On Two-Dimensional Model Representations of One Class of Commuting Operators
In the work by V. A. Zolotarev, Dokl. Akad. Nauk Arm. SSR, 63, No. 3, 136–140 (1976), a triangular model is constructed for a system of twice-commuting linear bounded completely nonself-adjoint operators {A 1, A 2} ([A 1, A 2] = 0, [A 1 ∗ , A 2] = 0) such that rank (A 1) I (A 2) I = 1 (2i(A k ) I...
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2014
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508046470414336 |
|---|---|
| author | Hatamleh, R. Zolotarev, V. A. Хатамлех, Р. Золотарьов, В. А. |
| author_facet | Hatamleh, R. Zolotarev, V. A. Хатамлех, Р. Золотарьов, В. А. |
| author_sort | Hatamleh, R. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T10:24:29Z |
| description | In the work by V. A. Zolotarev, Dokl. Akad. Nauk Arm. SSR, 63, No. 3, 136–140 (1976), a triangular model is constructed for a system of twice-commuting linear bounded completely nonself-adjoint operators {A 1, A 2} ([A 1, A 2] = 0, [A 1 ∗ , A 2] = 0) such that rank (A 1) I (A 2) I = 1 (2i(A k ) I = A k − A k ∗ , k = 1, 2) and the spectrum of each operator A k , k = 1, 2, is concentrated at zero. The indicated triangular model has the form of a system of operators of integration over the independent variable in L Ω 2 where the domain Ω = [0, a] × [0, b] is a compact set in ℝ2 bounded by the lines x = a and y = b and a decreasing smooth curve L connecting the points (0, b) and (a, 0). |
| first_indexed | 2026-03-24T02:18:59Z |
| format | Article |
| fulltext |
UDC 517.9
R. Hatamleh (Jadara Univ., Irbid-Jordan),
V. A. Zolotarev (Inst. Low Temperature Phys. and Eng. Nat. Acad. Sci. Ukraine; V. N. Karazin Kharkiv Nat. Univ.)
ON TWO-DIMENSIONAL MODEL REPRESENTATIONS
OF ONE CLASS OF COMMUTING OPERATORS
ПРО ДВОВИМIРНI МОДЕЛЬНI ЗОБРАЖЕННЯ ОДНОГО КЛАСУ
КОМУТУЮЧИХ ОПЕРАТОРIВ
In the work by Zolotarev V. A. “On triangular models of systems of twice commuting operators” (Dokl. Akad. Nauk
ArmSSR. – 1976. – 63, № 3. – P. 136 – 140 (in Russian)), a triangular model is constructed for the system of twice-
commuting linear bounded completely nonself-adjoint operators {A1, A2} ([A1, A2] = 0, [A∗
1, A2] = 0) such that
rank(A1)I(A2)I = 1 (2i(Ak)I = Ak −A∗
k, k = 1, 2) and the spectrum of each operator Ak, k = 1, 2, is concentrated at
zero. This triangular model has the form of a system of operators of integration over the independent variable in L2
Ω where
the domain Ω = [0, a]× [0, b] is a compact set in R2 bounded by the lines x = a and y = b and by a decreasing smooth
curve L connecting the points (0, b) and (a, 0).
У статтi Золотарьова В. О. „Про трикутнi моделi систем двiчi переставних операторiв” (Докл. АН АрмССР. – 1976.
– 63, № 3. – С. 136 – 140) для системи двiчi переставних лiнiйних обмежених цiлком несамоспряжених операторiв
{A1, A2} ([A1, A2] = 0, [A∗
1, A2] = 0) такої, що rank(A1)I(A2)I = 1 (2i(Ak)I = Ak − A∗
k, k = 1, 2) i спектр
кожного iз операторiв Ak, k = 1, 2, зосереджено в нулi, побудовано трикутну модель, яка є системою операторiв
iнтегрування по незалежнiй змiннiй в L2
Ω, де Ω = [0, a] × [0, b]. В данiй статтi одержано узагальнення цього
результату на випадок, коли область Ω модельного простору є компактом у R2, обмеженим прямими x = a, y = b i
спадною гладкою кривою L, що з’єднує точки (0, b) i (a, 0).
Triangular model [3 – 5] of nonself-adjoint bounded operator constructed first by M. S. Livs̆ic plays
an important role in several problems of spectral analysis for this operator class. In the simplest case,
this model represents the integration operator [3 – 5] acting in the space L2
(0,l). Generalization of
this result by M. S. Livs̆ic for the systems of twice-commuting nonself-adjoint operators {A1, A2},
A1A2 = A2A1, A
∗
1A2 = A2A
∗
1 is obtained in work [6]. Namely, it is specified that this class of
operator systems is realized by operators of integration by different variables in L2
Ω, where Ω =
= [0, a] × [0, b] is a rectangle (0 < a < ∞, 0 < b < ∞). This line of investigation receives its
development in the works [7, 8], where systems of nonself-adjoint operators, the commutators of
which C = [A1, A2] and D = [A∗1, A2] are nilpotent (Dm = 0, Cn = 0, n, m ∈ Z+), are studied.
In this case, the model operators are again the integrations by different variables in L2
Ω, besides, the
domain Ω represents the rectangle from which the series of rectangles adjoining the coordinate origin
and point (a, 0) are withdrawn. The problem of the construction of many-dimensional triangular
models when the domain Ω of model space is given by the smooth descending curve connecting the
points (0, b) and (a, 0) so far remained unsolved.
This paper is devoted to the solution of this problem, besides, we obtain generalization of the
well-known result by M. S. Livs̆ic (see Theorem 5).
I. Consider the continuous curve L in R2
+,
L = {(x, α(x)) : α(0) = b, α(a) = 0} ; (1)
specified by the smooth, monotonously decreasing function α(x) ∈ C1
[0,a] on [0, a] (0 < a, b < ∞).
Denote by ΩL the compact in R2
+ bounded by the curve L (1) and the lines x = a, y = b. Define the
c© R. HATAMLEH, V. A. ZOLOTAREV, 2014
108 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
ON TWO-DIMENSIONAL MODEL REPRESENTATIONS OF ONE CLASS OF COMMUTING OPERATORS 109
Hilbert space L2
ΩL
formed by the quadratically summable functions f(x, y),
L2
ΩL
df
=
f :
∫∫
ΩL
|f(x, y)|2dxdy <∞
. (2)
Specify the commutative system of linear bounded operators in L2
ΩL
,
(
Ã1f
)
(x, y) = i
a∫
x
f(t, y)dt,
(
Ã2f
)
(x, y) = i
b∫
y
f(x, s) ds. (3)
It is easy to see that
2
((
Ã1
)
I
f
)
(x, y) = χΩL
a∫
α−1(y)
f(t, y)dt,
2
((
Ã2
)
I
f
)
(x, y) = χΩL
b∫
α(x)
f(x, s) ds,
(4)
where α−1(y) is a smooth monotonously decreasing function on [0, b], which is reciprocal to α(x),
and χΩL
is the characteristic function of the set ΩL. (4) yields
L1
df
=
(
Ã1
)
I
L2
ΩL
=
{
f(y)χΩL
∈ L2
ΩL
}
,
L2
df
=
(
Ã2
)
I
L2
ΩL
=
{
g(x)χΩL
∈ L2
ΩL
}
.
