Функциональные модели алгебры Ли системы линейных операторов {A₁,A₂,A₃}
Построены функциональные модели неабелевой трехмерной нильпотентной алгебры Ли линейных операторов, действующих в гильбертовом пространстве H. Созданные алгебры {A₁,A₂,A₃} удовлетворяют соотношениям [A₁, A₃] = 0, [A₂,A₃] = 0, [A₁,A₂] = iA₃, причем A₁x₁+A₂x₂+A₃x₃ не является диссипативным для всех x...
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| Zitieren: | Функциональные модели алгебры Ли системы линейных операторов {A₁,A₂,A₃} / В.А. Кузнецова // Український математичний вісник. — 2008. — Т. 5, № 4. — С. 507-523. — Бібліогр.: 9 назв. — рос. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860261702121029632 |
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| author | Кузнецова, В.А. |
| author_facet | Кузнецова, В.А. |
| citation_txt | Функциональные модели алгебры Ли системы линейных операторов {A₁,A₂,A₃} / В.А. Кузнецова // Український математичний вісник. — 2008. — Т. 5, № 4. — С. 507-523. — Бібліогр.: 9 назв. — рос. |
| collection | DSpace DC |
| description | Построены функциональные модели неабелевой трехмерной нильпотентной алгебры Ли линейных операторов, действующих в гильбертовом пространстве H. Созданные алгебры {A₁,A₂,A₃} удовлетворяют соотношениям [A₁, A₃] = 0, [A₂,A₃] = 0, [A₁,A₂] = iA₃, причем A₁x₁+A₂x₂+A₃x₃ не является диссипативным для всех x = (x₁, x₂, x₃) ∊ R³, а пространство неэрмитовости G = span{(Ak − Ak*) h, k = 1, 2, 3, h ∊H} имеет размерность три.
Functional models are constructed for a non-Abelian nilpotent Lie algebra of linear operators acting in the Hilbert space H. The algebra generators {A₁,A₂,A₃} satisfy the relations [A₁, A₃] = 0, [A₂,A₃] = 0, [A₁,A₂] = iA₃, where A₁x₁+A₂x₂+A₃x₃ is not dissipative for all x = (x₁, x₂, x₃) ∊ R³, and the space of non-Hermiticity G = span {(Ak − Ak*) h, k = 1, 2, 3, h ∊ H} has dimension three.
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Ukrainian Mathematical Bulletin
Volume 5 (2008), № 4, 499 – 515 UMB
Functional models of the Lie algebra of a system
of linear operators {A1, A2, A3}
Victoriya A. Kuznetsova
(Presented by M. M. Malamud)
Abstract. Functional models are constructed for a non-Abelian nilpo-
tent Lie algebra of linear operators acting in the Hilbert space H.
The algebra generators {A1, A2, A3} satisfy the relations [A1, A3] = 0,
[A2, A3] = 0, [A1, A2] = iA3, where A1x1 + A2x2 + A3x3 is not dissi-
pative for all x = (x1, x2, x3) ∈ R
3, and the space of non-Hermiticity
G = span {(Ak − A∗
k)h, k = 1, 2, 3, h ∈ H} has dimension three.
2000 MSC. 47A48.
Key words and phrases. Functional models, Branges transformation,
Lie algebra.
Introduction
Functional models of contracting (dissipative) operators first con-
structed by B. Sz.-Nagy and C. Foiaš [5] represent the operators of mul-
tiplication by an independent variable in the special spaces of functions.
Construction of these models is associated with the Fourier transforma-
tion. For the non-dissipative operators, the construction of similar models
is based on the study of the Branges transformation [1, p. 152] [8, p. 126].
The characteristic function is the main analytic object, in terms of
which the functional models are constructed. L. L. Vaksman [7] showed
that if the structure constants of the Lie algebras of linear nonself-adjoint
operators are the same, and the corresponding characteristic functions
coincide, then these algebras are unitarily equivalent. Thus, the model
representations of a Lie algebra with assigned structure components built
by the characteristic function are unitarily isomorphic.
For the Lie algebra of linear operators {A1, A2} [A1, A2] = iA1 [6,
p. 10], the construction of functional models in the case where the oper-
ator A1, for example, is dissipative is also based on the Fourier transfor-
mation.
Received 17.10.2008
ISSN 1812 – 3309. c© Institute of Mathematics of NAS of Ukraine
500 Functional models...
In [3, pp. 54–60], the functional models for an arbitrary commutative
system of linear operators {A1, A2} were constructed, and the functional
models for an arbitrary Lie algebra of linear operators {A1, A2} were con-
structed in [4, pp. 176–185] without the assumption about the dissipative
property of the operators A1, A2. In this paper, we construct functional
models for the Lie algebra of linear operators {A1, A2, A3} satisfying the
relations [A1, A3] = 0, [A2, A3] = 0, [A1, A2] = iA3 in the case where
dimG = 3 [G = span{(Ak − A∗
k)h, k = 1, 2, 3, h ∈ H}] without the
assumption that the system contains dissipative operators.
