On Commutative Systems of Nonselfadjoint Unbounded Linear Operators
For a commutative system of nonselfadjoint unbounded operators A₁, A₂ the concept of colligation and associated open system is given. For these open systems, the consistency conditions are established and the conservation laws are obtained.
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2009
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irk-123456789-1065442016-10-01T03:01:56Z On Commutative Systems of Nonselfadjoint Unbounded Linear Operators Zolotarev, V.A. For a commutative system of nonselfadjoint unbounded operators A₁, A₂ the concept of colligation and associated open system is given. For these open systems, the consistency conditions are established and the conservation laws are obtained. 2009 Article On Commutative Systems of Nonselfadjoint Unbounded Linear Operators / V.A. Zolotarev // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 3. — С. 275-295. — Бібліогр.: 12 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106544 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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For a commutative system of nonselfadjoint unbounded operators A₁, A₂ the concept of colligation and associated open system is given. For these open systems, the consistency conditions are established and the conservation laws are obtained. |
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Zolotarev, V.A. |
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Zolotarev, V.A. On Commutative Systems of Nonselfadjoint Unbounded Linear Operators Журнал математической физики, анализа, геометрии |
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Zolotarev, V.A. |
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Zolotarev, V.A. |
| title |
On Commutative Systems of Nonselfadjoint Unbounded Linear Operators |
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On Commutative Systems of Nonselfadjoint Unbounded Linear Operators |
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On Commutative Systems of Nonselfadjoint Unbounded Linear Operators |
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On Commutative Systems of Nonselfadjoint Unbounded Linear Operators |
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On Commutative Systems of Nonselfadjoint Unbounded Linear Operators |
| title_sort |
on commutative systems of nonselfadjoint unbounded linear operators |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2009 |
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http://dspace.nbuv.gov.ua/handle/123456789/106544 |
| citation_txt |
On Commutative Systems of Nonselfadjoint Unbounded Linear Operators / V.A. Zolotarev // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 3. — С. 275-295. — Бібліогр.: 12 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT zolotarevva oncommutativesystemsofnonselfadjointunboundedlinearoperators |
| first_indexed |
2025-07-07T18:37:45Z |
| last_indexed |
2025-07-07T18:37:45Z |
| _version_ |
1837014414566359040 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2009, vol. 5, No. 3, pp. 275–295
On Commutative Systems of Nonselfadjoint Unbounded
Linear Operators
V.A. Zolotarev
Department of Mechanics and Mathematics, V.N. Karazin Kharkiv National University
4 Svobody Sq., Kharkiv, 61077, Ukraine
E-mail:Vladimir.A.Zolotarev@univer.kharkov.ua
Received September 8, 2008
For a commutative system of nonselfadjoint unbounded operators A1,
A2 the concept of colligation and associated open system is given. For these
open systems, the consistency conditions are established and the conserva-
tion laws are obtained.
Key words: commutative system, nonselfadjoint unbounded linear
operator.
Mathematics Subject Classification 2000: 47A48.
The study of nonselfadjoint unbounded operators originates from classic works
on the extensions of symmetric operators in Hilbert spaces by G. von Neumann
and M.G. Krein. However, only beginning with the work by M.S. Livs̆ic [1] the
study of this class of operators gained a proper technique — the characteristic
function. Further development of these methods was found in the works by
A.V. Kuzhel [2, 3] and A.V. Shtraus [4]. Nonselfadjoint unbounded operators
in rigged Hilbert spaces were studied by E.R. Tsekanovski and Yu.L. Shmul’yan
[5]. A somewhat different approach to the study of unbounded nonselfadjoint
operators, based on the analysis of the space of boundary values, was taken by
V.A. Derkach and M.M. Malamud [6] and resulted in the analytical formalism
for studying the properties of Weyl functions. The Shrödinger dissipative opera-
tor in the context of functional model was analyzed by B.S. Pavlov [7]. As for
commutative systems of unbounded nonselfadjoint operators, there have not been
appropriate approaches for studying. In the paper the methods of studying this
class of operators are presented.
For the commutative systems of nonselfadjoint bounded operators, M.S. Livs̆ic
suggested an effective method resulted in the construction of functional and tri-
angular models [8, 9]. The method is based on the generalization of the notion
of colligation of these systems of operators and on the study of the consistency
c© V.A. Zolotarev, 2009
V.A. Zolotarev
conditions for open systems. Therefore, it seems natural to give proper construc-
tions for the case of commutative systems of unbounded nonselfadjoint operators
as well. The paper is organized as follows. In Section 1 essential facts from
the theory of nonselfadjoint unbounded operators are given, in particular, the
notion of colligation and associated open system. In Section 2 these results are
generalized for the commutative systems of unbounded nonselfadjoint operators.
1. Preliminary Information
I. The pioneer work by M.S. Livs̆ic [1], where nonselfadjoint unbounded ope-
rators were studied, marked the beginning of the researches undertaken in this
branch of functional analysis which found its further fruitful development in the
papers [2–7]. The definition below plays an important role and it is an analogue
of those given in [10] for the unbounded case.
Definition 1. Let A be a linear operator acting in a separable Hilbert space H
such that: a) the domain D(A) of an operator A is dense in H, D(A) = H;
b) an operator A is dense in H; c) there exists a nonempty domain Ω (⊂ (C\R))
such that the resolvent Rα = (A − αI)−1 is regular for all α ∈ Ω. Consider
E±, the Hilbert spaces, ψ− : E− → H, ψ+ : H → E+, K : E− → E+, σ± :
E± → E±, linear bounded operators and selfadjoint operators σ±, σ± = σ∗±, that
are boundedly invertible. A collection
∆ = ∆(α) =
(
σ−,H ⊕ E−,
[
A ψ−
ψ+ K
]
,H ⊕ E+, σ+
)
(1.1)
is said to be the colligation of an unbounded operator A if there exists α ∈ Ω such
that
1. 2 Imα · ψ∗−ψ− = K∗σ+K − σ−, 2 Imα · ψ+ψ
∗
+ = Kσ−1
− K∗ − σ−1
+ ;
2. the operators
ϕ+ = ψ+(A− αI) : D(A)→ E+,
ϕ∗
− = ψ∗
− (A
∗ − ᾱI) : D (A∗)→ E− (1.2)
are such that
3. K∗σ+ϕ+ + ψ∗−(A− ᾱI) = 0, Kσ−1
− ϕ∗− + ψ+ (A∗ − αI) = 0;
4. 2 Im〈Ah, h〉 = 〈σ+ϕ+h, ϕ+h〉, ∀h ∈ D(A),
−2 Im
〈
A∗h̃, h̃
〉
=
〈
σ−1
− ϕ∗−h̃, ϕ̃∗−h̃
〉
, ∀h ∈ D (A∗) .