(5)
It is obvious that
dimL0 = 1, L0
df
= L1 ∩ L2, (6)
and, besides, (3) yields that
Ã1L2 ⊆ L2, Ã2L1 ⊆ L1. (7)
Specify smooth monotonously increasing functions
λ(y)
df
= a− α−1(y), µ(x)
df
= b− α(x). (8)
The equalities
∫∫
ΩL
|f(y)χΩL
|2 dxdy =
b∫
0
|f(y)|2λ(y)dy,
∫∫
ΩL
|g(x)χΩL
|2 dxdy =
a∫
0
|g(x)|2µ(x)dx
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
110 R. HATAMLEH, V. A. ZOLOTAREV
imply that the subspaces L1, L2 (5) are isomorphic to the weighted spaces
L2
(0,b)(λ(y)dy)
df
=
f :
b∫
0
|f(y)|2λ(y)dy <∞
,
L2
(0,a)(µ(x)dx)
df
=
g :
a∫
0
|g(x)|2µ(x)dx <∞
.
(9)
The biunique correspondences between subspaces (5) and spaces (9) are realized by the mappings
f(y)χΩL
→ f(y) and g(x)χΩL
→ g(x). Taking into account (4), we obtain that the operators
2
(
Ã1
)
I
, 2
(
Ã2
)
I
after this mapping act in the spaces (9) via the multiplication by the functions (8),
2
((
Ã1
)
I
f
)
(y) = λ(y)f(y)
(
f ∈ L2
(0,b)(λ(y)dy)
)
,
2
((
Ã2
)
I
g
)
(x) = µ(x)g(x)
(
g ∈ L2
(0,a)(µ(x)dx)
)
.
(10)
It is easy to show that the commutator C̃ =
[
Ã2, Ã
∗
1
]
and its adjoint C̃∗ are given by
(
C̃f
)
(x, y) =
b∫
y
ds
α−1(y)∫
α−1(s)
dtf(t, s) =
α−1(y)∫
0
dt
b∫
α(t)
dsf(t, s),
(
C̃∗f
)
(x, y) =
a∫
x
dt
α(x)∫
α(t)
dsf(t, s) =
α(x)∫
0
ds
a∫
α−1(s)
dtf(t, s).
(11)
Theorem 1. The operator C̃ is completely continuous, belongs to the Hilbert – Schmidt class
and its spectrum is concentrated at zero, σ
(
C̃
)
= {0}. Moreover, the equalities
C̃L2
ΩL
= L1, C̃∗L2
ΩL
= L2, (12)
are true, where L1 and L2 are given by (5).
Proof. The operator C̃ (11) is an integral operator,(
C̃f
)
(x, y) =
∫∫
ΩL
K(x, y, t, s)f(t, s) dtds,
the kernel of which is equalK(x, y, t, s) = χΩx,y(t, s), where χΩx,y(t, s) is the characteristic function
of the set Ωx,y =
{
(t, s) ∈ ΩL : 0 ≤ t ≤ α−1(y)
}
. The quadratic summability of K(x, y, t, s) in
L2
ΩL
× L2
ΩL
implies [1] the complete continuity of C̃ and the Hilbert – Schmidt class membership
of C̃.
To prove that C̃L2
ΩL
= L1, it is sufficient to ascertain that Ker C̃∗ = Ker
(
Ã1
)
I
. (4) implies that
Ker
(
Ã1
)
I
consists of such functions f(x, y) ∈ L2
ΩL
that
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
ON TWO-DIMENSIONAL MODEL REPRESENTATIONS OF ONE CLASS OF COMMUTING OPERATORS 111
a∫
α−1(y)
f(t, y)dt = 0 (∀y ∈ [0, b]). (13)
If f(x, y) ∈ Ker C̃∗, then
α(x)∫
0
ds
a∫
α−1(s)
dtf(t, s) = 0 (∀x ∈ [0, a]), (14)
in view of (11). Since (13) implies (14), then Ker
(
Ã1
)
I
⊆ Ker
(
C̃∗
)
. To prove the truth of the
inverse inclusion Ker C̃∗ ⊆ Ker
(
Ã1
)
, differentiate equality (14), then
α′(x)
a∫
x
f(t, α(x))dt = 0.
Taking into account α′(x) < 0 as x ∈ [0, a), we obtain relation (13) after the substitution x =
= α−1(y).
To show that σ
(
C̃
)
= {0}, it is necessary to establish that the function
(
I−zC̃
)−1
is holomorphic
for all z ∈ C. Let
(
I − zC̃
)−1
g = f, then f(x, y) is the solution of the integral equation
f(x, y)− z
b∫
y
ds
α−1(y)∫
α−1(s)
dtf(t, s) = g(x, y). (15)
Since C̃f depends only on y (in view of (11)), then
f(x, y) = g(x, y) + ψ(y)χΩL
,
where ψ(y) ∈ L2
(0,b)(λ(y)dy). (15) yields that ψ(y) satisfies the equation
ψ(y)− z
b∫
y
(α−1(y)− α−1(s))ψ(s) ds = z
b∫
y
ds
α−1(y)∫
α−1(s)
dtg(t, s).
The function
ϕ(y)
df
=
b∫
y
ds
α−1(y)∫
α−1(s)
dtg(t, s)
is continuous and is expressed via the known function g(x, y). Thus we obtain the integral equation
ψ(y)− zKψ(y) = zϕ(y, (16)
where the operator K is given by
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
112 R. HATAMLEH, V. A. ZOLOTAREV
(Kf)(y) =
b∫
y
(α−1(y)− α−1(s))f(s) ds =
b∫
y
(λ(s)− λ(y))f(s) ds. (17)
(16) implies that
ψ = zϕ+ z2Kϕ+ . . .+ zn+1Knϕ+ . . . . (18)
Show that this series converges for all z ∈ C. Since λ(y) is a monotonously increasing function, then
it is obvious that
|(Kϕ)(y)| ≤
b∫
y
λ(s)|ϕ(s)|ds ≤
b∫
0
λ(s)|ϕ(s)|ds ≤ d‖ϕ‖L2
(0,b)
(λ(y)dy),
where
d2 =
b∫
0
λ(s) ds.
Taking into account this estimation and that |α−1(y)− α−1(s)| ≤ a (∀y, s ∈ [0, b]), we obtain
∣∣(K2
)
(y)
∣∣ ≤
∣∣∣∣∣∣
b∫
y
(α−1(y)− α−1(s))(Kϕ)(s) ds
∣∣∣∣∣∣ < ad‖ϕ‖(b− y).
Repeating this procedure n times, we obtain
|(Knϕ) (y)| ≤ an−1d‖ϕ‖(b− y)n−1
(n− 1)!
(∀n ∈ N).
Therefore, for ψ(y) (18), we have
|ψ(y)| ≤ |z|ϕ(y)|+ |z|2d‖ϕ‖+ |z|3ad‖ϕ‖(b− y) + . . .
. . .+ |z|n+1an−1d‖ϕ‖(b− y)n−1
(n− 1)!
+ . . . = |z||ϕ(y)|+ |z|2d‖ϕ‖ exp{a · |z|(b− y)}.
Since the series in the right-hand side converges uniformly for all z ∈ C, the function
(
I − zC̃
)−1
is
holomorphic in the plane C.
Theorem 1 is proved.
Note that (12) implies that
C̃L2 = L1, C̃∗L1 = L2, (19)
besides,
C̃g(x)χΩL
= χΩL
α−1(y)∫
0
g(t)µ(t)dt,
C̃∗f(y)χΩL
= χΩL
α(x)∫
0
f(s)λ(s) ds,
(20)
in view of (11).