1. Preliminary information
I. Consider a linear bounded operator A acting in a Hilbert space H.
We recall that the family
∆ = (A,H,ϕ,E, J) (1.1)
is said to be the local colligation [2, p. 11], [8, p. 18] if the relation
A−A∗ = iϕ∗Jϕ (1.2)
holds, where E is a Hilbert space, and ϕ, J are operators such that
ϕ : H → E, J : E → E; moreover, J = J∗ = J−1.
The function
S(λ) = I − iϕ(A− λI)−1ϕ∗J (1.3)
is said to be the characteristic function [8, p. 24] of a colligation ∆ (1.1).
Consider the case where dimE = 3, and J is given by
J =
1 0 0
0 −1 0
0 0 −1
; (1.4)
moreover, the spectrum of the operator A is real. Then it is well known
[8, p. 66], [2, p. 71] that S(λ) has the multiplicative representation
S(λ) = Sl(λ), Sx(λ) =
x
x∫
0
exp
{
iJdFt
λ− αt
}
, (1.5)
where αx is a real bounded function non-decreasing on [0, l], 0 < l <
∞, and Ft is a matrix-valued (3 × 3) non-decreasing function such that
trFx = x. Suppose that
dFx = ax dx, (1.6)
V. A. Kuznetsova 501
where the matrix ax is such that ax ≥ 0, trax = 1,
ax =
a11(x) a12(x) a13(x)
a21(x) a22(x) a23(x)
a31(x) a32(x) a33(x)
, aij = aji, (1.7)
and aij(x), i, j = 1, 3, are functions on [0, l].
Consider the following integral equation for the matrix-function
Mx(z):
Mx(z) + iz
x∫
0
Mt(z) dFtJ = I, (1.8)
where x ∈ [0, l], z ∈ C. It is easy to see that Mx(z) can be represented
by
Mx(z) = JS∗
x(z̄−1)J. (1.9)
Define the row-vector Lx(z) =
[
L1
x(z), L2
x(z), L3
x(z)
]
as a solution of
the integral equation
Lx(z) + iz
x∫
0
Lt(z) dFtJ = (1, 1, 0) = Lx(0), (1.10)
where z ∈ C. It is obvious that
Lx(z) = (1, 1, 0)Mx(z) = (1, 1, 0)JS∗
x(z̄−1)J. (1.11)
Consider the Hilbert space L2
3, l(Ft) [8, pp. 66–67]
L2
3, l(Fx) =
{
fx ∈ E3;
l∫
0
ft dFt f
∗
t <∞
}
(1.12)
assuming that the proper factorization by the metric kernel is already
carried out.
Define the kernel
Kx(z, w) =
i
π(z − w̄)
Lx(z)JL∗
x(w̄). (1.13)
It is obvious that
Kx(z, w) =
i
π(z − w̄)
(
L1
x(z)L1
x(w)−L2
x(z)L2
x(w)−L3
x(z)L3
x(w)
)
. (1.14)
The following theorem [8, pp. 118–119] takes place.
502 Functional models...
Theorem 1.1. The row-vector Lx(z) =
[
L1
x(z), L2
x(z), L3
x(z)
]
, which is
a non-trivial solution (Lx(z) 6= (1, 1, 0)) of the integral equation (1.10),
is such that
1) Lx(z) ∈ L2
3,a(Ft) for all a ∈ [0, l] and z ∈ C;
2) for all z ∈ C and x ∈ [0, l]
∣∣L1
x(z)
∣∣−
∣∣L2
x(z)
∣∣−
∣∣L3
3(z)
∣∣ =
≥ 0, Im z > 0
= 0, Im z = 0
≤ 0, Im z < 0
(1.15)
is true.
II. Consider the following basis {ek}
3
1 in E3 :
e1 = (1, 1, 0);
e2 = (1, 0, 1); (1.16)
e3 = (5, 4, 3).
Similarly to (1.10), we define the vector-functions Nx(z) = [N1
x(z),
N2
x(z), N3
x(z)] and Rx(z) =
[
R1
x(z), R2
x(z), R3
x(z)
]
as solutions of the in-
tegral equations
Nx(z) + iz
x∫
0
Nt(z) dFtJ = (1, 0, 1) = Nx(0), (1.17)
Rx(z) + iz
x∫
0
Rt(z) dFtJ = (5, 4, 3) = Rx(0) (1.18)
when z ∈ C and x ∈ [0, l]. For Nx(z) and Rx(z), the relations
Nx(z) = (1, 0, 1)Mx(z) = (1, 0, 1)JS∗
x
(
z̄−1
)
J, (1.19)
Rx(z) = (5, 4, 3)Mx(z) = (5, 4, 3)JS∗
x
(
z̄−1
)
J (1.20)
hold, as well as (1.11).
