On isometric dilations of commutative systems of linear operators
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| citation_txt | On isometric dilations of commutative systems of linear operators / V.A. Zolotarev // Журнал математической физики, анализа, геометрии. — 2005. — Т. 1, № 2. — С. 192-208. — Бібліогр.: 9 назв. — англ. |
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Journal of Mathematical Physics, Analysis, Geometry
2005, v. 1, No. 2, p. 192�208
On isometric dilations of commutative systems
of linear operators
V.A. Zolotarev
Department of Mechanics and Mathematics, V.N. Karazin Kharkov National University
4 Svobody Sq., Kharkov, 61077, Ukraine
E-mail:Vladimir.A.Zolotarev@univer.kharkov.ua
Received October 28, 2004
The isometric dilation of two parameter semigroup T (n) = T
n1
1 T
n2
2 ,
where n = (n1; n2) 2 Z
2
+, for a commutative system fT1; T2g of linear
bounded operators, one of which is a contraction, kT1k � 1, is constructed.
The building of the dilation is based on characteristic qualities of the com-
mutative isometric expansion
�
Vs;
+
Vs
�2
s=1
, which was given in the previous
work by the author [8]. The isometric dilations U(n) and
+
U (n) of the
semigroups T (n) and T �(n) are shown to be unitarily linked.
Mathematics Subject Classi�cation 2000: 47A45.
Key words: dilation, commutative systems of linear operators.
The functional model of the contractive linear operator T is commonly con-
sidered as an analogue of the spectral decomposition for the nonunitary operator
T , [4, 9]. The construction of the functional models is based on the study of the
basic properties of the unitary dilation U of the operator T , [4].
In this work, the isometric dilation U(n) for the two-parameter semigroup
T (n) = T n11 T n22 where n = (n1; n2) 2 Z2
+ is constructed using the construction
of the commutative isometric expansion
�
Vs;
+
Vs
�2
1
for the commutative operator
system fT1; T2g such that kT1k � 1 (which was presented in the work [8]). The
construction of the dilation U(n) is based on consistency conditions for systems of
equations that are corresponding to the expansions fV1; V2g. Similarly, the isomet-
ric dilation fV1; V2g, n 2 Z
2
+, is constructed using corresponding consistency con-
ditions for equations that are corresponding to the expansions
�
+
V1;
+
V2
�
. It turns
The work has been done with support of the Weizmann Institute Warron Fund, Israel.
c
V.A. Zolotarev, 2005
On isometric dilations of commutative systems of linear operators
out that the dilations U(n) and
+
U (n) are acting in the separate Hilbert spaces
HN;� and HN�;��, besides, the spaces HN;� and HN�;�� are intersecting and their
intersection H = HN;� \HN�;�� has such property that U�(n1; 0)f =
+
U (n1; 0) f ,
where f 2 H and n1 2 Z+. Moreover, the restriction of the dilation U (n1; 0) on
H is a unitary operator such that PHU (n1; 0)jH = T n11 , n1 2 Z+.
I. Consider the commutative system of linear bounded operators fT1; T2g,
[T1; T2] = T1T2�T2T1 = 0; in the separable Hilbert space H. Hereinafter, we will
suppose that one of the operators of the system fT1; T2g, e.g., T1, is a contraction,
kT1k � 1. Following [6, 8], de�ne the commutative unitary expansion for the
system fT1; T2g.
De�nition 1. Let the commutative system of linear bounded operators fT1; T2g
be given in Hilbert space H where T1 is a contraction, kT1k � 1. The set of map-
pings
V1 =
�
T1 �
K
�
; V2 =
�
T2 �N
K
�
: H �E ! H � ~E;
+
V 1=
�
T �1 �
�� K�
�
;
+
V 2=
�
T �2 � ~N�
�� K�
�
: H � ~E ! H �E;
(1)
where E and ~E are Hilbert spaces, is called the commutative unitary expansion of
the commutative system of operators T1, T2 in H, [T1; T2] = 0, if there are such
operators �, � , N , � and ~�, ~� , ~N , ~� in the Hilbert spaces E and ~E, where �, � ,
~�, ~� are selfadjoint, that the following relations are taking place:
1)
+
V 1 V1 =
�
I 0
0 I
�
; V1
+
V 1=
�
I 0
0 I
�
;
2) V �
2
�
I 0
0 ~�
�
V2 =
�
I 0
0 �
�
;
+
V �
2
�
I 0
0 �
�
+
V 2=
�
I 0
0 ~�
�
;
3) T2�� T1�N = ��; T2 � ~N T1 = ~� ;
4) ~N �� �N = K�� ~�K;
5) ~NK = KN:
(2)
Consider the following class of commutative systems of linear operators
fT1; T2g.
Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2 193
V.A. Zolotarev
De�nition 2. The commutative system of operators T1, T2 is attributed to
the class C (T1) and is called the contracting T1 operator system if:
1)T1 is a contraction, kT1k � 1;
2)E = ~D1H � ~D2H; ~E = D1H � D2H;
3) dimT2 ~D1H = dimE; dimD1T2H = dim ~E;
4) operators D1j ~E ;
~D1T
�
2
���
T2
~D1H
; ~D1
���
E
; T �2D1jD1T2H
are boundedly invertible, where Ds = T �
s
Ts � I;
~Ds = TsT
�
s
� I; s = 1; 2:
(3)
It is easy to see that if fT1; T2g 2 C (T1) then unitary expansion (1) always
exists, [6, 8]. Indeed, let
=
p
~�1 =
p
�D1; � = ~D1T
�
2
q
��11 ; K =
p
~�1T
�
1 T
�
2
q
��11 ;
N = �
q
��11 T2 ~D2T
�
1
q
��11 ; ~N = �
q
~��11 T �1
~D2T
��1
1
q
~��11 ;
� =
q
��11 T2
�
~D2 �
~D1
�q
��11 ; ~� =
q
~��11 T ��12 (D2 �D1)
q
~��11 ;
� = �
q
��11 T1 ~D2T
�
1
q
��11 ; ~� = �
q
~��11 D2
q
~��11 :
� = �
q
��11 T2 ~D2T
�
2
q
��11 ; ~� = �
q
~��11 T ��12 T �1D2T1T
�1
2
q
~��11 ;
taking into account (3).
Then it is easy to see that relations 1)�5) (2) are true [8].
II. Following the work [8], de�ne the vector-functions of discrete argument
hn 2 H, un 2 E, vn 2 ~E at the points of integer-valued grid n = (n1;n2) 2 Z
2
+
(nk � 0; k = 1, 2; nk 2 Z). Consider [8] the system of equations8<
:
@1hn = T1hn +�un; h(0;0) = h0;
@2hn = T2hn +�Nun; n 2 Z
2
+;
vn = hn +Kun;
Vs
�
hn
un
�
=
�
@shn
vn
�
; s = 1; 2; (4)
where @1hn = h(n1+1;n2), @2hn = h(n1;n2+1) are the corresponding shifts by dif-
ferent variables. The next theorems are dedicated to the study of consistency
conditions for the discrete system of equations (4).
Theorem 1. The system (4) is consistent only if the vector-function un is
a solution of the equation
fN@1 � @2 + �gun = 0: (5)
194 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
On isometric dilations of commutative systems of linear operators
The proof of the theorem follows from the equality of the mixed shifts
@1@2hn = @2@1hn taking into account condition 3) (2), [8].
Theorem 2. Suppose that un is a solution of equation (5) and the vector-
functions hn and vn are given by relations (4). Then vn satis�es the following
equation n
~N@1 � @2 + ~�
o
vn = 0: (6)
The proof of the Theorem 2 is given in [8].
The following conservation laws
1)
k@1hnk
2 + kvnk
2 = khnk
2 + kunk
2 ;
k@2hnk
2 + h~�vn; vni = khnk
2 + h�un; uni ;
2) h(~�1 � ~�2) vn; vni+ h~�2@1vn; @1vni � h~�1@2vn; @2vni
= h(�1 � �2)un; uni+ h�2@1un; @1uni � h�1@2un; @2uni
(7)
are true for the discrete system of equations (4). Obviously, the relations 1) (7)
are a simple corollary of 1), 2) (2), while the equality 2) (7) follows from the
coincidence of the norms k@1@2hnk
2 = k@2@1hnk
2 and plays an important role
hereinafter.