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
ON TWO-DIMENSIONAL MODEL REPRESENTATIONS OF ONE CLASS OF COMMUTING OPERATORS 113
Denote by Ẽ1
s
df
= E1
λ(s) and Ẽ2
t
df
= E2
µ(t) the resolutions of identity of the self-adjoint operators
2
(
Ã1
)
I
∣∣∣
L1
and 2
(
Ã2
)
I
∣∣∣
L2
. Then, in view of (4), (10),
Ẽ1
sf(y)χΩL
= χ[0,s](y)f(y)χΩL
(s ∈ [0, b]),
Ẽ2
t g(x)χΩL
= χ[0,t](x)g(x)χΩL
(t ∈ [0, a]),
(21)
where χ[0,c](ξ) is the characteristic function of the set [0, c]. It is easy to see that
Ẽ1
sL1 ⊥ Ẽ2
α−1(s)L2 (∀s ∈ [0, b]). (22)
Obviously, the subspaces L1(s)
df
= E1
sL1 and L2(t)
df
= E2
t L2 form the maximal chains of invariant
subspaces Ã1
∣∣∣
L2
and Ã2
∣∣∣
L1
correspondingly,
Ã1L1(s) ⊆ L1(s) (∀s ∈ [0, b]), Ã2L1(t) ⊆ L1(t) (∀t ∈ [0, a]). (23)
Note that the commutator C̃ has the property
C̃
(
IL2 − E2
t
)
L2 = E1
α(t)L
1 (∀t ∈ [0, a]). (24)
Equality (24) can be written in the following form:(
IL1 − E1
α(t)
)
C̃
(
IL2 − E2
t
)
= 0 (∀t ∈ [0, a]). (25)
Remark 1. The orthogonality condition (22), as well as equality (25), can be taken as a definition
of the function α−1(y) (and so of α(x)) specifying the domain ΩL.
Denote by PL1 the orthoprojection on L1 (5) in L2
ΩL
,
(PL1f) (x, y)
df
=
χΩL
λ(y)
a∫
α−1(y)
f(t, y)dt, (26)
and specify the self-adjoint operator σ1 in L1,
σ1f(y)χΩL
df
=
a∫
α−1(y)
f(y)χΩL
dt = λ(y)f(y)χΩL
. (27)
(4) implies that 2
(
Ã1
)
I
= PL1σ1PL1 , and so the family
∆̃1 =
(
Ã1;L2
ΩL
;PL1 ;L1;σ1
)
(28)
is a colligation [4, 5], where the operators Ã1, PL1 , σ1 are given by (3), (26), (27), and the spaces
L2
ΩL
and L1 are equal to (2), (5) correspondingly.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
114 R. HATAMLEH, V. A. ZOLOTAREV
Theorem 2. The characteristic function S∆̃1
(z) (7) of the colligation ∆̃1 (28) is a scalar oper-
ator in L1,
S∆̃1
(z) = e
iλ(y)
z IL1 , (29)
where λ(y) equals (8).
Proof. The function f(x, y) =
(
Ã1 − zI
)−1
PL1σ1f(y)χΩL
satisfies the integral equation
i
a∫
x
f(t, y)dt− zf(x, y) = λ(y)f(y)χΩ, (30)
which is equivalent to the Cauchy problem
if(x, y) + z
∂
∂x
f(x, y) = 0,
f(a, y) = −λ(y)
z
f(y).
This implies
f(x, y) = −λ(y)
z
f(y)e
i(a−x)
z χΩL
.
Therefore
S∆̃1
(z)f(y)χΩL
− i χΩL
λ(y)
a∫
α−1(y)
f(t, y)dt =
=
{
f(y)− 1
λ(y)
[zf(x(y), y) + λ(y)f(y)]
}
χΩL
= e
iλ(y)
z f(y)χΩL
,
in view of equation (30).
Theorem 2 is proved.
Remark 2. The operator-function S∆̃1
(z) (29) commutes with the operator σ1 (27) for all z ∈ C,
z 6= 0.
Remark 3. Consider the restriction of the operator Ã2 (3) on the invariant (7) subspace L1 (5).
In spite of the fact that L0 (6) is one-dimensional, nevertheless, the closure of the operator image
PL12
(
Ã2
)
I
∣∣∣
L1
coincides with the whole of L1. Really, since
Ã∗2f(y)χΩL
= −i
y∫
α(x)
f(s)χΩL
ds,
then, taking into account the form of the orthoprojection PL1 (26), it is easy to show that
PL1Ã
∗
2f(y)χΩL
= −i χΩL
λ(y)
y∫
0
f(s)λ(s) ds.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
ON TWO-DIMENSIONAL MODEL REPRESENTATIONS OF ONE CLASS OF COMMUTING OPERATORS 115
Therefore
PL12
(
Ã2
)
I
f(y)χΩL
= χΩL
b∫
y
f(s) ds+
1
λ(y)
y∫
0
f(s)λ(s) ds
.
Let f(y)χΩL
∈ KerPL12
(
Ã2
)∣∣∣
L1
, then
λ(y)
b∫
y
f(s) ds+
y∫
0
f(s)λ(s) ds = 0.
Differentiating we obtain
λ′(y)
b∫
y
f(s) ds = 0,
and since λ′(y) 6= 0, this implies f(y) = 0. Thus KerPL12
(
Ã2
)
I
∣∣∣ = 0 and so PL12
(
Ã2
)
I
L1 = L1.
II. Consider a bounded self-adjoint operator in a Hilbert space H with the simple spectrum in
the segment [0, a]. Then [1] the operator B is unitary equivalent to the operator of multiplication by
an independent variable (
B̂f
)
(λ) = λf(λ)
(
f(λ) ∈ L2
(0,a)(dσ(λ))
)
, (31)
where σ(λ) = 〈Eλu, u〉 is nondecreasing on [0, a]; Eλ is the resolution of identity of B; and u ∈ H
is the generating vector of the operator B. This unitary equivalence is given by the map U [1],
Uf(λ) = f, f
df
=
a∫
0
f(λ)dEλu, (32)
besides, f(λ) ∈ L2
(0,a)(dσ(λ)) and f ∈ H. Suppose that the measure dσ(λ) is absolutely continuous
by the Lebesgue measure,
dσ(λ) = m(λ)dλ (m(λ) = σ′(λ) ≥ 0). (33)
Definition 1. An absolutely continuous measure dσ(λ) (33) is said to have the AC0-property if
λ∫
0
dσ(t)
t
<∞, (34)
for all λ ∈ [0, a].
Requirement (34), per se, is conditioned by the convergence of the given improper integral at
zero. Define the smooth monotonously increasing function y(λ),
y(λ)
df
=
λ∫
0
dσ(t)
t
. (35)
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116 R. HATAMLEH, V. A. ZOLOTAREV
Remark 4. ‘A priori’ we can suppose that the function y(λ) maps [0, a] onto [0, b], where
b is a preset finite positive number. If y(λ) : [0, a] → [0, b] (d > 0), then setting the measure
dσ1(t) =
b
d
dσ(t) (b > 0) and realizing the substitution f(λ) → f(λ)
√
d
b
in L2
(0,a) (dσ(λ)) we
obtain the Hilbert space L2
(0,a) (dσ1(λ)) isomorphic to L2
(0,a)(dσ(λ)), besides, the function y1(λ)
constructed by dσ1(λ) (35) already possesses the values on [0, b]. This procedure signifies renormal-
ization of the generating vector u→
√
b
d
u since σ(λ) = 〈Eλu, u〉 .