For the functions Nx(z) and Rx(z), the analog of Theorem 1.1 is true.
V. A. Kuznetsova 503
Definition 1.1. Denote, by B(L(z)), the linear space of the entire func-
tions F (z), z ∈ C, such that
A)
F (z) = BLft =
1
π
l∫
0
ft dFt L
∗
t (z̄), (1.21)
where BL is the Branges transform [8, p. 125] of the function ft ∈
L2
3,l(Ft);
B) and let
‖F (z)‖B(L(z)) = ‖ft‖L2
3,l
(Ft). (1.22)
Theorem 1.2 ([1, p. 152], [8, pp. 126–127]). Consider the family
of Hilbert spaces B(La(z)), where Lx(z) is the vector-function which is
a solution of the integral equation (1.10) on the interval [0, l] for some
matrix-valued measure Ft. Match every function ht =
(
h1(t), h2(t), h3(t)
)
from L2
3,l(Ft) with the function given by
F (z) =
1
π
a∫
0
ht dFt L
∗
t (z̄), (1.23)
where a is the inner point of the interval [0, l], 0 < a < l. Then F (z) ∈
B (La(z)) .
Definition 1.2. The transform F (z) (1.21) of the function ht ∈ L2
3,l(Ft)
is said to be the Branges transform of the function ht by the measure Ft.
Remark 1.1. Similarly, the Hilbert spaces B(N(z)) and B(R(z)) are
defined. The Branges transformation of the function ht ∈ L2
3,l(Ft) in the
space B(N(z)) is given by
Φ1(z) = BNht =
1
π
l∫
0
ht dFtN
∗
t (z) (1.24)
and the Branges transformation of the function ht ∈ L2
3,l(Ft) in the space
B(R(z)), correspondingly, is
Φ2(z) = BRht =
1
π
l∫
0
ht dFtR
∗
t (z), (1.25)
where z ∈ C.
504 Functional models...
III. Consider the matrix T1
T1 =
1 −1 0
0 1 1
0 0 1
. (1.26)
Apply T1 from the right to Eq. (1.10),
LxT1 + iz
x∫
0
Lt(z) dFt JT1 = Lx(0)T1.
Since Lx(0)T1 = Nx(0), this relation can be rewritten as
Lx(z)T1 + iz
x∫
0
Lt(z)T1T
−1
1 dFt JT1 = Nx(0). (1.27)
Obviously, T−1
1 exists and is equal
T−1
1 =
1 1 −1
0 1 −1
0 0 1
.
It is easy to see that
JT1 = T̃1J, (1.28)
where
T̃1 =
1 1 0
0 1 1
0 0 1
, (1.29)
therefore
Lx(z)T1 + iz
x∫
0
Lt(z)T1T
−1
1 atT̃1J dt = Nx(0). (1.30)
Suppose
atT̃1 = T1at. (1.31)
Then relation (1.30) implies that Lx(z)T1 satisfies Eq. (1.17), and this
signifies in view of the uniqueness of the solution of (1.17) that
Lx(z)T1 = Nx(z), (1.32)
for all x ∈ [0, l], z ∈ C.
Consider Φ1(z) = BNft.
V. A. Kuznetsova 505
BNft =
1
π
l∫
0
ftat dtN
∗
t (z) =
1
π
l∫
0
ftat dt T̃
∗
1L
∗
1(z)
=
1
π
l∫
0
ftT̃
∗
1 at dtL
∗
t (z̄) = BL(ftT̃
∗
1 )
by virtue of (1.31).
Thus,
BNft = BL(ftT̃
∗
1 ). (1.33)
Denote, by ϕ1(t), the function
ϕ1(t) = ftT̃
∗
1 =
(
f1(t), f2(t), f3(t)
)
T̃ ∗
1 . (1.34)
It is obvious that ϕ1(t) belongs to the space L2
3,l (Ft), if ft ∈ L2
3,l(Ft). So,
Φ1(z) = BNft = BL(ftT̃
∗
1 ) = BLϕ1(t). (1.35)
Therefore, there exists the transformation ψ1 : B(L(z)) → B(N(z)),
given by the formula
(ψ1G)(z) = G1(z). (1.36)
Here, G(z) ∈ B(L(z)), and G1(z) ∈ B(N(z)), i.e. G(z) = BLft, where
ft ∈ L
2
3,l(Ft) and ψ1G(z) = ψ1BLft = G1(z). Since G1(z) ∈ B(N(z)),
we have G1(z) = BNft, where ft ∈ L2
3,l(Ft), ψ1BLft = BNft. Thus, by
virtue of (1.33),
ψ1BLft = BLT̃
∗
1 ft, (1.37)
i.e., ψ1BL = BLT̃
∗
1 and
ψ1 = BLT̃
∗
1 B−1
L . (1.38)
Definition 1.3. The transformation B−1
L is said to be inverse to the
Branges transformation BL for the function ft ∈ L2
3,l(Ft).