First of all, show that an arbitrary operator A satisfying the conditions a)–c)
of the given definition may always be included in the colligation ∆ (1.1). Really,
let Bα and B̃α be selfadjoint bounded operators (see [2, 3])
Bα = iRα − iR∗
α + 2 Imα · R∗
αRα,
B̃α = iRα − iR∗
α + 2 Imα · RαR
∗
α.
(1.3)
276 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3
On Commutative Systems of Nonselfadjoint Unbounded Linear Operators
Consider the subspace
E+ = E+(α) = span {Bαh : h ∈ H} ,
E− = E−(α) = span
{
B̃αh : h ∈ H
}
.
(1.4)
It is obvious that the bounded operator
Tα = I + i2 Imα ·Rα (1.5)
maps the subspace E+ in E− since [2, 3]
TαBα = B̃αTα. (1.6)
Moreover, it is easy to see that
〈Ah, f〉 − 〈h,Af〉 = i 〈Bα(A− αI)h, (A − αI)f〉 , ∀h, f ∈ (A), (1.7)〈
A∗h̃, f̃
〉
−
〈
h̃, A∗f̃
〉
= −i
〈
Bα (A∗ − ᾱI) h̃, (A∗ − ᾱI) f̃
〉
, forallh̃, f̃ ∈ D (A∗) .
Now specifying the operators
ψ∗
− =
√∣∣∣B̃α
∣∣∣, ψ+ =
√
|Bα|, σ− = signBα, σ+ = signBα, (1.8)
where
√|B| and signB for a selfadjoint bounded operator are understood in
terms of the spectral decomposition B [10], from (1.6) we obtain
Tα
√
|Bα| =
√∣∣∣B̃α
∣∣∣Tα, Tα · signBα = sign B̃α · Tα. (1.9)
Setting K = −σ+T
∗
α, it is easy to verify that the colligation relations (1.2) follow
from (1.3), (1.7), and (1.9).
R e m a r k 1.1. If the operators Bα, B̃α (1.1) are boundedly invertible on
E+ and E− (1.4), then setting
ψ− = B̃α, ψ+ = Pα, σ− = B̃α, σ+ = Bα, K = −T ∗
α, (1.10)
where Pα is an orthoprojector on E+ (1.4), it is easy to verify that the conditions
of colligation (1.2) also take place.
II. Open systems associated with colligations [10] play an important role in
the study of nonselfadjoint operators. Let u−(t) be a vector-function from E−
defined on [0, T ], and h be a vector from H. The open system F∆ = {R∆, S∆}
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 277
V.A. Zolotarev
associated with the colligation ∆ (1.1) is the pair of maps [10], R∆ : H⊕E− → H,
S∆ : H ⊕ E− → H ⊕ E+,
F∆ :
{
R∆ (h, u−(t)) = h(t),
S∆ (h, u−(t)) = (hT , u+(t)) ,
defined as follows. The operator R∆ is specified by using the Cauchy problem
R∆ :
i
d
dt
h(t) +Ay(t) = αψ−u(t),
y(t) = h(t) + ψ−u−(t) ∈ D(A),
h(0) = h, t ∈ [0, T ],
(1.11)
and the transfer mapping S∆ has the form:
S∆ :
{
u+(t) = Ku−(t)− iϕ+y(t),
hT = h(T ), t ∈ [0, T ], (1.12)
where h(t) is a solution of (1.11), and y(t) ∈ D(A) is defined by h(t) and u−(t)
using formula (1.11).
R e m a r k 1.2. If u−(t) ≡ 0 in (1.11), then y(t) = h(t) ∈ D(A), and we
obtain the Cauchy problem{
i
d
dt
h(t) +Ah(t) = 0,
h(0) = h ∈ D(A), t ∈ [0, T ].
The solvability of this Cauchy problem is equivalent to the existence of the
strongly continuous semigroup Zt = exp{itA}, with h(t) = Zth. So, the so-
lutions of the Cauchy problem (1.11) exist if the operator A is an infinitesimal
operator of the strongly continuous semigroup Zt. The well-known theorem by
Miyader–Feller–Fillips [11] gives the necessary and sufficient conditions for the
closed densely defined operator A, when the resolvent Rλ = (A−λI)−1 is regular
in the semiplane C−(ω) = {λ ∈ C : ω + Imλ < 0}, ω ∈ R (|ω| < ∞), and,
moreover, when λ ∈ C−(ω), the estimations
‖Rn
λ‖ ≤M |ω + Imλ|−n, ∀n ∈ Z+,
take place.
R e m a r k 1.3. Let u−(t) be differentiable. Then (1.11) yields that y(t) also
has the derivative and satisfies the nonhomogenous equation
i
d
dt
y(t) +Ay(t) = ψ−
(
iu′−(t) + αu−(t)
)
.
The solution of this equation exists if:
278 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3
On Commutative Systems of Nonselfadjoint Unbounded Linear Operators
1) the operator A meets the conditions of the Miyader–Feller–Fillips theorem;
2) y(0) = h+ ψ−u−(0) ∈ D(A);
3) the function f(t) = ψ−
(
iu′−(t) + αu−(t)
)
is twice continuously differen-
tiable, and f(0) ∈ D(A).
So, if the conditions 1)–3) are met, then there always exists y(t), and thus
h(t) (1.11) exists also.
Theorem 1.1. The conservation law
‖h‖2 +
T∫
0
〈σ−u−(t), u−(t)〉 dt = ‖hT ‖2 +
T∫
0
〈σ+u+(t), u+(t)〉 dt (1.13)
holds for the open system F∆ = {R∆, S∆} (1.11), (1.12) associated with the
colligation ∆ (1.1).
P r o o f. Equation (1.11) yields
d
dt
‖h(t)‖2 = 〈iAy(t)− iαψ−u−(t), , y(t) − ψ−u−(t)〉+ 〈y(t)− ψ−u−(t),
iAy(t)− iαψ−u−(t)〉 = −2 Im〈Ay(t), y(t)〉 − 2 Imα ‖ψ−u−(t)‖2
−〈i(A− ᾱI)y(t), ψ−u−(t)〉 − 〈ψ−u−(t), i(A − ᾱI)y(t)〉 .
Using the relations 1–4 (1.2), we get
d
dt
‖h(t)‖2 = −〈σ+ϕ+y(t), ϕ+y(t)〉+ 〈σ−u−(t), σ−u−(t)〉 − 〈J+Ku−(t),
Ku−(t)〉 + 〈iσ+ϕ+y(t),Ku−(t)〉+ 〈J+Ku−(t), iϕ+y(t)〉
= 〈σ−u−(t), u−(t)〉 − 〈σ+ [Ku−(t)− iϕ+y(t)] , [Ku−(t)− iϕ+y(t)]〉 .