Denote by λ−1(y) the function reciprocal to y(λ) (35). Since dσ(λ) = λdy(λ), then the change
of variable λ → λ(y) translates the space L2
(0,a)(dσ(λ)) into L2
(0,b)(λ(y)dy) where the operator B̂
(31) acts as a multiplication by the function λ(y),(
B̂f
)
(y) = λ(y)f(y)
(
f(y) ∈ L2
(0,b)(λ(y)dy)
)
. (36)
Theorem 3. Let B be a bounded self-adjoint operator with the simple spectrum in H, besides,
the spectrum of B belongs to the segment [0, a]. If the spectral measure σ(λ) of the operator B is
absolutely continuous (33) and has the AC0-property (34), then the operator B is unitary equivalent
to the operator of multiplication B̃ (36) by the smooth monotonously increasing function λ(y) (recip-
rocal to y(λ) (35)) in L2
(0,b)(λ(y)dy), besides, the finite positive number b can be chosen arbitrarily.
The following statement gives the description of the commutant of the operator B̂ (31).
Theorem 4. An arbitrary linear bounded operator  in L2
(0,a) (dσ(λ)) commuting with B̂ (31)
is the operator of multiplication,(
Âf
)
(λ) = a(λ)f(λ)
(
f ∈ L2
(0,a)(dσ(λ))
)
, (37)
where a(λ) is a complex-valued function from L2
(0,a)(dσ(λ)), besides, ‖A‖ = ‖a(λ)‖L2
(0,a)
(dσ(λ)).
Proof. Let A be a linear bounded operator in H commuting with B where B is a self-adjoint
operator with the simple spectrum and σ(B) ⊆ [0, a]. The permutability of A and B implies [1] that
[A,Eλ] = 0 (∀λ ∈ [0, a]) where Eλ is the resolution of identity of the operator B. (32) yields that
Af =
a∫
0
f(λ)dEλAu.
To the vector Au from H in view of the mapping U (32) there corresponds such function a(λ) ∈
∈ L2
(0,a)(dσ(λ)) that
Au =
a∫
0
a(λ)dEλu,
and so
〈EλAu, u〉 =
λ∫
0
a(t)dσ(t).
This implies that
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ON TWO-DIMENSIONAL MODEL REPRESENTATIONS OF ONE CLASS OF COMMUTING OPERATORS 117
〈Au,Eλu〉 =
λ∫
0
f(t)a(t)dσ(t) =
a∫
0
f(t)dt 〈Etu,Eλu〉 =
〈 a∫
0
f(t)a(t)dEtu,Eλu
〉
.
And since the linear manifold of the vectors Eλu is dense in H, we obtain that
Af =
a∫
0
f(λ)a(λ)dEλu.
Thus AU = U where  is given by (37), and U is given by formula (32).
Theorem 4 is proved.
So, the commutant B̂′ coincides with the set of the operators given by (37) and is isomorphic to
the space L2
(0,a)(dσ(λ)).
The following generalization of the M. S. Livs̆ic Theorem is true.
Theorem 5. Let a linear bounded dissipative completely nonself-adjoint operator with the spec-
trum at zero, σ(A) = {0}, be given in a Hilbert space H, and the following conditions be met:
1) operator 2AI restricted on H1 = AIH has a simple spectrum filling the finite segment
[0, a], 0 < a < ∞, and its spectral function σ(λ) is absolutely continuous (33) and has the AC0-
property (34);
2) for all z ∈ C (z 6= 0),
[
PH1(A− zI)−1PH1 , AI
]
= 0 takes place, where PH1 is the ortho-
projection on H1.
Then the operator A is unitary equivalent to the integration operator,
(
Ãf
)
(x, y) = i
a∫
x
f(t, y)dt, (38)
in the space L2
ΩL
(2), besides, the curve L (1) is given by the function α−1(y) = a− λ−1(y), where
λ−1(y) is the reciprocal to y(λ) (35) function.
Proof. Theorem 3 implies that there exists a unitary operator U : H1 → L2
(0,b)(λ(y)dy) such that
U2AI = B̃U, where B̃ is given by (36). Construct a colligation
∆ =
(
A;H;UPH1 ;L2
(0,b)(λ(y)dy); B̃
)
.
Condition 2 of the Theorem implies that the characteristic function S∆(z) of this colligation com-
mutes with B̃. Using Theorem 4, we obtain that S∆(z) is an operator of multiplication by the
function exp
{
iz−1c(y)
}
in the space L2
(0,b)(λ(y)dy), in view of the standard type of the character-
istic function, if one takes into account that σ(A) = {0} and B̃ ≥ 0. Note that c(y) = λ(y) since
limz→∞ iz (I − S∆(z)) = B̃.
Knowing λ(y), from formula (8) we find the smooth decreasing function α−1(y) specifying the
curve L (1) and so the domain ΩL also, for the functions from the space L2
ΩL
. After this we construct
the colligation ∆1 (28) and observe that the characteristic functions of the colligations ∆1 and ∆
coincide in view of Theorem 2. Application of the Theorem on unitary equivalence [4] concludes the
proof.
Note that condition 2 in the M. S. Livs̆ic Theorem is met automatically since rankAI = 1.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
118 R. HATAMLEH, V. A. ZOLOTAREV
III. Consider two absolutely continuous measures
dσ(λ) = m(λ)dλ, dω(µ) = n(µ)dµ (39)
defined on the finite segments, λ ∈ [0, a], µ ∈ [0, b], where m(λ) ≥ 0 and n(µ) ≥ 0. Supposing
that dσ(λ) and dω(λ) has the AC0-property (34) and taking into account (35), we define the positive
increasing functions
y(λ)
df
=
λ∫
0
dσ(t)
t
, x(µ)
df
=
µ∫
0
dω(s)
s
. (40)
Remark 4 yields that we can suppose that y(λ): [0, a] → [0, b] and x(µ): [0, b] → [0, a]. Denote
by λ−1(y) and µ−1(x) the functions reciprocal to y(λ) and x(µ) (40). Differentiating (40), we obtain
that
m(λ) = λy′(λ), n(µ) = µx′(µ). (41)
Suppose that the functions λ(y) and µ(x) are given by (8), then taking into account that α−1(y) is
the function reciprocal to α(x), we have
x = a− λ(b− µ(x)),
and after the substitution x = x(µ) we obtain that
a− x(µ) = λ(b− µ). (42)
This implies
x′(µ) = λ′(b− µ),
or, using (41), we obtain the equality
λ′(b− µ) =
n(µ)
µ
. (43)
Since y = y(λ−1(y)), then
1 =
dy(λ−1(y))
dλ
dλ−1(y)
dy
=
m(λ−1(y))
λ(y)
n(b− y)
b− y
,
in view of (41) and (43). Thus
n(µ)m(λ−1(b− µ)) = µλ(b− µ) (∀µ ∈ [0, b]). (44)
Lemma 1. Let two absolutely continuous measures dσ(λ) and dω(µ) (39) have the AC0-
property (34). Then in order that (42) take place, where λ−1(y) is the function reciprocal to y(λ)
(40) and x(µ) is given by (40), it is necessary and sufficient that the fitting condition (44) is met.