Similarly to (4), consider (see [8]) the vector-functions ~hn 2 H, ~un 2 E,
~vn 2 ~E at the integer-valued grid points n = (n1;n2) 2 Z
2
�
(nk < 0; k = 1, 2;
nk 2 Z). De�ne the two-variable dual type of system of equations (4)
8<
:
~@1~hn = T �1
~hn + �~vn; ~h(�1;�1) = ~h�1;
~@2~hn = T �2
~hn + � ~N�~vn; n 2 Z
2
�
;
~un = ��~hn +K�~vn;
+
V s
�
~hn
~vn
�
=
�
~@s~hn
~un
�
; s = 1; 2;
(8)
where ~@1~hn = ~h(n1�1;n2),
~@2~hn = ~h(n1;n2�1) are shifts by di�erent variables for-
mally adjoint to @1 and @2, so that ~@s = @�
s
, s = 1; 2, in the metric of the space
l2. Statements similar to the Theorems 1 and 2 are true for the system (8).
Theorem 3. Consistency of the system of equations (8) takes place only if
~vn is the solution of the equationn
~N� ~@1 � ~@2 + ~��
o
~vn = 0: (9)
Theorem 4. Vector-function ~un (8) satis�es the following equation
fN�@1 � @2 + ��g ~un = 0 (10)
Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2 195
V.A. Zolotarev
under the conditions that ~vn is the solution of (9) and ~hn are given by relations
(8).
Similarly to (7), the following conservation laws
1)
~@1~hn
2 + k~unk
2 =
~hn
2 + k~vnk
2 ;
~@2~hn
2 + h� ~un; ~uni =
~hn
2 + h~� ~vn; ~vni ;
2) h(�1 � �2) ~un; ~uni+
D
�2 ~@1~un; ~@1~un
E
�
D
�1 ~@2~un; ~@2~un
E
= h(~�1 � ~�2) ~vn; ~vni+
D
~�2 ~@1~vn; ~@1~vn
E
�
D
~�1 ~@2~vn; ~@2~vn
E
(11)
are true for the dual system (8) in view of 2) (2).
III. Turn to the construction of the dilation for the operator systems fT1; T2g
of the class C (T1) (3). First of all, construct the unitary dilation [4, 6, 9] for the
contraction T1. As usually [6, 8], we will denote by l2
M
(G) the Hilbert space of G-
valued functions uk 2 G, where k 2M ðnd M � Z are such that
P
k2M
kukk
2
<1.
Let H be the Hilbert space of the following type
H = D� �H �D+; (12)
where D� = l2
Z�
(E) and D+ = l2
Z+
( ~E). Specify the dilation U on the vector-
functions f = (uk; h; vk) from H (12) in the following way:
Uf =
�
PD�uk�1;
~h; ~vk
�
; (13)
where ~h = T1h + �u�1, ~v0 = h +Ku�1, ~vk = vk�1 (k = 1; 2 : : :,) and PD� is
the operator of contraction on D�. The unitary property of U (13) in H follows
from 1) (2). Take advantage now of equations (5) and (6) as a way to continue
the incoming D� and outgoing D+ subspaces
D� = l2Z�(E); D+ = l2Z+(
~E) (14)
by the second variable �n2�. At �rst, continue functions un1 2 l2
Z�
(E) from the
semiaxis Z� into the domain
~Z2
�
= Z�� (Z� [ f0g) =
�
n = (n1;n2) 2 Z
2 : n1 < 0;n2 � 0
; (15)
using the following Cauchy problem(
~@2un =
�
N ~@1 + �
�
un; n = (n1; n2) 2 ~Z2
�
;
unjn2=0 = un1 2 l2
Z�
(E):
(16)
196 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
On isometric dilations of commutative systems of linear operators
As a result, we obtain the Hilbert space D�(N;�) which is formed by un, the
solutions of (16), at the same time the norm in D�(N;�) is induced by the norm
of initial data kunk = kun1kl2
Z
�
(E).
