIENTS 629
space and Γ1, Γ2 are the linear maps from Dom (T ∗) to H, is called a boundary triplet for T, if the
following two conditions are satisfied:
(1) for any f, g ∈ Dom (L∗) there holds
(T ∗f, g)H − (f, T ∗g)H = (Γ1f,Γ2g)H − (Γ2f,Γ1g)H ,
(2) for any g1, g2 ∈ H there is a vector f ∈ Dom (T ∗) such that Γ1f = g1 and Γ2f = g2.
The above definition implies that f ∈ Dom (T ) if and only if Γ1f = Γ2f = 0. A boundary triplet
(H,Γ1,Γ2) with dimH = k exists for any symmetric operator T with equal non-zero deficiency
indices (k, k) (k ≤ ∞), but it is not unique [12 – 14].
The following result is crucial for the rest of the paper.
Lemma 3. Let n be a positive integer. Define linear maps Γ[1], Γ[2] from Dom(Lmax) to Cms
as follows: for m = 2n we set
Γ[1]y := i2n
−D[2n−1]y(a)
. . .
(−1)nD[n]y(a)
D[2n−1]y(b)
. . .
(−1)n−1D[n]y(b)
, Γ[2]y :=
D[0]y(a)
. . .
D[n−1]y(a)
D[0]y(b)
. . .
D[n−1]y(b)
(6)
and for m = 2n+ 1 and we set
Γ[1]y := i2n+1
−D[2n]y(a)
. . .
(−1)nD[n+1]y(a)
D[2n]y(b)
. . .
(−1)n−1D[n+1]y(b)
αD[n]y(b) + βD[n]y(a)
, Γ[2]y :=
D[0]y(a)
. . .
D[n−1]y(a)
D[0]y(b)
. . .
D[n−1]y(b)
γD[n]y(b) + δD[n]y(a)
,
where
α = 1, β = 1, γ =
(−1)n
2
+ i, δ =
(−1)n+1
2
+ i.
Then (Cms,Γ[1],Γ[2]) is a boundary triplet for Lmin.
Remark 2. The values of the coefficients α, β, γ, δ may be replaced by an arbitrary quadruple
of numbers satisfying the conditions
αγ + αγ = (−1)n, βδ + βδ = (−1)n+1, αδ + βγ = 0,
βγ + αδ = 0, αδ − βγ 6= 0.
Proof of Lemma 3. The proof follows from Lemmas 1 and 2. It repeats the arguments of [1, 2]
in the case of scalar differential operators (s = 1).
For any bounded operator K in Cms denote by LK the restriction of Lmax onto the set of the
functions y ∈ Dom (Lmax) satisfying the homogeneous boundary condition in the canonical form
(see [12])
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 5
630 O. O. KONSTANTINOV
(K − I) Γ[1]y + i (K + I) Γ[2]y = 0. (7)
Similarly, denote by LK the restriction of Lmax onto the set of the functions y ∈ Dom (Lmax)
satisfying the boundary condition
(K − I) Γ[1]y − i (K + I) Γ[2]y = 0. (8)
Recall that a densely defined linear operator T on a complex Hilbert space H is called dissipative
(resp. accumulative) if
= (Tx, x)H ≥ 0 (resp. ≤ 0), for all x ∈ Dom(T )
and it is called maximal dissipative (resp. maximal accumulative) if, in addition, T has no nontrivial
dissipative (resp. accumulative) extensions in H. Every symmetric operator is both dissipative and
accumulative, and every self-adjoint operator is a maximal dissipative and maximal accumulative
one. According to Phillips’ theorem (see [12, p. 154]) every maximal dissipative or accumulative
extension of a symmetric operator is a restriction of its adjoint operator. Abstract results of [12] and
Lemma 3 lead to the following description of dissipative, accumulative and self-adjoint extensions of
Lmin.
Theorem 2. Every LK with K being a contracting operator in Cms, is a maximal dissipative
extension of Lmin. Similarly every LK with K being a contracting operator in Cms, is a maximal
accumulative extension of the operator Lmin. Conversely, for any maximal dissipative (respectively,
maximal accumulative) extension L̃ of the operator Lmin there exists a contracting operator K such
that L̃ = LK (respectively, L̃ = LK). The extensions LK and LK are self-adjoint if and only if K is
a unitary operator on Cms. These correspondences between operators {K} and the extensions {L̃}
are all bijective.
Remark 3. It follows from [10] (Theorem 7.2) that in the case of even m Lmin and therefore all
its extensions are bounded below. Otherwise, for odd m the operator Lmin is unbounded below and
above (see, e.g., [10], Theorem 10.3).
Remark 4. Analogously to [2] one can prove that the mapping K → LK is not only bijective but
also continuous. More accurately, if unitary operators Kn strongly converge to an operator K, then∥∥∥(LK − λ)−1 − (LKn − λ)−1
∥∥∥→ 0, n→∞, Imλ 6= 0.