As a result, we obtain the following conservation law:
d
dt
‖h(t)‖2 = 〈σ−u−(t), u−(t)〉 − 〈σ+u+(t), u+(t)〉 , (1.14)
which yields (1.13) after integration.
Consider an open system dual to F∆ = {R∆, S∆}. Denote by ũ+(t) a vector
function from E+ defined on [0, T ] (0 < T < ∞), and by h̃ — a vector from H.
A pair of mappings R+
∆ : H + E+ → H, S+
∆ : H + E+ → H + E−,
F+
∆ :
R+
∆
(
h̃, ũ+(t)
)
= h̃(t),
S∆
(
h̃, ũ+(t)
)
=
(
h̃0, ũ−(t)
)
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 279
V.A. Zolotarev
is said to be the dual open system F+
∆ =
{
R+
∆, S
+
∆
}
associated with the colligation
∆ (1.1). Besides, R+
∆ is specified by the Cauchy problem
R+
∆ :
i
d
dt
h̃(t)−A∗ỹ(t) = −ᾱψ∗
+ũ+(t),
ỹ(t) = ψ∗
+ũ+(t)− h̃(t) ∈ D (A∗) ,
h̃(T ) = h̃, t ∈ [0, T ];
(1.15)
and S+
∆ is given by
S+
∆ :
{
ũ−(t) = K∗ũ+(t) + iϕ∗−ỹ(t),
h̃0 = h̃(0), t ∈ [0, T ], (1.16)
h̃(t) and ỹ(t) can be found from (1.15). Similarly to Remarks 1.2 and 1.3, it is
easy to establish the solvability of (1.15) under natural restrictions on the class
of operators A∗ and the class of functions ũ+(t).
Theorem 1.2. Let F+
∆ =
{
R+
∆, S
+
∆
}
be the dual open system (1.15), (1.16)
of the colligation ∆ (1.1). Then
‖h̃‖2 +
T∫
0
〈
σ−1
+ ũ+(t), ũ+(t)
〉
dt =
∥∥∥h̃0
∥∥∥2
+
T∫
0
〈
σ−1
− ũ−(t), ũ−(t)
〉
dt. (1.17)
P r o o f. Equation (1.15) yields
d
dt
‖h̃(t)‖2 =
〈−iA∗ỹ(t) + iᾱψ∗
+ũ+(t), ψ∗
+ũ+(t)− ỹ(t)
〉
+
〈
ψ∗
+ũ+(t)− ỹ(t),−iA∗ỹ(t) + iᾱψ∗
+ũ+(t)
〉
= −2 Im 〈A∗ỹ(t), ỹ(t)〉
+2 Imα
∥∥ψ∗
−ũ+(t)
∥∥2− 〈
i (A∗− αI) ỹ(t), ψ∗
+ũ+(t)
〉−〈ψ∗
+ũ+(t), i (A∗− αI) ỹ(t)
〉
.
Using the second relations of 1–4 (1.2), we have
d
dt
‖h(t)‖2 =
〈
σ−1
− ϕ∗
−ỹ(t), ϕ
∗
−ỹ(t)
〉
+
〈
σ−1
− K∗ũ+(t),K∗ũ+(t)
〉
− 〈σ−1
+ ũ+(t), ũ+(t)
〉
+
〈
iσ−1
− ϕ∗
−ỹ(t),K
∗ũ+(t)
〉
+
〈
σ−1
− K∗ũ+(t), iϕ∗
−ỹ(t)
〉
=
〈
σ−1
−
[
K∗ũ+(t) + iϕ∗
−ỹ(t)
]
,
[
K∗ũ+(t) + iϕ∗
−ỹ(t)
]〉− 〈
σ−1
+ ũ+(t), ũ+(t)
〉
.
Therefore
d
dt
∥∥∥h̃(t)∥∥∥2
=
〈
σ−1
− ũ−(t), ũ−(t)
〉− 〈
σ−1
+ ũ+(t), ũ+(t)
〉
, (1.18)
which yields (1.17) after integration.
280 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3
On Commutative Systems of Nonselfadjoint Unbounded Linear Operators
The following theorem establishes an important correlation between the open
systems F∆ (1.11), (1.12) and F+
∆ (1.15), (1.16).
Theorem 1.3. Let h(t) and u+(t) be defined by u−(t) using the relations
(1.11), (1.12) of the open system F∆, and the vector functions h̃(t) and ũ−(t)
be constructed by equalities (1.15), (1.16) of the dual open system F+
∆ . Then the
equality
〈
hT , h̃
〉
+
T∫
0
〈u+(t), ũ+(t)〉 dt =
〈
h, h̃0
〉
+
T∫
0
〈u−(t), ũ−(t)〉 dt (1.19)
is true.
P r o o f. From (1.11) and (1.15) it follows that
d
dt
〈
h(t), h̃(t)
〉
=
〈
iAy(t)− iαψ−u−(t), ψ∗
+ũ+(t)− ỹ(t)
〉
+
〈
y(t)− ψ−u−(t),−iAỹ(t) + iᾱψ∗
+ũ+(t)
〉
=
〈
i(A− αI)y(t), ψ∗
+ũ+(t)
〉
+ 〈ψ−u−(t), i (A∗ − ᾱI) ỹ(t)〉 .
Taking into account 2 (1.2) and (1.12), (1.16), we obtain
d
dt
〈
h(t), h̃(t)
〉
= 〈iϕ+y(t), ũ+(t)〉+
〈
u−(t), iϕ∗
−ỹ(t)
〉
= 〈Ku−(t)− u+(t), ũ+(t)〉+ 〈u−(t), ũ−(t)−K∗ũ+(t〉
= 〈u−(t), ũ−(t)〉 − 〈u+(t), ũ+(t)〉
and, consequently,
d
dt
〈
h(t), h̃(t)
〉
= 〈u−(t), ũ−(t)〉 − 〈u+(t), ũ+(t)〉 . (1.20)
Equality (1.19) follows from (1.20) after integration.
III. Let u−(t) in the open system F∆ (1.11) be the plane wave u−(t) =
eiλtu−(0). And let the vector functions h(t), y(t), and u+(t) depend on t in
a similar way: h(t) = eiλth, y(t) = eiλty, u+(t) = eiλtu+(0), where h, y ∈ H, and
u+(0) do not depend on t. Then (1.11), (1.12) yield
−λh+Ay = αψ−u−(0),
h− y = −ψ−u−(0),
u+(0) = Ku−(0) − iϕ+y,
(1.21)
where y ∈ D(A).