N o t e 1. Note that the formal continuation of the function un1 2 l2
Z�
(E) from
the semiaxis Z� using the Cauchy problem (16) has wider domain of existence
then ~Z2
�
(15). Really, if we continue un1 with nulls on Z+ then using recurrent
relation, we obtain un that is given in the cone K�:
K� =
�
n = (n1; n2) 2 Z
2 : n2 � 0;n1 + n2 < 0
: (17)
Similarly, continue functions vn1 2 l2
Z+
( ~E) from the semiaxis Z+ into the
domain Z
2
+ = Z+� Z+ using the Cauchy problem(
~@2vn =
�
~N ~@1 + ~�
�
vn; n = (n1; n2) 2 Z
2
+;
vnjn2=0 = vn1 2 l2
Z+
(E):
(18)
Thus, we obtain Hilbert space D+( ~N; ~�) that is made of solutions vn (18),
besides kvnk = kvn1kl2
Z+
( ~E). Unlike the evident recurrent scheme (16) of the
layer-to-layer calculation of n2 ! n2�1 for un, in this case, while constructing vn
in Z2
+, we are dealing with the implicit linear system of equations for layer-to-layer
calculation of n2 ! n2 + 1 for the function vn. Therefore it is necessary to study
solvability and uniqueness of Cauchy problem (18). First, study reversibility of
linear pencils of operators Nz + � and ~Nz + ~�.
Lemma 1. Suppose the commutative unitary expansion Vs,
+
V s (1) is such
that
Ker� = Ker � = f0g (19):
Then KerN\Ker� = f0g given KerK� = f0g, and respectively Ker ~N�\Ker ~�� =
0 given KerK = f0g.
P r o o f. Let G = KerN \ Ker� then it follows from the equality T2� =
T1�N + �� that the subspace L = span
�
T k1 �g : g 2 G; k 2 Z+
from H has
properties T1L � L, T2L = 0. It follows from the equality T �2 T2 + �~� = I
that h = �~� h takes place for all h 2 L, therefore �g = �~g = �~� �g
and so ~g = ~� �g, in view of Ker � = 0 (19). Since T �2�N + �~�K = 0 then
K�~g = K�~� �g = �N���T2�g = 0; then ~g = 0 because of KerK� = 0. So
�g = �~g = 0 and thus g = 0 in view of Ker� = 0 (19). Similarly, one proves
the second statement of the lemma.
N o t e 2. Note that if the suppositions of the Lemma 1 are true and the
spaces E and ~E are �nite dimensional, then the linear pencils Nz+� and ~N�z+~��
Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2 197
V.A. Zolotarev
are reversible operators for all z 2 C , except for the �nite number of points that
are zeroes of polynomials det(Nz +�) = 0 and det
�
~N�z + ~��
�
= 0 respectively.
Since reversibility of ~Nz + ~� and of the adjoint to it operator ~N��z + ~�� are
equivalent in the �nite dimensional space ~E, then reversibility of ~Nz + ~� follows
from the Lemma 2 when dim ~E <1.
Turn to the solvability of Cauchy problem (18).
Statement 1. Let dim ~E <1 and the assumptions of the Lemma 1 be true,
then the solution vn of Cauchy problem (18) exists and is unique in the domain
Z
2
+ for all initial data vn1 from l2
Z+
( ~E).
P r o o f. First, consider the case of the �nite initial data vn1 , i.e., let vn1 = 0
when n1 > n, where n 2 Z+. Show that the vector-function v (n1; 1) which is a
solution of problem (18) that also turns to zero when n1 > n, is uniquely de�ned
by initial data vn1 . It is necessary to prove that the homogeneous linear system
of equations generated by (18) has only trivial solution. It follows from (18) when
vn1 = 0, that function v (n1; 1) satis�es the system of equations
8>>>><
>>>>:
~�v(0; 1) = 0;
~Nv(0; 1) + ~�v(1; 1) = 0;
:::
~Nv(n� 1; 1) + ~�v(n; 1) = 0;
~Nv(n; 1) = 0:
(20)
Multiply the second equality in (20) by z, the third one � by z2, and so on, �nally,
the last one � by zn+1 (z 2 C ); then after summation we obtain that
( ~Nz + ~�) fv(0; 1) + zv(1; 1) + � � �+ znv(n; 1)g = 0:
It follows from the Note 2, in view of reversibility of ~Nz + ~�, that
nX
k=0
zkv(k; 1) = 0
for all z 2 C except for a �nite number of points. Therefore v(k; 1) = 0 for all
k, 0 � k � n. Thus, the �rst layer v(k; 1) is de�ned from equations (18) by
the initial data vk, 0 � k � n unambiguously. Realizing in that way layer-to-
layer reconstruction of v(k; p + 1) by v(k; p), we will obtain the unique solution
of the Cauchy problem (18) in the domain Z
2
+. The general case follows from the
considered case of the �nite initial data as a result of natural approximation.