Consider ψ−1
1 : B(N(z)) → B(L(z)) and ψ−1
1 = BLT̃
∗−1
1 B−1
L , i.e.,
(
ψ−1
1 Φ1
)
(z) = ψ−1
1 BNft = ψ−1
1 BLT̃
∗
1 ft
= BLT̃
∗−1
1 B−1
L BLT̃
∗
1 ft = BLft = F̂1(z)
takes place for all functions Φ1(z) ∈ B(N(z)), where F̂1(z) ∈ B(L(z)).
Thus, there exists ĥt from L2
3,l(Ft) such that
506 Functional models...
F̂1(z) =
1
π
l∫
0
ĥt dFt L
∗
t (z), (1.39)
F̂1(z) = BLĥt. (1.40)
IV. Similar considerations can be carried out for the space B(R(z)).
Namely, there exists the matrix T2 given by
T2 =
2 3 0
3 1 3
0 0 1
. (1.41)
By applying T2 to Eq. (1.10) from the right, we get
Lx(z)T2 + iz
x∫
0
Lt(z)T2T
−1
2 dFtJT2 = Rx(0). (1.42)
Obviously, the matrix T−1
2 exists. It is easy to see that
JT2 = T̃2J, (1.43)
where
T̃2 =
2 −3 0
−3 1 3
0 0 1
. (1.44)
Therefore, supposing that
atT̃2 = T2at, (1.45)
we obtain that Lx(z)T2 satisfies Eq. (1.18). This signifies in view of the
uniqueness of the solution of (1.18) that
Lx(z)T2 = Px(z), (1.46)
for all x ∈ [0, l], z ∈ C.
In exactly the same way, let us consider the function ϕ2(t) given by
ϕ2(t) = ftT̃2. (1.47)
Similarly,
Φ2(z) = BRft =
1
π
l∫
0
ftat dtR
∗
t (z)
V. A. Kuznetsova 507
=
1
π
l∫
0
ftat dt T̃
∗
2L
∗
t (z) =
1
π
l∫
0
ftT̃
∗
2 at dt T1L
∗
t (z);
Φ2(z) = BL(ftT̃
∗
2 ), (1.48)
by virtue of (1.45).
That is, BRft = BLftT̃2 or BRft = BL(ftT̃
∗
2 ) = BLϕ2(t). Car-
rying out similar considerations, we obtain that there exists the map
ψ2 : B(R(z)) → B(L(z)) given by the formula
(ψ2G)(z) = G2(z), (1.49)
where G2(z) ∈ B(R(z)), G2(z) = BRft, ft ∈ L2
3,l(Ft) and
Ψ2BLft = BLT̃
∗
2 ft, (1.50)
i.e., ψ2BL = BLT̃
∗
2 и
ψ2 = BLT̃
∗
2 B−1
L . (1.51)
Consider ψ−1
2 : B(R(z)) → B(L(z)) and ψ−1
2 = BLT̃
∗−1
2 B−1
L , i.e.,
(
ψ−1
2 Φ2
)
(z) = ψ−1
2 BRft = ψ−1
2 BLT̃
∗
2 ft
= BLT̃
∗−1
2 (t)B−1
L BLT̃
∗
2 ft = BLft = F̂2(z),
where F̂2(z) ∈ B(L(z)) takes place for every function Φ2(z) ∈ B(R(z)).
Thus,
F̂2(z) =
1
π
l∫
0
ĥt dFt L
∗
t (z), (1.52)
F̂1(z) = BLĥt, (1.53)
where ĥt ∈ L2
3,l(F1).
Definition 1.4. The function ĥt =
(
ĥ1(t), ĥ2(t), ĥ3(t)
)
∈ L2
2,l(Ft) con-
structed by this rule is said to be the dual function to the function ht =(
h1(t), h2(t), h3(t)
)
∈ L2
3,l(Ft).
Remark 1.2.
Φ1(z) = (ψ1F̂1)(z), (1.54)
Φ2(z) = (ψ2F̂2)(z), (1.55)
takes place.