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 281
V.A. Zolotarev
Thus, if λ ∈ Ω, then
y = (α− λ)(A− λI)−1ψ−u−(0),
h = −(A− αI)(A− λI)−1ψ−u−(0),
u+(0) = S∆(λ)u−(0),
(1.22)
where S∆(λ) is a characteristic function of the colligation ∆ (1.1),
S∆(λ) = K + i(λ− α)ψ+(A− αI)(A − λI)−1ψ−. (1.23)
The function S∆(λ) is normalized at the point λ = α, S∆(α) = K. Consider the
operator function
Tλ,α = (A− αI)(A− λI)−1 = I + (λ− α)Rλ. (1.24)
Then S∆(λ) can be written in the form
S∆(λ) = K + i(λ− α)ψ+Tλ,αψ−. (1.25)
From (1.14) it follows easily (see, for instance, [10]) that
σ− − S∗
∆(w)σ+S∆(λ)
i(λ− w̄)
= ψ∗
−T
∗
w,αTλ,αψ−. (1.26)
Analogously, if ũ+(t) in the dual open system F+
∆ =
{
R+
∆, S
+
∆
}
(1.15), (1.16)
is given by ũ+(t) = eiλ̄(t−T )ũ+(T ), where ũ+(T ) is an independent of t vector from
E+, and if h̃(t), ỹ(t), ũ−(t) also have the same dependency on t, h̃(t) = eiλ̄(t−T )h̃,
ỹ(t) = eiλ̄(t−T )ỹ, ũ−(t) = eiλ̄(t−T )u−(T ), then (1.15), (1.16) yield
λ̄h̃+A∗ỹ = ᾱψ∗
+ũ+(T ),
h̃+ ỹ = ψ∗
+ũ+(T ),
ũ−(T ) = K∗ũ+(T ) + iϕ∗−ỹ,
(1.27)
where ỹ ∈ D (A∗). Hence, if λ ∈ Ω, this implies
ỹ =
(
ᾱ− λ̄
) (
A∗ − λ̄I
)−1
ψ∗
+ũ+(T ),
h̃ = (A∗ − ᾱI)
(
A∗ − λ̄I
)−1
ψ∗
+ũ+(T ),
ũ−(T ) =
+
S∆ (λ)ũ+(T ),
(1.28)
where the function
+
S∆ (λ) is given by
+
S∆ (λ) = K∗ − i
(
λ̄− ᾱ
)
ψ∗
− (A
∗ − ᾱI)
(
A∗ − λ̄I
)−1
ψ∗
+. (1.29)
282 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3
On Commutative Systems of Nonselfadjoint Unbounded Linear Operators
It is obvious that the functions S∆(λ) (1.25) and
+
S∆ (λ) (1.29) satisfy the relation
+
S∆ (λ) = S∗
∆(λ). (1.30)
Using (1.12), it is easy to show that
S∆(w)σ−1
− S∗
∆(λ)− σ−1
+
i
(
λ̄− w
) = ψ+Tw,αT
∗
λ,αψ
∗
+. (1.31)
Finally, (1.18), with (1.29) being taken into account, implies
S∆(λ)− S∆(w)
i(λ− w)
= ψ+Tw,αTλ,αψ−. (1.32)
IV. Consider the operator function K(λ,w) : E− ⊕E+ → E− ⊕ E+,
K(λ,w) =
σ1 − S∗
∆(w)σ+S∆(λ)
i (λ− w̄)
S∗
∆(λ)− S∗
∆(w)
i
(
w̄ − λ̄
)
S∆(λ)− S∆(w)
i(λ− w)
S∆(w)σ−1
− S∗
∆(λ)− σ−1
+
i
(
λ̄− w
)
, (1.33)
assuming that λ, w, α ∈ Ω. The formulae (1.26), (1.31), (1.32) imply that the
kernel K(λ,w) (1.33) is positively defined [10].
A subspace H1 ⊆ H is said to be reducing for a densely defined operator A,
if at every point of regularity λ ∈ Ω of the resolvent Rλ = (A−λI)−1 there takes
place RλP1 = P1Rλ, where P1 is an orthoprojector on H1. For the colligation ∆
(1.1), define the subspace
H1 = span
{
Rλψ−u− +R∗
wψ
∗
+u+ : u± ∈ E±, λ, w ∈ Ω} . (1.34)
Theorem 1.4. The subspace H1 (1.34) reduces the operator A, besides, the
contraction of A on H0 = H �H1 is a selfadjoint operator.
The proof of the statement follows from the colligation relations (1.2) and it
is standard [10].
A colligation ∆ (1.1) is said to be simple if H = H1 (1.34).
Let two colligations ∆ and ∆′ be given such that E± = E′±, σ± = σ′±, K = K ′,
and, moreover, α = α′ ∈ Ω ∩ Ω′ (�= ∅). These colligations are called unitarily
equivalent [10] if there exists a unitary operator U : H → H ′ such that
UA = A′U, UD(A) = D
(
A′) , UA∗ =
(
A′)∗ U, UD
(
A′)∗ = D
((
A′)) ,
Uψ− = ψ′
−, ψ′
+U = ψ+.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 283
V.A. Zolotarev
Theorem 1.5. Let ∆ and ∆′ be simple colligations, E± = E′±, σ± = σ′±
of which are invertible, and α = α′ ∈ Ω ∩ Ω′ (�= ∅). Then if in some neighbor-
hood Uδ(α) ⊂ Ω ∩ Ω′ of the point α the characteristic functions (1.23) coincide,
S∆(λ) = S∆′(λ), then the colligations ∆ and ∆′ are unitarily equivalent.
The proof of the theorem follows easily from (1.26), (1.31), and (1.32).
Thus, the characteristic function S∆(λ) (1.23) defines the colligation ∆ (1.1)
up to the unitary equivalency.
V. Let us describe a class of functions generated by the characteristic func-
tions S∆(λ) of the colligations ∆ (1.1). Consider the following functions from H1
(1.34):
F (λ, u−) = Tλ,αψ−u−, F̃ (λ, u+) = T ∗
λ,αψ
∗
+u+, (1.35)
where u± ∈ E±, and λ, α ∈ Ω.
Theorem 1.6. The operator Tw,λ (1.24) (λ, w ∈ Ω) acts on F (λ, u−) and
F̃ (λ, u+) in the following way:
1) Tw,λF (λ, u−) = F (w, u−) ;
2) T ∗
w,λF̃ (λ, u+) = F̃ (w, u+) ;
3) Tw,λ̄F̃ (λ, u+) = −F (w, σ−1
− S∆(λ)u+
)
;
4) T ∗
w,λ̄
F (λ, u+) = −F̃ (w, σ+S∆(λ)u−) ,
(1.36)
where S∆(λ) is the characteristic function of (1.25), and λ, w, α ∈ Ω.