Remark 5. The fitting condition (44) for the measures dσ(λ) and dω(µ) (39) provides realization
of the functions λ−1(y) and µ−1(x) in the form of (8) where α(x) and α−1(y) are mutually reciprocal
functions. Since λ(y) is explicitly constructed by m(λ) in view of (40), then (44) implies that n(µ)
is uniquely defined by the function m(λ). Finally, the truth of the AC0-property for dω(µ) in this
case signifies that
µ∫
0
n(s)
s
ds =
µ∫
0
λ(b− s)
m(λ−1(b− s))
ds <∞
for all µ ∈ [0, b].
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ON TWO-DIMENSIONAL MODEL REPRESENTATIONS OF ONE CLASS OF COMMUTING OPERATORS 119
IV. Consider a commutative system of linear bounded operators {A1, A2} in a Hilbert space H.
Denote by H1 and H2 two subspaces in H,
H1
df
= (A1)IH, H2
df
= (A2)IH. (45)
Theorem 6. Let two self-adjoint bounded nonnegative operators 2(A1)I and 2(A2)I be given
in a Hilbert space H such that:
1) the restrictions 2(Ak)I |Hk
on the subspaces Hk, k = 1, 2, (45) are operators with simple
absolutely continuous spectrum, besides, σ
(
2(A1)I |H1
)
= [0, a] and σ
(
2(A2)I |H2
)
= [0, b];
2) the spectrum measures σ(λ) and ω(µ) corresponding to 2(A1)I |H1
and 2(A2)I |H2
respec-
tively are absolutely continuous (39), have the AC0-property (34), and (44) takes place, where λ−1(y)
is the function reciprocal to y(λ) (40).
Then there exist the Hilbert space L2
ΩL
(2) and the isometric operator U : H1+H2 → L1+L2 (Lk
are given by (5), k = 1, 2) realizing unitary equivalence between the operators PH1+H22(Ak)I |H1+H2
and PL1+L22
(
Ãk
)
I
∣∣∣
L1+L2
, k = 1, 2, besides,
(
2
(
Ã1
)
I
f
)
(x, y) =
a∫
α−1(y)
f(t, y)dt,
(
2
(
(Ã2
)
I
f
)
(x, y) =
b∫
α(x)
f(x, s) ds, (46)
where f(x, y) = [f(y) + g(x)]χΩL
∈ L1 + L2.
Proof. The conditions 1, 2, and Theorem 3 imply that the operator 2(A1)|H1
restricted on H1
is unitary equivalent to the operator of multiplication by the function λ(y) in the function space
L2
(0,b)(λ
−1(y)dy). Similarly, 2(A2)I |H2
on H2 is unitary equivalent to the operator of multiplication
by the function µ(x) in the space L2
(0,a)(µ
−1(x)dx). The functions λ−1(y): [0, b] → [0, a] and
µ−1(x): [0, a] → [0, b] are the reciprocal to y(λ) and x(µ) (40). The fitting condition (44) implies
that λ−1(y) and µ−1(x) are given by (8), where α−1(y) and α(x) are mutually reciprocal functions.
Knowing α(x), we construct the curve L (1) and define the Hilbert space L2
ΩL
(2) of the functions
f(x, y) in the domain ΩL. As was noted before (see Section I), the mappings f(y) → f(y)χΩL
,
g(x)→ g(x)χΩL
set an isomorphism between the spaces
L2
(0,b)(λ
−1(y)dy)↔ L1, L2
(0,a)(µ
−1(x)dx)↔ L2,
where L1, L2 are given by (5). Besides, the operator of multiplication by λ(y) in L2
(0,b)(λ
−1(y)dy)
transforms into the operator 2
(
Ã1
)
=
∫ a
α−1(y)
.dt on L1, and the operator of multiplication by µ(x) in
L2
(0,a)(µ
−1(x)dx) transforms correspondingly into the operator 2
(
Ã2
)
I
=
∫ b
α(x)
.ds on L2. So, each
subspace Hk (45) from H is isomorphic to Lk (5) in L2
ΩL
, k = 1, 2, and the operators 2(Ak)I |Hk
appear to be unitary equivalent to 2
(
Ãk
)
I
∣∣∣
Lk
, k = 1, 2,
2
(
Ã1
)
I
f(y)χΩL
=
a∫
α−1(y)
f(y)χΩL
dt, 2
(
Ã2
)
I
g(x)χΩL
=
b∫
α(x)
g(x)χΩL
ds, (47)
where f(y)χΩL
∈ L1 and g(x)χΩL
∈ L2.
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120 R. HATAMLEH, V. A. ZOLOTAREV
Show that knowing these mappings Hk → Lk, k = 1, 2, one can realize the unitary equivalence
between H1 +H2 from H and L1 + L2 from L2
ΩL
. To do this, consider
〈h1, h2〉 = 〈h1, PH1h2〉 ,
where hk ∈ Hk, k = 1, 2. Let h1 → f(y)χΩL
∈ L1 and h2 → g(x)χΩL
∈ L2, then
〈f(y)χΩL
, g(x)χΩL
〉L2
ΩL
=
b∫
0
dy
a∫
α−1(y)
dxf(y)g(x) =
b∫
0
dyf(y)
a∫
α−1(y)
g(x)dx.
Since operator (26)
(PL1f) (x, y) =
χΩL
λ(y)
a∫
α−1(y)
f(t, y)dt
is the orthoprojection on L1 in L2
ΩL
,
〈f(y)χΩL
, g(x)χΩL
〉L2
ΩL
=
〈
f(y),
1
λ(y)
a∫
α−1(y)
g(x)dx
〉
L2
(0,b)
(λ−1(y)dy)
=
= 〈f(y)χΩL
, PL1gχΩL
〉L2
ΩL
.
Thus
〈h1, h2〉H = 〈f(y)χΩL
, g(x)χΩL
〉L2
ΩL
,
and so the correspondence h1 + h2 → [f(y) + g(x)]χΩL
is a unitary isomorphism between H1 +H2
and L1 + L2.
To complete the proof of the theorem, it is left for us to ascertain that the formulas (46) are true.
For 2
(
Ã1
)
I
(for example) formula (46) on functions of the type f(y)χΩL
is already proved (47). It
is left to ascertain the truth of (46) on the functions g(x)χΩL
. Really, since
2(A1)Ih2 = 2(A1)PH1h2, h2 ∈ H,
then
2
(
Ã1
)
I
PL1g(x)χΩL
= 2
(
Ã1
)
I
χΩL
λ(y)
a∫
α−1(y)
g(t)dt =
a∫
α−1(y)
g(t)χΩL
dt,
in view of (47), where h2 → g(x)χΩL
.
Theorem 6 is proved.
Define now the class of linear operators K∞, which in some sense is close to the class Kn [6]
when n =∞, but cannot be obtained from Kn as n→∞.
The class K∞. A system of linear bounded operators {A1, A2} in a Hilbert space H is said to
belong to the class K∞ if
1) [A1, A2] = 0; (48)
2) CH = H1, C∗H = H2, where C = [A2, A
∗
1] , and Hk, k = 1, 2, are given by (45);
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ON TWO-DIMENSIONAL MODEL REPRESENTATIONS OF ONE CLASS OF COMMUTING OPERATORS 121
3) the operator C is completely continuous and belongs to the Hilbert – Schmidt class, and its
spectrum lies at zero, σ(C) = {0}.
Theorem 1 implies that the operator system
{
Ã1, Ã2
}
(3) in L2
ΩL
(2) belongs to the class K∞.