508 Functional models...
2. Triangular models of an operator system
V. Consider the commutative system of linear bounded operators
{A1, A2} acting in a Hilbert space H, i.e., the relation
[A1, A2] = A1A2 −A2A1 = 0. (2.1)
holds.
As is well known [2, pp. 11–15], the family
∆ = (A1, A2, H, ϕ,E, σ1, σ2, γ, γ̃) , (2.2)
where E is some Hilbert space, ϕ, σ1, σ2, γ, γ̃ are operators such that
ϕ : H → E, σ1 : E → E, σ2 : E → E, γ : E → E, γ̃ : E → E,
and σk = σ∗k, k = 1, 2, γ = γ∗, γ̃ = γ̃∗, is said to be the commutative
colligation if the relations
1. Ak −A∗
k = iϕ∗σϕ, k = 1, 2
2. γϕ = σ1ϕA
∗
2 − σ2ϕA
∗
1 (γ̃ϕ = σ1ϕA2 − σ2ϕA1)
3. γ − γ̃ = i(σ1ϕϕ
∗σ2 − σ2ϕϕ
∗σ1)
(2.3)
hold.
Definition 2.1. The matrix-function S(λ1) given by
S(λ1) = I − iϕ (A1 − λ1I)
−1 ϕ∗σ1, (2.4)
is said to be the characteristic function of colligation (2.2) corresponding
to the operator A1. If dim E = 3, and the spectrum of the operator A1
is real, then, for S(λ1) [2, p. 71], the multiplicative representation (1.5)
takes place.
Let σ1 = J, where J (1.4) and σ2 = σ. Then the intertwining condition
[9, p. 117]
(σλ1 + γ)JS(λ1) = S(λ1)(σλ1 + γ̃)J (2.5)
takes place for function (2.4).
Suppose that dF1 = atdt, where the matrix at is given by (1.7) and
is such that at ≥ 0 and trat = 1. Then the following theorem takes
place [9, p. 118].
Theorem 2.1. In order that the intertwining condition
(σλ+ γx)JSx(λ) = Sx(λ)(σλ+ γ̃)J, (2.6)
for the matrix-function Sx(λ) hold, it is necessary and sufficient that
1)
d
dx
γxJ = i[Jax, σJ ]γ0 = γ̃, (2.7)
2) [Jax, (σαx + γx)J ] = 0. (2.8)
V. A. Kuznetsova 509
VI. Consider now the system of linear bounded operators {A1, A2,
A3} in H such that
[A1, A3] = 0,
[A2, A3] = 0, (2.9)
[A1, A2] = 0.
The triangular model realization of the Lie algebra (2.9) in the space
L2
3,l(Ft) (1.12) is given by
Â1fx = fxJ(γx,1 + αxσ1) + i
l∫
x
ftat dt σ1,
Â2fx = fxJ(γx,2 + αxσ2) + i
l∫
x
ftat dt σ2, (2.10)
Â3fx = αxfx + i
l∫
x
ftat dt σ3.
In this case, we suppose that
σ3 = J,
σ2 =
0 1 0
1 0 1
0 1 0
, (2.11)
σ1 =
0 b 0
b 0 b
0 b 0
,
γx,1 =
β11(x) β12(x) β13(x)
β̄12(x) β22(x) β23(x)
β̄13(x) β̄23(x) β33(x)
, (2.12)
where b ∈ R, βij(x) are some functions, and γ0,1 = γ1. In addition,
γx,2 =
d11(x) d12(x) d13(x)
d12(x) d22(x) d23(x)
d13(x) d23(x) d33(x)
, (2.13)
where dij(x) are some functions, and γ0,2 = γ2. Moreover, the relation
γ2 − γ∗2 = iσ3 (2.14)
510 Functional models...
holds for γ2.
In order that the conditions of Theorem 2.1 hold for the commutative
operators {A1, A3} and {A2, A3}, namely, in order that (2.7) and (2.8)
take place and condition (2.9) hold, the matrix ax must be given by
ax =
1 − a2(x) ia1(x) a2(x)
−ia1(x) 1 − 2a2(x) −ia1(x)
a2(x) ia1(x) 3a2(x) − 1
, (2.15)
and γx,1 and γx,2 must satisfy the relation
γx,1 = bγx,2 + c, (2.16)
where c is a constant matrix given by
c =
−β − ib/2 −i/2 0
i/2 β + ib/2 −i/2
0 i/2 β + ib/2
. (2.17)
In this case, γ1, γ2 are
γ1 =
−β bα− i/2 0
bᾱ+ i/2 β bα− i/2
0 bᾱ+ i/2 β
,
γ2 =
i/2 α 0
ᾱ −i/2 α
0 ᾱ −i/2
,
(2.18)
where β ∈ R, α = ik, k ∈ R. The matrix γx,2 is such that
d
dx
γx,2 =
2ia1(x) 2(1 − a2(x)) 0
−2(1 − a2(x)) 4ia1(x) 2(3a2(x) − 1)
0 −2(3a2(x) − 1) −2ia1(x)
.