P r o o f. Equations 1), 2) from (1.36) follow from the chain identity Tw,λTλ,α =
Tw,α. Since
Tw,λ̄F̃ (λ, u+) = Tw,λ̄T
∗
λ,αψ
∗
+u+ = Tw,αTα,λ̄T
∗
λ,αψ
∗
+u+,
we have to find the expression
Tα,λ̄T
∗
λ,αψ
∗
+u+ =
(
I +
(
α− λ̄
)
Rα
)
T ∗
λ,αψ
∗
+u+ = T ∗
λ,αψ
∗
+u+
+(α− ᾱ)RαT
∗
λ,αψ
∗
+u+ +
(
ᾱ− λ̄
)
RαT
∗
λ,αψ
∗
+u+.
Taking into account the equality
Rα = −iψ−σ−1
− ψ− +R∗
α + (α− ᾱ)RαR
∗
α,
which follows from 4. (1.2) and (1.7), and using the Hilbert identity Rλ = Tλ,αRα
for the resolvents Rλ, we obtain
Tα,λ̄T
∗
λ,αψ
∗
+u+ = ψ∗
+u+ + (α− ᾱ)Rαψ
∗
+u+ + ψ−σ−1
− [K∗ − S∗
∆(λ)] u+.
284 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3
On Commutative Systems of Nonselfadjoint Unbounded Linear Operators
Finally, since ψ∗
+ + (α− ᾱ)Rαψ
∗
+u+ + ψ+σ
−1
− K∗ = 0 (see 2, 3 (1.2)), we obtain
that
Tα,λ̄T
∗
λ,αψ
∗
+u+ = −ψ−σ−1
− S∗
∆(λ)u+,
which proves 3) from (1.36). The proof of 4) from (1.36) is analogous.
This theorem implies the statement below.
Theorem 1.7. The family of operators Tw,z (1.24) (w, z ∈ Ω) acts on the
functions F (λ, u−) and F (λ, u+) (1.35) in the following way:
1) Tw,zF (λ, u−) =
w − z
w − λ
F (w, u−) +
λ− z
λ−w
F (λ, u−) ;
2) T ∗
w,zF̃ (λ, u+) =
w̄ − z̄
w̄ − λ̄
F̃ (w, u+) +
λ̄− z̄
λ̄− w̄
F̃ (λ, u+) ;
3) Tw,zF̃ (λ, u+) =
λ̄− z
λ̄− w
F̃ (λ, u+)− w − z
w − λ̄
F
(
w, σ−1
− S∗
∆(λ)u+
)
;
4) T ∗
w,zF (λ, u−) =
λ− z̄
λ− w̄
F (λ, u−)− w̄ − z̄
w̄ − λ
F̃ (w, σ+S∆(λ)u−) ;
(1.37)
for all u± ∈ E± and all λ, w, z, α ∈ Ω.
The class Ωα (σ−, σ+). Let E± be Hilbert spaces, σ± be selfadjoint invertible
operators in E and α ∈ C \ R. An operator function S(λ): E− → E+ belongs to
the class Ωα (σ−, σ+) if:
1) the function S(λ) is holomorphic in some neighborhood Uδ(α) = {λ ∈ C :
|λ− α| < δ} of the point α and S(α) �= 0;
2) the kernel K(λ,w) (1.33) is Hermitian positive for all λ, w ∈ Uδ(α).
It is obvious that the characteristic function S∆(λ) (1.25) belongs to the class
Ωα (σ−, σ+).
Theorem 1.8. Let an operator function S(λ): E− → E+ belong to the class
Ωα (σ−, σ+). Then there exists such a colligation ∆ (1.1) that its characteristic
function S∆(λ) (1.25) coincides with S(λ), S∆(λ) = S(λ), for all λ ∈ Uδ(α).
P r o o f. Following [10], denote by eλf the “δ-function”, the support of
which is concentrated at the point λ ∈ Uδ(α) and eλf in this point takes the
value f = (u−, u+) ∈ E− ⊕ E+. Consider the manifold L generated by the finite
linear combinations
N∑
1
eλk
fk (N ∈ N). Using K(λ,w) (1.33), on L specify the
nonnegative bilinear form
〈eλf, ewg〉K def= 〈K(λ,w)f, g〉E−⊕E+ . (1.38)
Closing L by the norm generated by form (1.38) and factorizing by the kernel of
metric (1.38), we obtain the Hilbert space HK [10].
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 285
V.A. Zolotarev
Proceeding from (1.37), in HK specify the family of operators Tw,z:
Tw,zeλf = ew
(
w − z
w − λ
u− − w − z
w − λ̄
σ−1
− S∗(λ)u+, 0
)
+eλ
(
λ− z
λ− w
u−,
λ̄− z
λ̄− w
u+
)
,
(1.39)
where λ, w ∈ Uδ(α), and f = (u−, u+). It is easy to show that
T ∗
w,zeλf = ew
(
0,
w̄ − z̄
w̄ − λ̄
u+ − w̄ − z̄
w̄ − λ
σ+S(λ)u−
)
+eλ
(
λ− z̄
λ− w̄
u−,
λ̄− z̄
λ̄− w̄
u+
)
.
(1.40)
Let
K = S(λ), ψ−u− = eαu−, ψ∗
+u+ = eαu+. (1.41)
It is obvious that the colligation relations 1 (1.2) are true. Really,
〈
ψ−u−, ψ−u′−
〉
=
〈
K(α,α)u−, u′−
〉
=
〈
σ− −K∗σ+K
i (α− ᾱ)
u−, u′−
〉
,
which proves the first condition in 1. (1.2). Simple calculations show that
ψ∗−eλf =
σ− −K∗σ+S(λ)
i(λ− ᾱ)
u− +
S∗(λ)−K∗
i
(
ᾱ− λ̄
) u+,
ψ+eλf =
S(λ)−K
i(λ− α)
u− +
Kσ−1
− S∗(λ)− σ−1
+
i
(
λ̄− α
) u+.
(1.42)
Since the first relation in 3 (1.2) can be written as K∗σ+ψ+ + ψ∗−Tα,ᾱ = 0, its
proof follows easily from (1.39) and (1.42).
Relations 4 (1.2) are proven in a similar way after being rewritten in terms of
Tα,ᾱ.
From (1.39), it is easy to calculate how the resolvent Rw = (w−z)−1 (Tw,z − I)
acts on the elements eλf , which, in virtue of the relation ARw = I+wRw, results
in the conclusion that the operator A in HK has the form
Aeλf = eλ
(
λu−, λ̄u+
)
, (1.43)
where the domain D(A) is
D(A) =
{ N∑
p=1
eλpfp ∈ HK : λp ∈ Uδ(α), fp =
(
up
−, u
p
+
) ∈ E− ⊕ E+,
up
− = σ−1
− S∗ (λp) u
p
+, 1 ≤ p ≤ N,N ≤ ∞
}
.