Theorem 7. Suppose that the operator system {A1, A2} belongs to the class K∞ (48) and
conditions 1, 2 of Theorem 6 are met. Let[
IH1 − iE1
λ−1(α(t))
]
C
[
IH2 − E2
µ−1(t)
]
= 0; (49)
takes place for all t, where E1
λ and E2
µ are the resolutions of identity of the operators 2(A1)I |H1
and
2(A2)|H2
, the functions µ−1(x) and λ−1(y) are reciprocal to x(µ) and y(λ) (40) correspondingly.
Then the operator C is unitary equivalent to the operator C̃ : L2 → L1,
C̃g(x)χΩL
= χΩL
α−1(y)∫
0
dtg(t)µ(t) = χΩL
b∫
y
ds
α−1(y)∫
α−1(s)
dtg(t), (50)
for all g(x)χωL ∈ L2.
Proof. The isometric mapping H1 + H2 on L1 + L2 constructed in the proof of Theorem 6
transforms the operator C into the operator C̃ mapping surjectively L2 onto L1. The Hilbert – Schmidt
operator C̃ always can be represented as
C̃ =
∞∑
k=1
sk 〈., ϕk(x)χΩL
〉ψk(y)χΩL
, sk > 0,
besides, the series converges by the norm of the space L2
ΩL
× L2
ΩL
, the functions ϕk(x)χΩL
form
the complete orthonormal system in L2 of the eigenvectors of the operator
√
C̃∗C̃, and ψk(y)χΩL
=
= Uϕk(x)χΩL
is the orthonormal basis in L1, where U is a unitary operator from L2 onto L1
corresponding to the polar decomposition C̃ = U
√
C̃∗C̃. The last formula implies that
C̃g(x)χΩL
=
∫∫
ΩL
K(t, y)g(t)χΩL
dtds = χΩL
a∫
0
dtK(t, y)g(t)µ(t), (51)
besides, the kernel K(t, y) is given by
K(t, y) =
n∑
k=1
skϕk(t)χΩL
(t, s)ψk(y)χΩL
(x, y).
Since the condition of theorem (49) can be represented as (25) where the spectral projectors E1
t and
E2
s are equal to (21), (49) implies that the operator C̃ (51) equals
C̃gxχΩL
= χΩL
α−1(y)∫
0
dtK(t, y)g(t)µ(t) =
α−1(y)∫
0
dt
b∫
α(t)
dsK(t, y)g(t), (52)
and so
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122 R. HATAMLEH, V. A. ZOLOTAREV
C̃∗fyχΩL
= χΩL
α(x)∫
0
dsK(x, s)f(s)λ(s) =
α(x)∫
0
ds
a∫
α−1(s)
dtK(x, s)f(s). (53)
Use the following formula:
4
[(
Ã1
)
I
,
(
Ã2
)
I
]
= C̃∗ − C̃. (54)
It is obvious that
C̃∗f(y)χΩL
− C̃f(y)χΩL
= χΩL
α(x)∫
0
dsK(x, s)f(s)λ(s)− C̃PL2f(y)χΩL
=
= χΩL
α(x)∫
0
dsK(x, s)f(s)λ(s)− χΩL
α−1(y)∫
0
dtK(t, y)
b∫
α(t)
dsf(s),
in view of (52), (53) and the fact that
PL2f(y)χΩL
=
χΩL
µ(x)
b∫
α(x)
dsf(s).
Since
PL1g(x)χΩL
=
χΩL
λ(y)
a∫
α−1(y)
dtg(t),
it is easy to see that
4
{(
Ã1
)
I
(
Ã2
)
I
−
(
Ã2
)
I
(
Ã1
)
I
}
f(y)χΩL
=
= 4
(
Ã1
)
I
(
Ã2
)
I
PL2f(y)χΩL
− 2
(
Ã2
)
I
λ(y)f(y)χΩL
=
= 2
(
Ã1
)
I
PL1χΩL
b∫
α(x)
dsf(s)− 2
(
Ã2
) χΩL
µ(x)
b∫
α(x)
dsf(s)λ(s) =
= χΩL
a∫
α−1(y)
dt
b∫
α(t)
dsf(s)− χΩL
b∫
α(x)
dsf(s)λ(s).
Using equality (54), we obtain that
α(x)∫
0
dsK(x, s)f(s)λ(s) +
b∫
α−1(y)
dsf(s)λ(s) =
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ON TWO-DIMENSIONAL MODEL REPRESENTATIONS OF ONE CLASS OF COMMUTING OPERATORS 123
=
a∫
α−1(y)
dt
b∫
α(t)
dsf(s) +
α−1(y)∫
0
dtK(t, y)
b∫
α(t)
dsf(s).
In connection with the fact that the left-hand side of this equality does not depend on y, then
supposing that y = b and taking into account that α−1(b) = 0, we obtain the relation
α(x)∫
0
ds
{
K(x, s)− χ[0,b](s)
}
f(s)λ(s) +
b∫
0
dsf(s)λ(s) =
=
a∫
0
dt
a∫
α(t)
dsf(s) =
b∫
0
dsf(s)λ(s).
So,
a∫
0
ds
{
K(x, s)− χ[0,b](s)
}
f(s)λ(s) = 0,
for all f(y)χΩL
∈ L1 and all x ∈ [0, a]. This easily implies that K(x, y) = χΩL
, this concludes the
proof of the theorem.
(50) and (53) imply that
C̃∗f(y)χΩL
= χΩL
α(x)∫
0
dsf(s)λ(s) = χΩL
α(x)∫
α(t)
dsf(s), (55)
where f(y)χΩL
∈ L1.
Formulate the main theorem.
Theorem 8. Suppose that the operator system {A1, A2} belongs to the class K∞ (48), besides,
each operator Ak is completely nonself-adjoint, dissipative, and the spectrum σ(Ak) = {0}, k = 1, 2,
and let the operator system {A1, A2} be such that
1) the subspace H0 = H1 ∩H2 is one-dimensional, where Hk, k = 1, 2, are given by (45);
2) restrictions 2(Ak)I |Hk
on the subspaces Hk, k = 1, 2, (45) are operators with the simple
completely continuous spectrum, and σ
(
PH12(A1)I |H1
)
= [0, a], σ
(
PH22(A2)I |H2
)
= [0, b];
3) the spectral measures σ(λ) and ω(λ) corresponding to PH12(A1)I |H1
and PH2 2(A2)I |H2
are absolutely continuous (39) and have the AC0-property (34), besides, (44) takes place, where
λ−1(y) is the function reciprocal to y(λ) (40);
4) for all t ∈ [0, a] condition (49) takes place where E1
λ and E2
µ are the resolutions of identity of
the operators 2(A1)I |H1
and 2(A2)I |H2
, the functions µ−1(x) and λ−1(y) are reciprocal to x(µ)
and y(λ) (40).
Then the operator system {A1, A2} in H is unitary equivalent to the system
{
Ã1, Ã2
}
(3) in the
function space L2
ΩL
(2).