(2.19)
3. Functional models of the Lie algebra of
operators {Â1, Â2, Â3}
VII. Consider the operator system {Â1, Â2, Â3} (2.10) acting in
L2
3,l(Ft) (1.12); moreover, {σ1, σ2, σ3} (2.11), γ1, γ2 (2.18) respectively,
γx,1, γx,2 satisfy relation (2.16); moreover, relation (2.19) holds for γx,2.
Let us study how the action of each of the operators {Â1, Â2, Â3}
changes after the Branges transformation (1.21)
V. A. Kuznetsova 511
πÂ3F (z) =
l∫
0
( l∫
t
fs dFs J
)
dFtL
∗
t (z̄)
=
l∫
0
ft dFt
L∗
t (z̄) − L∗
t (0)
z
= π
F (z) − F (0)
z
.
That is,
Ã3F (z) =
F (z) − F (0)
z
, (3.1)
Ã3F (z) ∈ B(Ll(z)).
We now calculate πÂ1ft (πÂ2ft can be obtained similarly)
πÂ1F (z) =
l∫
0
(A1ft) dFt L
∗
t (z̄) =
l∫
0
ft dFt (A∗
1Lt(z))
∗
=
l∫
0
ft dFt
(
αtLt(z)Jσ1 + Lt(z)Jγt,1 − i
x∫
0
Ls(z) dFs σ1
)∗
.
By virtue of the integral equation (1.10), we obtain
πÂ1F (z) =
l∫
0
ft dFt
(
Lt(z) − Lt(0)
z
Jσ1 + Lt(z)Jγt,1
)∗
=
1
z
l∫
0
ft dFt (Lt(z)J(σ1 + γt,1z) − Lt(0)Jσ1)
∗ .
Remark 3.1. It is easy to see that
Lt(z)J(σ1 + γt,1z)|z=0 = Lt(0)Jσ1. (3.2)
Remark 3.2. It is shown earlier that, for the pair of the operators
{A1, A3} forming the commutative operator system, the conditions of
Theorem 2.1 are true, and, thus, the following intertwining property
takes place, namely:
(σ1λ+ γx,1)JSx(λ) = Sx(λ)(σ1λ+ γ1)J, (3.3)
and setting λ = 1
z
in this relation we obtain
(σ1 + γx,1z)JSx
(
z−1
)
= Sx
(
z−1
)
(σ1 + γ1z)J.
512 Functional models...
In view of relations (1.11), (1.17), and (1.18), we obtain
Lx(z)J(σ1 + γx,1z)
= (1, 1, 0)M(z)J(σ1 + γx,1z)
= (1, 1, 0)JS∗
x
(
z̄−1
)
JJ(σ1 + γx,1z)J
= (1, 1, 0)J(σ1 + γ1z)JS
∗
x
(
z̄−1
)
J
= (1, 1, 0)J(σ1 + γ1z)Mx(z).
This relation can be represented in the form
(1, 1, 0)J(σ1 + γ1z)Mx(z) =
3∑
j=1
ζj(z)ejMx(z), (3.4)
where ej (j = 1, 2, 3) are given by (1.16), and ζj(z), (j = 1, 2, 3) are
some functions from z, z ∈ C. Taking relations (1.11), (1.19), and (1.20)
into account, we obtain
3∑
j=1
ζj(z)ejMx(z) = ζ1(z)Lx(z) + ζ2(z)Nx(z) + ζ3(z)Rx(z),
i.e.,
Lx(z)J(σ1 + γx,1z) = ζ1(z)Lx(z) + ζ2(z)Nx(z) + ζ3(z)Rx(z). (3.5)
In the case where σ1 and γ1 are given by formulas (2.11) and (2.18),
ζj(z), (j = 1, 2, 3) are given by
ζ1(z) = pz − b; ζ2(z) = pz + b; ζ3(z) = −idz − b; (3.6)
where p = −β + id, d = (2bk − 1)/2, k : α = ik, k ∈ R. In addition,
ζj(z) (j = 1, 2, 3) at the point z = 0 are equal to
ζ1(0) = −b; ζ2(0) = b; ζ3(0) = −b. (3.7)
Thus,
πÂ1F (z) =
1
z
l∫
0
ft dFt (ζ1(z)Lt(z) − ζ1(0)Lt(0) + ζ2(z)N(z)
− ζ2(0)N(0) + ζ3(z)R(z) − ζ3(0)R(0))∗
=
1
z
{ζ̄1(z)F (z) − ζ̄1(0)F (0) + ζ̄2(z)Φ1(z)
V. A. Kuznetsova 513
− ζ̄2(0)Φ1(0) + ζ̄3(z)Φ2(z) − ζ̄3(0)Φ2(0)}.