The converse is also true, because the set of unitary operators in the space Cms is a compact set.
This means that the mapping
K → (LK − λ)−1 , Imλ 6= 0,
is a homeomorphism for any fixed λ ∈ C \ R.
5. Real extensions. Recall that a linear operator L acting in L2([a, b],Cs) is called real if for
every function y ∈ Dom(L) the complex conjugate function y also lies in Dom(L) and L(y) = L(y).
If the minimal quasidifferential operator is real, one arrives at the natural question on how to
describe its real extensions. The following theorem is valid.
Theorem 3. Let m = 2n, and let the entries of the Hermitian matrix distribution q be real-
valued, then the maximal and minimal quasi-differential operators Lmax and Lmin generated by
Shin – Zettle matrix (5) are real. All real maximal dissipative and maximal accumulative extensions
of Lmin are self-adjoint. The self-adjoint extensions LK or LK are real if and only if the unitary
matrix K is symmetric.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 5
TWO-TERM DIFFERENTIAL EQUATIONS WITH MATRIX DISTRIBUTIONAL COEFFICIENTS 631
Proof. By Remark 1 one can assume that Q is real all the coefficients of the quasi-derivatives
are real matrix functions. Therefore
D[i]y = D[i]y, i = 1, 2, . . . ,m,
which implies l(y) = l(y). Thus for any y ∈ Dom (Lmax) we have y ∈ Dom(Lmax) and Lmax(y) =
= Lmax(y). It follows that the operator Lmax is real. Analogously, the operator Lmin is a also real.
Let LK be an arbitrary real maximal dissipative extension given by the boundary conditions (7), then
for any y ∈ Dom(LK) the complex conjugate y satisfies (7) too, that is
(K − I) Γ[1]y + i (K + I) Γ[2]y = 0.
Due to the real-valuedness of the coefficients of the quasiderivatives, the equalities (6) imply
Γ[1]y = Γ[1]y, Γ[2]y = Γ[2]y.
By taking the complex conjugates we obtain(
K − I
)
Γ[1]y − i
(
K + I
)
Γ[2]y = 0,
and LK ⊂ LK due to Theorem 2. Thus, the dissipative extension LK is also accumulative, which
means that it is symmetric. As LK is a maximal dissipative extension of Lmin we have that the
operator LK = LK is self-adjoint. It follows that K is a unitary operator. In this case the boundary
condition (7) is equivalent to(
K−1 − I
)
Γ[1]y − i
(
K−1 + I
)
Γ[2]y = 0.
It follows that LK = LK
−1
. On the other hand LK = LK and therefore K−1 = K. As K is unitary,
we have K−1 = KT , which gives K = KT . Here KT is the transpose of the matrix K. In a similar
way one can show that a maximal accumulative extension LK is real if and only if it is self-adjoint
and K = KT .
Theorem 3 is proved.
6. Separated boundary conditions. In this section we discuss the extensions of Lmin defined
by the so-called separated boundary conditions. Denote by fa the germ of a continuous function f
at the point a. We recall that the boundary conditions that define an operator L ⊂ Lmax are called
separated if for any y ∈ Dom(L) and any g, h ∈ Dom(Lmax) with
ga = ya, gb = 0, ha = 0, hb = yb
we have g, h ∈ Dom(L).
The following theorem gives a description of the operators LK and LK with separated boundary
conditions in the case of an even order m = 2n .
Theorem 4. The boundary conditions (7) and (8) defining LK and LK respectively are separated
if and only if the matrix K has the block form
K =
(
Ka 0
0 Kb
)
, (9)
where Ka and Kb are (ns× ns)-matrices.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 5
632 O. O. KONSTANTINOV
Proof. We consider the operators LK only, the case of LK can be considered in a similar way.
Denote
Γ[1] =: (Γ1a,Γ1b) , Γ[2] =: (Γ2a,Γ2b) ,
where
Γ1ay = i2n
(
−D[2n−1]y(a), . . . , (−1)nD[n]y(a)
)
,
Γ1by = i2n
(
D[2n−1]y(b), . . . , (−1)n−1D[n]y(b)
)
,
Γ2ay =
(
D[0]y(a), . . . , D[n−1]y(a)
)
,
Γ2by =
(
D[0]y(b), . . . , D[n−1]y(b)
)
.
Let y, g ∈ Dom (Lmax). Clearly, for any c ∈ [a, b] the equality yc = gc implies that D[k]yc = D[k]gc,
k = 0, 1, . . . ,m− 1. In particular, the equality ya = ga implies Γ1ay = Γ1ag and Γ2ay = Γ2ag, and
the equality yb = hb implies Γ1by = Γ1bh and Γ2by = Γ2bh.