(1.44)
286 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3
On Commutative Systems of Nonselfadjoint Unbounded Linear Operators
Construct the colligation
∆K =
(
σ−,HK ⊕ E−,
[
A ψ−
ψ+ K
]
,HK ⊕ E+, σ+
)
, (1.45)
where K, ψ−, ψ+, and A are given by formulae (1.41) and (1.43), (1.44) corre-
spondingly. Finally, show that the characteristic function S∆K
(λ) (1.25) of the
colligation ∆K (1.45) coincides with S(λ). Equations (1.39), (1.41) imply
Tλ,αψ−u− = Tλ,αeα (u−, 0) = eλ (u−, 0) .
Using the structure of the operator ψ+ (1.42), we obtain that
ψ+Tλ,αψ−u− =
S(λ)−K
i(λ− α)
u−,
which concludes the proof.
Conservative and not only (passive and others) systems from one variable
were studied in [12].
2. Commutative Colligations and Open Systems.
Systems of Unbounded Operators
I. In a Hilbert space, consider a commutative system of the linear unbounded
operators {A1, A2}, where the domain D (Ap) of each operator Ap is dense in H,
D (Ap) = H, p = 1, 2, and the commutativity of the operators A1, A2 is un-
derstood in terms of interchangeability of resolvents, [R1, R2] = 0, where
Rp = Rp(α) = (Ap − αI)−1, p = 1, 2, assuming that α is a point of regularity of
resolvents R1(λ), R2(λ). Obviously, [R1, R2] = 0 yields [R1(λ), R2(w)] = 0 for all
λ and w belonging to the joint domain of regularity of R1(λ) and R2(λ). The fol-
lowing definition plays an important role hereinafter and it is a generalization of
Definition 1 (see Sect. 1) for the commutative case.
Definition 2. Let a system of the linear unbounded operators {A1, A2} be
given in a Hilbert space H such that: a) the domain D (Ap) of the operator Ap
is dense in H, D (Ap) = H, p = 1, 2; b) every operator Ap is closed in H,
p = 1, 2; c) there exists a nonempty domain Ω ⊂ C\R such that the resolvents
Rp(λ) = (Ap − λI)−1 are regular for all λ ∈ Ω, p = 1, 2; d) the resolvents R1
(= R1(α)), R2 (= R2(α)) commute at least at one point α ∈ Ω.
And let the Hilbert spaces E±, the linear bounded operators ψ− : E− → H,
ψ+ : H → E+ and
{
σ−p
}2
1
,
{
τ−p
}2
1
, {Np}2
1, Γ : E− → E−,
{
σ+
p
}2
1
,
{
τ+
p
}2
1
,
{
Ñp
}2
1
,
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 287
V.A. Zolotarev
Γ̃ : E+ → E+ be given, where
{
σ±p
}2
1
and
{
τ±p
}2
1
are selfadjoint. The totality
∆ = ∆(α) =
({
σ−p
}2
1
,
{
τ−p
}2
1
, {Np}2
1 ,Γ,H ⊕ E−,
{[
Ap ψ−
ψ+ K
]}2
1
,
H ⊕ E+, Γ̃,
{
Ñp
}2
1
,
{
τ+
p
}2
1
,
{
σ+
p
}2
1
) (2.1)
is said to be the commutative colligation if there exists α ∈ Ω such that:
1) 2 Imα ·N∗
pψ
∗−ψ−Np = K∗σ+
p K − σ−p , 2 Imα · Ñpψ+ψ
∗
+Ñ
∗
p = Kτ−p K∗ − τ+
p ;
2) the operators
ϕp
+ = ψ+ (Ap − αI) : D (Ap)→ E+,(
ϕp
−
)∗ = ψ∗
−
(
A∗
p − ᾱI
)
: D
(
A∗
p
)→ E−
are such that:
3) K∗σ+
p ϕ
p
+ +N∗
pψ
∗− (Ap − ᾱI) = 0, Kτ−p
(
ϕp
−
)∗ + Ñpψ1
(
A∗
p − αI
)
= 0;
4) 2 Im 〈Aphp, hp〉 =
〈
σ+
p ϕ
p
+hp, ϕ
p
+hp
〉
, ∀hp ∈ D (Ap) ,
−2 Im
〈
A∗
ph̃p, h̃p
〉
=
〈
τ−p
(
ϕp
−
)∗
h̃p,
(
ϕp
−
)∗
h̃p
〉
, ∀h̃p ∈ D (Ap) , (2.2)
where p = 1, 2. And, moreover, the relations:
5) R2ψ−N1 −R1ψ−N2 = ψ−Γ, Ñ1ψ+R2 − Ñ2ψ+R1 = Γ̃ψ̃+;
6) Γ̃K −KΓ = i
(
Ñ1ψ+ψ−N2 − Ñ2ψ+ψ−N1
)
;
7) KNp = ÑpK, are true, where Rp = Rp(α), p = 1, 2.
Show that for every operator system {A1, A2} satisfying the suppositions
a)–d) there always exist such Hilbert spaces E± and corresponding operators ψ±,
K,
{
σ±p
}2
1
,
{
τ±p
}2
1
, {Np}2
1,
{
Ñp
}2
1
, Γ, Γ̃, that the relations 1–7 (2.1) hold. To do
this, similarly to (1.5), consider two commuting bounded operators
Tp = I + i2 Imα ·Rp, p = 1, 2, (2.3)
and let (see (1.3))
Bp = iRp − iR∗
p + 2 Imα ·R∗
pRp, p = 1, 2,
B̃p = iRp − iR∗
p + 2 Imα · RpR
∗
p , p = 1, 2.
(2.4)
It is easy to see that
TpBp = B̃pTp, p = 1, 2. (2.5)
Analogously as in (1.7),
2 Im 〈Aphp, hp〉 = 〈Bp (Ap − αI)hp, (Ap − αI)hp〉 ,∀hp ∈ D (Ap) , (2.6)
288 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3
On Commutative Systems of Nonselfadjoint Unbounded Linear Operators
−2 Im
〈
A∗
ph̃p, h̃p
〉
=
〈
B̃p
(
A∗
p − ᾱI
)
h̃p,
(
A∗
p − ᾱI
)
h̃p
〉
,∀h̃p ∈ D
(
A∗
p
)
take place, where p = 1, 2. Define the bounded operators in H
σ+
p = Bp, p = 1, 2, σ−1 = T2B̃1T
∗
2 , σ−2 = T1B2T
∗
1 ,
N1 = B̃1T
∗
2 , N2 = B̃2T
∗
1 , Γ = B̃1R
∗
2 − B̃2R
∗
1,
τ−p = B̃p, p = 1, 2, τ+
1 = T ∗
2B1T2, τ+
2 = T ∗
1B2T1,
(2.7)
Ñ1 = T ∗
2B1, Ñ2 = T ∗
1B2, Γ̃ = R∗
2B1 −R∗
1B2.