Taking relations (1.54) and (1.55) into consideration, we obtain
Ã1F (z) = b
F (0) − F (z)
z
+ p̄F (z)
+ b
(
Ψ1F̂1
)
(z) −
(
Ψ1F̂1
)
(0)
z
+ p̄
(
Ψ1F̂1
)
(z)
+ b
(
Ψ2F̂2
)
(0) −
(
Ψ2F̂2
)
(z)
z
+ id
(
Ψ2F̂2
)
(z). (3.8)
By carrying on similar considerations for the operator Â2, we get
πÂ2F (z) =
l∫
0
(A2ft) dFt L
∗
t (z̄) =
l∫
0
ft dFt (A∗
1Lt(z))
∗
=
l∫
0
ft dFt
(
αtLt(z)Jσ2 + Lt(z)Jγt,2 − i
x∫
0
Ls(z) dFs σ2
)
.
In this case, the corresponding analogs of Remarks 3.1 and 3.2 are also
valid.
Consider Lt(z)J(σ2 + γt,2z):
Lx(z)J(σ2 + γx,2z) = (1, 1, 0)Mx(z)J(σ2 + γx,2z)
= (1, 1, 0)JS∗
x(z̄−1)JJ(σ2 + γx,2z)
∗J
= (1, 1, 0)J(σ2 + γ2z̄ − iσ3z̄)JS
∗
x(z̄−1)J.
Remark 3.3. (σ2 + γx,2z)
∗ = (σ2 + γ∗x,2z̄) and, consequently, there is γ∗2
in the relations. Since γ2 and γ∗2 satisfy relation (2.14), we have
(σ2 + γ∗2 z̄) = (σ2 + γ2z̄ − iσ3z̄). (3.9)
Taking σ3 from (2.11), we get (σ2 + γ∗2z) = (σ2 + γ2z̄) − iJz̄,
Lx(z)J(σ2 + γx,2z) = (1, 1, 0)J(σ2 + γ2z − iJz)Mx(z). (3.10)
As before, we have
Lx(z)J(σ2 + γx,2z) =
3∑
j=1
ηj(z)ejMx(z), (3.11)
514 Functional models...
where ej (j = 1, 2, 3) are given by (1.16), and ηj(z), (j = 1, 2, 3) are
some functions from z, z ∈ C
Lx(z)J(σ2 + γx,2z) = η1(z)Lx(z) + η2(z)Nx(z) + η3(z)Rx(z). (3.12)
In the case where σ2 and σ3 are defined by (2.11), ηj(z) (j = 1, 2, 3) are
given by
η1(z) = −1−iz (k + 1/2) ; η2(z) = 1+iz (k − 1/2) ; η3(z) = −1−ikz,
(3.13)
where k : α = ik, k ∈ R. Note that, when z = 0,
η1 = −1; η2 = 1; η3 = −1. (3.14)
Thus, similarly to the aforesaid for the operator Â1, we obtain
Ã2F (z) =
F (0) − F (z)
z
+
i
2
(1 + 2k)F (z)
+
(
Ψ1F̂1
)
(z) −
(
Ψ1F̂1
)
(0)
z
+
i
2
(1 − 2k)
(
Ψ1F̂1
)
(z)
+
(
Ψ2F̂2
)
(0) − (Ψ2F̂2)(z)
z
+ ik
(
Ψ2F̂2
)
(z). (3.15)
So, we obtain the following result.
Theorem 3.1. Let {Â1, Â2, Â3} be a system of the model operators
(2.10) acting in the space L2
3,l(Ft) (1.12) (dFt = atdt, (1.31), (1.45) take
place for at) satisfying the commutative relations (2.9); in addition, let
{σ1, σ2, σ3} be given by (2.11) and γ1, γ2, correspondingly, by (2.18); γx,1
and γx,2 satisfy relation (2.16), and let relation (2.19) be true for γx,2.
If F (z) ∈ B(L(z)) is the Branges transform of the function ht from
L2
3,l(Ft), and if F̂1(z) and F̂2(z) are the Branges transforms [by (1.36)
and (1.49), correspondingly] for the dual function ĥt (by Definition 1.4),
then the Branges transform (1.21) establishes the unitary equivalence be-
tween the triangular models {Â1, Â2, Â3} (2.10) and the functional models
{Ã1, Ã2, Â3} (3.14), (3.15), (3.1).