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
124 R. HATAMLEH, V. A. ZOLOTAREV
Proof. First of all, the equality
iC = A22(A1)I − 2(A1)IA2 (56)
implies that A2H1 ⊆ H1, in view of 2 (48). And similarly A1H2 ⊆ H2. Denote by Ã2 and Ã1
the operators restricted on L1 and L2 (5) correspondingly, which are unitary equivalent to A2|H1
and A1|H2
, besides, this equivalence specifies the isometric mapping from H1 + H2 onto L1 + L2
constructed in the proof of Theorem 6. Theorem 7 implies that this correspondence transforms the
commutator C into the operator C̃ (50), therefore
iC̃f(y)χΩL
= iC̃PL2f(y)χΩL
= iχΩL
α−1(y)∫
0
dt
b∫
α(t)
dsf(s) =
= iχΩL
b∫
y
dsf(s)
α−1(y)∫
α−1(s)
dt = iχΩL
b∫
y
dsf(s)[λ(s)− λ(y)],
in view of (8). Equality (56) for C̃, Ã2 and 2
(
Ã1
)
I
is
i
b∫
y
dsf(s)[λ(s)− λ(y)] = Ã2f(y)λ(y)− 2
(
Ã1
)
I
Ã2f(y).
Taking into account that Ã2f(y) ∈ L1 and that the operator 2
(
Ã1
)
I
on L1 acts as a multiplication
by λ(y), we obtain that
λ(y)
Ã2f(y)− i
b∫
y
dsf(s)
= Ã2f(y)λ(y)− i
b∫
y
dsf(s)λ(s).
Thus the operator
B2f(y)
df
= Ã2f(y)− i
b∫
y
dsf(s) (f(y)χΩL
∈ L1)
maps L1 onto L1 and commutes with the operator of multiplication by λ(y). Theorem 4 implies that
the operator B2 is the operator of multiplication by the function B2f(y) = Φ(λ(y))f(y), therefore
Ã2f(y) = i
b∫
y
dsf(s) + ϕ(y)f(y),
where ϕ(y) = Φ(λ(y)). Elementary calculations show that
(
Ã2 − zIL1
)−1
f(y) =
f(y)
ϕ(y)− z
− 1
ϕ(y)− z
b∫
y
ds
f(s)
ϕ(s)− z
exp
i
s∫
y
dξ
z − ϕ(ξ)
.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
ON TWO-DIMENSIONAL MODEL REPRESENTATIONS OF ONE CLASS OF COMMUTING OPERATORS 125
Thus spectrum of the operator Ã2 consists of the range of values of the function ϕ(y) and since
σ
(
Ã2
)
= {0}, then ϕ(y) = 0, and we finally obtain that
Ã2f(y) = i
b∫
y
dsf(s). (57)
Similarly,
Ã1g(x) = i
a∫
x
dtg(t). (58)
So the restrictions A2|H1
and A1|H2
are unitary equivalent to the integration operators: Ã2 (57), on
L1, and Ã1 (58), correspondingly, on L2.
The one-dimensional subspace L0 = L1 ∩ L2 isomorphic to H0 = H1 ∩ H2 is formed by the
constant functions from L2
ΩL
. The form of the operator Ã2 (57) obviously implies that the linear
span of the vectors Ãn2f0 (n ∈ Z+, f0 ∈ L0) is dense in L1 (5). And since H1 and L1 are unitary
isomorphic and the operators A2|H1
and Ã2
∣∣∣
L1
are unitary equivalent, then
H1 = span {An2h0 : n ∈ Z+;h0 ∈ H0} , (59)
and the equalities
〈An2h0, A
m
2 h0〉 =
〈
Ãn2f0, Ã
m
2 f0
〉
(∀n,m ∈ Z+) (60)
take place, where f0 is the image of h0 under the correspondence H1 → L1. Similar considerations
for H2 and L2, and the operators A1
∣∣∣
H2
and Ã1
∣∣∣
L2
(58) give to us
H2 = span {An1h0 : n ∈ Z+;h0 ∈ H0} , (61)
besides,
〈An1h0, A
m
1 h0〉 =
〈
Ãn1f0, Ã
m
1 f0
〉
(∀n,m ∈ Z+). (62)
Moreover, the equalities
〈An1h0, A
m
2 h0〉 =
〈
Ãn1f0, Ã
m
2 f0
〉
(∀n,m ∈ Z+) (63)
are true in view of unitary isomorphism (Theorem 6) between the subspaces H1 +H and L1 + L2.
Taking into account complete nonself-adjointness of the operator A1,
H = span {An1h1 : n ∈ Z+; h1 ∈ H1} ,
and form of H1 (59), we obtain that
H = span {An1Am2 h0 : n,m ∈ Z+; h0 ∈ H0} . (64)
Continue the operators Ã2 (57) and Ã1 (58) on the whole L2
ΩL
(2) using the formulas
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
126 R. HATAMLEH, V. A. ZOLOTAREV
(
Ã1f
)
(x, y) = i
a∫
x
f(t, y)dt,
(
Ã2f
)
(x, y) = i
b∫
y
f(x, s) ds. (65)
It is obvious that operators (65) commute and
L2
ΩL
= span
{
Ãn1 Ã
m
2 f0 : n,m ∈ Z+; f0 ∈ L0
}
. (66)
To conclude the proof of the theorem it is necessary to ascertain that
〈An1Am2 h0, A
p
1A
q
2h0〉 =
〈
Ãn1 Ã
m
2 f0, Ã
p
1Ã
q
2f0
〉
(∀n,m, p, q ∈ Z+). (67)
Really, if (67) takes place, we can specify unitary in view of (67) operator U,
UAn1A
m
2 h0 = Ãn1 Ã
m
2 f0 (n,m ∈ Z+),
mapping the whole H (64) onto L2
ΩL
(66), besides, it is obvious that UAk = ÃkU, k = 1, 2.
Proof of the relations (67) can be realized in several stages. Show first that the equalities (60),
(62), and (63) imply that
〈An1Am2 h0, A
p
1h0〉 =
〈
Ãn1 Ã
m
2 f0, Ã
p
1f0
〉
(∀n,m, p ∈ Z+) (68)
take place. Use the method of induction by the parameter n ∈ Z+, for all m, p ∈ Z+. When n = 0,
equality (68) follows from (63). Let n = 1, then
〈A1A
m
2 h0, A
p
1h0〉 = 〈(A∗1 + 2i(A1)I)A
m
2 h0, A
p
1h0〉 =
=
〈
Am2 h0, A
p+1
1 h0
〉
+ 2i 〈(A1)IA
m
2 h0, A
p
1h0〉 .
Since for the first summand (63) is true, then one ought to consider the second summand, which can
be written in the form
i 〈2(A1)IA
m
2 h0, PH1A
p
1h0〉 .
Note that 〈PH1A
p
1h0, h1〉 =
〈
PL1Ã
p
1f0, f1
〉
in view of (63) and (59), where h1 ∈ H1, and f1
is the image of h1 under the correspondence H1 → L1 and f1 ∈ L1. And in accordance with
2(A1)Am2 h0 ∈ H1, the equality 2
〈
(A1)I , A
m
2 h0, h1
〉
=
〈
2
(
Ã1
)
I
Ãm2 f0, f1
〉
also holds in view of
Theorem 6 and formulas (46), (65). Thus
i
〈
2(A1)IA
m
2 h0, A
p
1h0
〉
= i
〈
2
(
Ã1
)
I
Ãm2 f0, Ã
p
1f0
〉
,
and so the equalities (68) for n = 1 are proved. Let for all n = 0, 1, . . . , q the statement be proved,
show that it also is true for n = q + 1. Consider〈
Aq+1
1 Am2 h0, A
p
1h0
〉
= 〈(A∗1 + 2i(A1)I)A
q
1A
m
2 h0, A
p
1h0〉 =
=
〈
Aq1A
m
2 h0, A
p+1
1 h0
〉
+ 2i 〈(A1)IA
q
1A
m
2 h0, A
p
1h0〉 .