If K has the form (9), then the boundary condition (7) can be rewritten as a system
(Ka − I)Γ1ay + i(Ka + I)Γ2ay = 0,
− (Kb − I)Γ1by + i(Kb + I)Γ2by = 0,
and these boundary conditions are obviously separated. Conversely, let the boundary conditions (7)
be separated. The matrix K ∈ C2ns×2ns can be written in the block form
K =
(
K11 K12
K21 K22
)
with ns×ns blocks Kjk. We need to show that K12 = K21 = 0. Let us rewrite boundary conditions
(7) in the form of the system
(K11 − I)Γ1ay +K12Γ1by + i(K11 + I)Γ2ay + iK12Γ2by = 0,
K21Γ1ay + (K22 − I)Γ1by + iK21Γ2ay + i(K22 + I)Γ2by = 0.
As the boundary conditions are separated, any function g with ga = ya and gb = 0 also satisfies this
system, which gives
K11 [Γ1ay + iΓ2ay] = Γ1ay − iΓ2ay,
K21 [Γ1ay + iΓ2ay] = 0.
This means that
Γ1ay + iΓ2ay ∈ Ker(K21), y ∈ Dom(LK). (10)
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TWO-TERM DIFFERENTIAL EQUATIONS WITH MATRIX DISTRIBUTIONAL COEFFICIENTS 633
For any z = (z1, z2) ∈ C2ns consider the vectors −i (K + I) z and (K − I) z. Due to Lemma 3
there is a function yz ∈ Dom(Lmax) such that
−i (K + I) z = Γ[1]yz,
(K − I) z = Γ[2]yz.
(11)
A simple calculation shows that yz satisfies the boundary conditions (7) and, therefore, yz ∈ Dom(LK).
We can rewrite (11) as a system
−i(K11 + I)z1 − iK12z2 = Γ1ayz,
−iK21z1 − i(K22 + I)z2 = Γ1byz,
(K11 − I)z1 +K12z2 = Γ2ayz,
K21z1 + (K22 − I)z2 = Γ2byz.
It follows from the first and the third equations of the system above that
Γ1ayz + iΓ2ayz = −2iz1
for any z = (z1, z2) ∈ C2ns. By (10) we have that Ker(K21) = Cns or equivalently K21 = 0.
Similarly one can prove that K12 = 0.
Theorem 4 is proved.
7. Generalized resolvents. Let us recall [15] that a generalized resolvent of a closed symmetric
operator L in a Hilbert space H is an operator-valued function λ 7→ Rλ defined on C \ R which can
be represented as
Rλx = P+
(
L+ − λI+
)−1
x, x ∈ H,
where L+ is a self-adjoint extension L which acts in a certain Hilbert space H+ containing H as a
subspace, I+ is the identity operator on H+, and P+ is the orthogonal projection operator from H+
onto H. It is known [15] that an operator-valued function Rλ (Imλ 6= 0) is a generalized resolvent
of a symmetric operator L if and only if it can be represented as
(Rλx, y)H =
+∞∫
−∞
d (Fµx, y)
µ− λ
, x, y ∈ H,
where Fµ is a generalized spectral function of the operator L, i.e., µ 7→ Fµ is an operator-valued
function Fµ defined on R and taking values in the space of continuous linear operators in H with the
following properties:
(1) for µ2 > µ1, the difference Fµ2 − Fµ1 is a bounded nonnegative operator;
(2) Fµ+ = Fµ for any real µ;
(3) for any x ∈ H there holds
lim
µ→−∞
||Fµx||H = 0, lim
µ→+∞
||Fµx− x||H = 0.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 5
634 O. O. KONSTANTINOV
The following theorem provides a description of all generalized resolvents of the operator Lmin.
Theorem 5. 1. Every generalized resolvent Rλ of the operator Lmin in the half-plane Imλ < 0
acts by the rule Rλh = y, where y is the solution of the boundary-value problem
l(y) = λy + h,
(K(λ)− I) Γ[1]f + i (K(λ) + I) Γ[2]f = 0.
Here h(x) ∈ L2([a, b],Cs) and K(λ) is an (ms ×ms)-matrix-valued function which is holomorph
in the lower half-plane and satisfy ||K(λ)|| ≤ 1.
2. In the half-plane Imλ > 0, every generalized resolvent of Lmin acts by Rλh = y, where y is
the solution of the boundary-value problem
l(y) = λy + h,
(K(λ)− I) Γ[1]f − i (K(λ) + I) Γ[2]f = 0.
Here h(x) ∈ L2([a, b],Cs) and K(λ) and K(λ) is an (ms ×ms)-matrix-valued function which is
holomorph in the upper half-plane and satisfy ||K(λ)|| ≤ 1.
The parametrization of the generalized resolvents by the matrix-valued functions K is bijective.
Proof. The result directly follows from Lemma 3 and [14] (Theorem 1) which prove a description
of generalized resolvents in terms of boundary triplets. We need only to take as an auxiliary Hilbert
space Cms and as the operator γy := {Γ[1]y,Γ[2]y}.
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Received 26.01.15
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