References
[1] L. de Branges, Hilbert Spaces of Entire Functions, Prentice-Hall, Englewood
Cliffs, 1968, 326 p.
[2] M. S. Livshits, A. A. Yantsevich, Theory of Operator Colligations in Hilbert
Spaces. Kharkov, 1971, 160 p. (in Russian).
[3] V. A. Naboka, Functional models of commutative system of linear operators //
Mat., Prikl. Mat. Mekh., (2003), N 602, 46–60 (in Russian).
V. A. Kuznetsova 515
[4] V. A. Naboka, On functional models of Lie algebra of nonself-adjoint operators //
Mat., Prikl. Mat. Mekh., (2004), N 645, 172–186 (in Russian).
[5] B. Sz.-Nagy, C. Foiaš, Analyse Harmonique des Operateurs de l’Espace de Hilbert,
Masson, Szeged, 1967.
[6] J.-P. Serre, Lie Algebras and Lie Groups, Benjamin, New York, 1965.
[7] L. L. Vaksman, On characteristic operator-functions of Lie algebras // Vestnik
Khark. Univ., Ser. Mat. Mekh., (1972), N 33, Iss. 37, 41–45 (in Russian).
[8] V. A. Zolotarev, Analytical Methods of Spectral Representations of Nonself-
Adjoint and Nonunitary Operators, Kharkov Nats. Univ., Kharkov, 2003, 342 p.
(in Russian).
[9] V. A. Zolotarev, Lax–Fillips scattering scheme on groups and functional models
of Lie algebras // Mat. Sbor. 183 (1992), N 5, 115–144 (in Russian).
Contact information
Victoriya
Aleksandrovna
Kuznetsova
V. N. Karazin Kharkov
National University,
Svobody sq., 4,
61077, Kharkov,
Ukraine
E-Mail: victoriya979@mail.ru
|
| id | nasplib_isofts_kiev_ua-123456789-10949 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1810-3200 |
| language | Russian |
| last_indexed | 2025-12-07T18:56:07Z |
| publishDate | 2008 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Кузнецова, В.А. 2010-08-10T10:16:23Z 2010-08-10T10:16:23Z 2008 Функциональные модели алгебры Ли системы линейных операторов {A₁,A₂,A₃} / В.А. Кузнецова // Український математичний вісник. — 2008. — Т. 5, № 4. — С. 507-523. — Бібліогр.: 9 назв. — рос. 1810-3200 https://nasplib.isofts.kiev.ua/handle/123456789/10949 Построены функциональные модели неабелевой трехмерной нильпотентной алгебры Ли линейных операторов, действующих в гильбертовом пространстве H. Созданные алгебры {A₁,A₂,A₃} удовлетворяют соотношениям [A₁, A₃] = 0, [A₂,A₃] = 0, [A₁,A₂] = iA₃, причем A₁x₁+A₂x₂+A₃x₃ не является диссипативным для всех x = (x₁, x₂, x₃) ∊ R³, а пространство неэрмитовости G = span{(Ak − Ak*) h, k = 1, 2, 3, h ∊H} имеет размерность три. Functional models are constructed for a non-Abelian nilpotent Lie algebra of linear operators acting in the Hilbert space H. The algebra generators {A₁,A₂,A₃} satisfy the relations [A₁, A₃] = 0, [A₂,A₃] = 0, [A₁,A₂] = iA₃, where A₁x₁+A₂x₂+A₃x₃ is not dissipative for all x = (x₁, x₂, x₃) ∊ R³, and the space of non-Hermiticity G = span {(Ak − Ak*) h, k = 1, 2, 3, h ∊ H} has dimension three. ru Інститут прикладної математики і механіки НАН України Функциональные модели алгебры Ли системы линейных операторов {A₁,A₂,A₃} Functional models of the Lie algebra of a system of linear operators {A₁,A₂,A₃} Article published earlier |
| spellingShingle | Функциональные модели алгебры Ли системы линейных операторов {A₁,A₂,A₃} Кузнецова, В.А. |
| title | Функциональные модели алгебры Ли системы линейных операторов {A₁,A₂,A₃} |
| title_alt | Functional models of the Lie algebra of a system of linear operators {A₁,A₂,A₃} |
| title_full | Функциональные модели алгебры Ли системы линейных операторов {A₁,A₂,A₃} |
| title_fullStr | Функциональные модели алгебры Ли системы линейных операторов {A₁,A₂,A₃} |
| title_full_unstemmed | Функциональные модели алгебры Ли системы линейных операторов {A₁,A₂,A₃} |
| title_short | Функциональные модели алгебры Ли системы линейных операторов {A₁,A₂,A₃} |
| title_sort | функциональные модели алгебры ли системы линейных операторов {a₁,a₂,a₃} |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/10949 |
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