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
ON TWO-DIMENSIONAL MODEL REPRESENTATIONS OF ONE CLASS OF COMMUTING OPERATORS 127
For the first summand, (68) takes place in view of the induction supposition, as for the second
summand, we write it as
〈Aq1A
m
2 h0, hp〉 ,
where hp = 2(A1)IA
p
1h0 ∈ H1. If m = 0, then Theorem 6 and (63) imply that〈
Ap1h0, 2(A1)IA
p
1h0
〉
=
〈
Ãp1f0, 2
(
Ã1
)
I
Ãp1f0
〉
.
When m > 0, we obtain that〈
Aq1A
m
2 h0, hp
〉
=
〈
Aq1A
m−1
2 h0, A2hp
〉
+ i
〈
Aq1A
m−1
2 h0, 2(A2)shp
〉
.
Since 2(A2)Ihp ∈ H2 andH2 is given by (61), for the second summand (68) is true by the supposition
of induction. In accordance with A2hp ∈ H1, repeating this process, ‘transfer’ of the operator A2
on the second place in the scalar product, proper number of times, we at last receive the expression
〈Aq1h0, A
m
2 hp〉 , for which (63) is true. Thus truth of the equalities (68) is proved.
To prove that (67) take place, consider〈
An1A
m
2 h0, A
p
1A
q
2h0
〉
=
〈
An1A
m
2 h0, (A
∗
2 + 2i(A2)I)A
q−1
2 Ap1h0
〉
=
=
〈
An1A
m+1
2 h0, A
q−1
2 Ap1h0
〉
− 2i
〈
(A2)IA
n
1A
m
2 h0, A
q−1
2 Ap1h0
〉
.
Taking into account that (A2)IA
n
1A
m
2 h0 ∈ H2 and H2 is given by (61), we in view of (68) have that〈
2(A2)IA
n
1A
m
2 h0, A
q−1
2 Ap1h0
〉
=
〈
2
(
Ã2
)
I
Ãn1 Ã
m
2 f0, Ã
q−1
2 Ãp1f0
〉
.
For the first summand, again repeat this procedure of ‘transfer’ of grades of the operator A2 from
the second place in the scalar product to the first. After the finite number of steps, we obtain the
expression
〈
An1A
m+q
2 h1, A
p
1h0
〉
, for which the truth of equalities (68) is already proven.
Theorem 8 is proved.
1. Akhiezer N. I., Glazman I. M. Theory of linear operators in Hilbert space. – Boston etc.: Pitman, 1981. – Vol. 1.
2. Akhiezer N. I. The classical moment problem and some related questions in analysis. – Oliver & Boyd, 1965.
3. Livs̆ic M. S. On spectral decomposition of linear nonself-adjoint operators // Mat. Sb. – 1954. – Issue 34 (76), № 1. –
P. 145 – 198.
4. Livs̆ic M. S., Yantsevich A. A. Theory of operator colligations in Hilbert spaces. – New York: Wiley, 1979.
5. Zolotarev V. A. Analitic methods of spectral representations of nonself-adjoint and nonunitary operators. – Kharkov:
MagPress, 2003 (in Russian).
6. Zolotarev V. A. On triangular models of systems of twice commuting operators // DAN ArmSSR. – 1976. – 63, № 3. –
P. 136 – 140 (in Russian).
7. Zolotarev V. A. On structure and triangular models of systems of one class of commuting operators // Vestnik Khark.
Univ. – 1978. – Issue 43. – P. 69 – 76 (in Russian).
8. Hatamleh R. Triangular model of systems of linear operators with nilpotent commutators [A,B] and [A∗, B∗] //
Vestnik Khark. Univ. Ser. Mat., Prikl. Mat. i Mech. – 2004. – № 645. – P. 79 – 84.
Received 25.09.11,
after revision — 14.10.13
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
|
| id | umjimathkievua-article-2115 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:18:59Z |
| publishDate | 2014 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/47/13acdefa472de5e2326db262aa4a9347.pdf |
| spelling | umjimathkievua-article-21152019-12-05T10:24:29Z On Two-Dimensional Model Representations of One Class of Commuting Operators Про двовимірні модельні зображення одного класу комутуючих операторів Hatamleh, R. Zolotarev, V. A. Хатамлех, Р. Золотарьов, В. А. In the work by V. A. Zolotarev, Dokl. Akad. Nauk Arm. SSR, 63, No. 3, 136–140 (1976), a triangular model is constructed for a system of twice-commuting linear bounded completely nonself-adjoint operators {A 1, A 2} ([A 1, A 2] = 0, [A 1 ∗ , A 2] = 0) such that rank (A 1) I (A 2) I = 1 (2i(A k ) I = A k − A k ∗ , k = 1, 2) and the spectrum of each operator A k , k = 1, 2, is concentrated at zero. The indicated triangular model has the form of a system of operators of integration over the independent variable in L Ω 2 where the domain Ω = [0, a] × [0, b] is a compact set in ℝ2 bounded by the lines x = a and y = b and a decreasing smooth curve L connecting the points (0, b) and (a, 0). У статті Золотарьова В. О. „Про трикутні модєлі систем двічі переставних onepaTopiB" (Докл. АН АрмССР. - 1976. - 63, № 3. - С. 136-140) для системи двічі переставних лінійних обмежених цілком несамоспряжених операторів $\{A_1, A_2\} ([A_1, A_2] = 0, [A_1 ∗ , A_2] = 0)$ такої, що ранг $(A_1)_I (A_2)_I = 1 (2i(A_k )_I = A_k − A_k ∗,\; k = 1, 2)$ i спектр кожного із операторів $A_k, k = 1, 2$, зосереджено в нулі, побудовано трикутну модель, яка є системою операторів інтегрування по незалежній змінній в $L_{Ω^2}$, де $Ω = [0,a] x [0,b]$. В даній статті одержано узагальнення цього результату на випадок, коли область О модельного простору є компактом у $ℝ^2 $, обмеженим прямими $x = a,\; y = b$ i спадною гладкою кривою $L$, що з'єднує точки $(0, b)$ i $(a, 0)$. Institute of Mathematics, NAS of Ukraine 2014-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2115 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 1 (2014); 108–127 Український математичний журнал; Том 66 № 1 (2014); 108–127 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2115/1232 https://umj.imath.kiev.ua/index.php/umj/article/view/2115/1233 Copyright (c) 2014 Hatamleh R.; Zolotarev V. A. |
| spellingShingle | Hatamleh, R. Zolotarev, V. A. Хатамлех, Р. Золотарьов, В. А. On Two-Dimensional Model Representations of One Class of Commuting Operators |
| title | On Two-Dimensional Model Representations of One Class of Commuting Operators |
| title_alt | Про двовимірні модельні зображення одного класу комутуючих операторів |
| title_full | On Two-Dimensional Model Representations of One Class of Commuting Operators |
| title_fullStr | On Two-Dimensional Model Representations of One Class of Commuting Operators |
| title_full_unstemmed | On Two-Dimensional Model Representations of One Class of Commuting Operators |
| title_short | On Two-Dimensional Model Representations of One Class of Commuting Operators |
| title_sort | on two-dimensional model representations of one class of commuting operators |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2115 |
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