Consider the Hilbert spaces
E− = span
{
B̃1H + B̃2H +N∗
1H +N∗
2H
}
,
E+ = span
{
B1H +B2H + Ñ1H + Ñ2H
}
,
(2.8)
and let
K = −T ∗
1T
∗
2 , ψ− = P−, ψ+ = P+, (2.9)
where P± are the orthoprojectors on E± (2.8). It is easy to see that the relations
1–4 (2.2) follow from equalities (2.3)–(2.7). Consequently, 7 (2.2) follows from
(2.5). By simple calculations we may check the conditions 5, 6. Finally, it is easy
to show that the operator K (2.9) maps E− in (2.8).
R e m a r k 2.1. Equations 2, 4 (2.2) imply
Bp = ψ∗
+σ
+
p ψ+, B̃p = ψ−τ−p ψ
∗
−, p = 1, 2, (2.10)
by virtue of the density of the domains D (Ap), D
(
A∗
p
)
, p = 1, 2.
II. Before turning to the open system associated with the commutative col-
ligation ∆ (2.1), which is a two-variable analogue of the system F∆ = {R∆, S∆}
(1.11), (1.12), write the main equations (1.11), (1.12) in other form. Since (1.11),
(1.12) are given by
i∂th(t) +Ay(t) = αψ−u−(t),
y(t) = h(t) + ψ−u−(t) ∈ D(A),
h(0) = h, t ∈ [0, T ],
u+(t) = Ku−(t)− iϕ+y(t),
(2.11)
where ∂t =
∂
∂t
, by multiplying the second equality by α and subtracting it from
the first one, we obtain
Lh(t) + ŷ(t) = 0, (2.12)
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 289
V.A. Zolotarev
where the operator L and the function ŷ(t) are such that
L = i∂t + α, y(t) = Rαŷ(t) ∈ D(A). (2.13)
Therefore equations (2.11) can be written in the following form:
Lh(t) + ŷ(t) = 0,
Rαŷ(t) = h(t) + ψ−u−(t) ∈ D(A),
h(0) = h, t ∈ [0, T ],
u+(t) = Ku−(t)− iψ+ŷ(t).
(2.14)
The first two equalities yield that ŷ(t) is a solution of the equation
LRαŷ(t) + ŷ(t) = ψ−Lu−(t). (2.15)
Applying the operator L to equalities (2.11), we obtain
−i∂tŷ(t) +ALy(t) = αψ−Lu−(t),
Ly(t) = −ŷ(t) + ψ−Lu−(t) ∈ D(A),
Lu+(t) = KLu−(t)− iϕ+Ly(t).
(2.16)
Since these equalities coincide with the relations (2.11) after substitutions
h(t) → −ŷ(t), y(t) → Ly(t), u±(t) → Lu±(t), by using the conservation law
(1.14), we obtain
∂t ‖ŷ(t)‖2 = 〈σ−Lu−(t), Lu−(t)〉 − 〈σ+Lu+(t), Lu+(t)〉 . (2.17)
III. Denote by D = [0, T1] × [0, T2] the rectangle in R
2
+, 0 < Tp < ∞,
p = 1, 2, and let u−(t) be a vector function in E− specified when t = (t1, t2) ∈ D.
The system of the relations
R∆ :
i∂1h1(t) +A1y1(t) = αψ−N1u−(t),
y1(t) = h1(t) + ψ−N1u−(t) ∈ D (A1) ,
i∂2h2(t) +A2y2(t) = αψ−N2u−(t),
y2(t) = h2(t) + ψ−N2u−(t) ∈ D (A2) ,
h1(0) = h1, h2(0) = h2, t = (t1, t2) ∈ D,
(2.18)
where ∂p = ∂/∂tp, p = 1, 2, is said to be the open system F∆ = {R∆, S∆}
associated with the colligation ∆ (2.1). Let the vector functions y1(t) and y2(t)
be such that
y1(t) = R1y(t), y2(t) = R2y(t), (2.19)
and y(t) be a vector function from H. Thus, the functions {yp(t)}2
1 have a joint
generatrix y(t), moreover, (2.19) implies that
R1y2(t) = R2y1(t). (2.20)
290 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3
On Commutative Systems of Nonselfadjoint Unbounded Linear Operators
As for the initial data h1 and h2, we assume that
hp = Rpy(0) − ψ−Npu−(0), p = 1, 2. (2.21)
The mapping S∆ is given by
S∆ : u+(t) = Ku−(t)− iψ+y(t). (2.22)
Similarly to (2.13), consider the differential operators
Lp = i∂p + α, p = 1, 2. (2.23)
Then the main equations (2.18) can be written in the following form:
L1h1(t) + y(t) = 0,
R1y(t) = h1(t) + ψ−N1u−(t) ∈ D (A1) ,
L2h2(t) + y(t) = 0,
R2y(t) = h2(t) + ψ−N2u−(t) ∈ D (A2) ,
(2.24)
which is similar to (2.14). Consequently, L1h1(t) = −y(t) = L2h2(t). Therefore,
taking into account (2.23) and (2.18), we obtain that (cf. (2.15))
R1L1y(t) + y(t) = ψ−N1L1u−(t),
R2L2y(t) + y(t) = ψ−N2L2u−(t),
y(0) = y0, t = (t1, t2) ∈ D,
u+(t) = Ku−(t)− iψ+y(t).
(2.25)
So, if the vector function y(t) satisfies the relations (2.25), then the functions
h1(t), h2(t) (2.24) as well as y1(t), y2(t) (2.19) can be defined by it.
Theorem 2.1. The system of equations (2.18) is consistent if the vector
function u−(t) is a solution of the equation
{N1L1 −N2L2 + ΓL1L2}u−(t) = 0, (2.26)
given that (2.19), (2.21) hold and Lp has the form of (2.23), p = 1, 2.
P r o o f. The consistency condition follows from (2.25) if one takes into
account the commutativity [R1L1, R2L2] = 0. Since
L1R1L2R2y(t) = y(t)− ψ−N1L1u−(t) +R−ψ−N2L1L2u−(t)
and similarly
L2R2L1R1y(t) = y(t)− ψ−N2L2u−(t) +R2ψ−N1L1L2u−(t),
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 291
V.A. Zolotarev
then, subtracting these equalities, we obtain
ψ− (N1L1 −N2L2) u−(t) + (R2ψ−N1 −R1ψ−N2)L1L2u−(t) = 0.
Taking into account 5 (2.2), we obtain
ψ− {N1L1 −N2L2 + ΓL1L2}u−(t) = 0,
which proves (2.26).
Theorem 2.2. If for the vector function y(t) equation (2.25) takes place,
and u−(t) is a solution of (2.26), then u+(t) (2.22) satisfies the equation{
Ñ1L1 − Ñ2L1 + Γ̃L1L2
}
u+(t) = 0. (2.27)
P r o o f. Calculate[
Ñ1L1 − Ñ2L2
]
u+(t) = K [N1L1 −N2L2] u−(t)
−i
[
Ñ1ψ+L1 − Ñ2ψ+L2
]
y(t) = −KΓL1L2u−(t)
−iÑψ+L1 (ψ−N2L2u−(t)− L2R2y(t)) + iN2ψ+L2 (ψ−N1L1u−(t)− L1R1y(t))
=
{
−KΓ− iÑ1ψ+ψ−N2 + iÑ2ψ+ψ−N1
}
L1L2u−(t)
+i
(
Ñ1ψ+R2 − Ñ2ψ+R1
)
L1L2y(t) = −Γ̃KL1L2u−(t) + iΓ̃ψ+L1L2y(t)
= −Γ̃L1L2u+(t),
which proves (2.27) in virtue of (2.26), (2.25) and 5, 6 (2.2).
Theorem 2.3. For the open system F∆ = {R∆, S∆} (2.18), (2.22) associated
with the colligation ∆ (2.1), the following conservation laws are true:
1) ∂p ‖hp(t)‖2 =
〈
σ−p u−(t), u−(t)
〉− 〈
σ+
p u+(t), u+(t)
〉
, p = 1, 2;
2) ∂2
{〈
σ−1 L1u−(t), L1u−(t)
〉− 〈
σ+
1 L1u+(t), L1u+(t)
〉}
= ∂1
{〈
σ−2 L2u−(t), L2u−(t)
〉− 〈
σ+
2 L2u+(t), L2u+(t)
〉}
.
(2.28)
P r o o f. The relations 1) (2.27) are proved in the same way as equality
(1.14). Since the conservation laws 1) (2.28) can be written as (see (2.17))
∂p‖y(t)‖2 =
〈
σ−p Lpu−(t), Lpu−(t)
〉− 〈
σ+
p Lpu+(t), Lpu+(t),
〉
, p = 1, 2,
then, taking into account the equality of mixed derivatives ∂2∂1‖y(t)‖2 =
∂1∂2‖y(t)‖2, we obtain 2) (2.28).
292 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3
On Commutative Systems of Nonselfadjoint Unbounded Linear Operators
IV. Along with the open system F∆ = {R∆, S∆} (2.18), (2.22) characterizing
the evolution generated by {A1, A2}, consider also the dual situation responding
to the dynamics specified by the adjoint operator system {A∗
1, A
∗
2}.
Let a vector function ũ+(t) in E+, t = (t1, t2), be specified in the rectangle
D = [0, T1]× [0, T2] from R
2
+, 0 < Tp <∞, p = 1, 2. The equation system
R+
∆ :
i∂1h̃1(t)−A∗
1ỹ1(t) = −ᾱψ∗
+Ñ
∗
1 ũ+(t),
ỹ1(t) = ψ∗
+Ñ
∗
1 ũ+(t)− h̃1(t) ∈ D (A∗
1) ,
i∂2h̃2(t)−A∗
2ỹ2(t) = −ᾱψ∗
+Ñ
∗
2 ũ+(t),
ỹ2(t) = ψ∗
+Ñ
∗
2u+(t)− h̃2(t) ∈ D (A∗
2) ,
h̃1(T ) = h̃1, h̃2(T ) = h̃2, t = (t1, t2) ∈ D,
(2.29)
where, as usually, ∂p = ∂/∂tp, p = 1, 2 and ỹ1(t), ỹ2(t) are such that
ỹ1(t) = R∗
1ỹ(t), ỹ2(t) = R∗
2ỹ(t) (2.30)
is said to be the dual open system F+
∆ =
{
R+
∆, S
+
∆
}
associated with the colligation
∆ (2.1). Thus the vector functions {ỹp(t)}2
1 have the joint generatrix ỹ(t) ∈ H,
besides,
R∗
1ỹ2(t) = R∗
2ỹ1(t). (2.31)
The initial data h̃1, h̃2 of problem (2.28) can be found from the equalities
h̃p = ψ∗
+Ñ
∗
pu+(T )−R∗
pỹ(T ), p = 1, 2. (2.32)
The mapping S+
∆ is given by
S+
∆ : ũ−(t) = K∗ũ+(t) + iψ∗
−ỹ(t). (2.33)
Consider (see (2.23)) the differential operators
L+
p = i∂p + ᾱ, p = 1, 2. (2.34)
Similarly to the considerations in Section 3, we obtain that the vector function
ỹ(t) satisfies the relations
R∗
1L
+
1 ỹ(t) + ỹ(t) = ψ∗
+Ñ
∗
1L
+
1 ũ+(t),
R∗
2L
∗
2ỹ(t) + ỹ(t) = ψ∗
+Ñ
∗
2L
+
2 ũ+(t),
ỹ(T ) = ỹT , t = (t1, t2) ∈ D,
ũ−(t) = K∗ũ+(t) + iψ∗−ỹ(t).
(2.35)
It is not difficult to obtain the analogues of Theorems 2.1–2.3 using the equa-
lities above.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 3 293
V.A. Zolotarev
Theorem 2.4. The system of the equations (2.29) of the dual open system
F+
∆ =
{
R+
∆, S
+
∆
}
(2.29)–(2.33) corresponding to the commutative colligation ∆
(2.1) is consistent if ũ+(t) satisfies the equation{
Ñ∗
1L
+
1 − Ñ∗
2L
+
2 + Γ̃
∗L+
1 L
+
2
}
ũ+(t) = 0 (2.36)
under the condition that (2.30) and (2.32) take place.
The proof of this theorem is similar to that of Theorem 2.1.
Theorem 2.5. Let ỹ(t) be the solution of (2.35), and ũ+(t) satisfy equation
(2.36). Then for the vector functions ũ−(t) (2.33), we have{
N∗
1L
+
1 −N∗
2L
+
2 + Γ
∗L+
1 L
+
2
}
ũ−(t) = 0. (2.37)
Theorem 2.6. For the dual open system F+
∆ =
{
R+
∆, S
+
∆
}
(2.29)–(2.33),
the conservation laws
1) ∂p
∥∥∥h̃p(t)
∥∥∥2
=
〈
τ−p ũ−(t), ũ−(t)
〉− 〈
τ+
p ũ+(t), ũ+(t)
〉
, p = 1, 2;
2) ∂2
{〈
τ−1 L
+
1 ũ−(t), L
+
1 ũ−(t)
〉− 〈
τ+
1 L
+
1 ũ+(t), ũ+(t)
〉}
= ∂1
{〈
τ−2 L
+
2 ũ−(t), L
+
1 ũ−(t)
〉− 〈
τ+
2 L
+
2 ũ+(t), L+
2 ũ+(t)
〉} (2.38)
hold.
As it will be shown later, relations 2) (2.28) and 2) (2.38) play an important
role in the study of the properties of characteristic functions for commutative
systems of the linear unbounded operators {Ak}n
1 for any n ∈ N.
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