198 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
On isometric dilations of commutative systems of linear operators
N o t e 3. It is not di�cult to establish (similarly to the Note 1) that the
solution of Cauchy problem (18) exists in the conic domain K+:
K+ =
�
n = (n1; n2) 2 Z
2 : n2 � 0;n1 + n2 � 0
: (21)
Consider now the operator-function of discrete argument
~�� =
�
I : � = (1; 0);
~�; � = (0; 1):
(22)
Let Ln0 be the nonincreasing broken line that connects points O = (0; 0) and
n = (n1; n2) 2 Z
2
+ and linear segments of which are parallel to the axes OX
(n2 = 0) and OY (n1 = 0). Denote by fPkg
N
0 all integer-valued points from
Z
2
+, Pk 2 Z
2
+ (N = n1 + n2) that lay on Ln0 , beginning with (0; 0) and �nishing
with the point (n1; n2), that are numbered in nondescending order (of one of the
coordinates of Pk). Assuming that P�1 = (�1; 0), establish the quadratic form
h~�vki
2
L
n
0
=
NX
k=0
~�P
k
�P
k�1
vP
k
; vP
k
�
; (23)
on the vector-functions vk 2 D+( ~N; ~�).
Similarly, consider the nondecreasing broken line L�1
m
in ~Z2
�
(15) that connects
pointsm = (m1;m2) 2 ~Z2
�
and (�1; 0), the straight segments of which are parallel
to OX and OY . Let fQsg
�1
M
(M = m1+m2) be all integer-valued points on L�1
m
,
beginning with m = (m1;m2) and �nishing with (�1; 0), that are numbered
in nondescending order (of one of the coordinates of Qs). De�ne the metric in
D�(N;�),
h�uki
2
L
�1
m
=
�1X
s=M
�Qs�Qs�1uQs ; uQs
�
; (24)
besides QM � QM�1 = (1; 0), and the operator-function �� is de�ned similarly
to ~�� (22). Denote by ~L�1
�n
the broken line in ~Z2
�
that is obtained from the curve
Ln0 in Z
2
+ (n 2 Z
2
+) using the shift by �n�:
~L�1
�n
=
n
Qs = (l1; l2) 2 ~Z2
�
: (l1 + n1 + 1; l2 + n2) = Pk 2 Ln0
o
: (25)
IV. Having now the Hilbert space D�(N;�), that is formed by the solutions
of Cauchy problem (16), and space D+( ~N; ~�), that is formed by the solutions of
(18) respectively, we can de�ne Hilbert space
HN;� = D�(N;�)�H �D+( ~N; ~�); (26)
Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2 199
V.A. Zolotarev
the norm in which is de�ned by the norm of the initial space H = D� �H �D+
(12). Denote by Ẑ2
+ the subset in Z
2
+,
Ẑ
2
+ = Z
2
+n(f0g � N) = f(0; 0)g [ (N �Z+); (27)
that obviously is an addition semigroup.
For every n 2 Ẑ
2
+ (27), de�ne an operator-function U(n) that acts on the
vectors f = (uk; h; vk) 2 Hn;� (26)in the following way:
U(n)f = f(n) = (uk(n); h(n); vk(n)) ; (28)
where uk(n) = PD�(N;�)uk�n (PD�(N;�) is an orthoprojector that corresponds
with the restriction on D�(N;�)); h(n) = y0, besides yk 2 H (k 2 Z
2
+) is a
solution of the Cauchy problem8<
:
~@1yk = T1yk +�u~k;
~@2yk = T2yk +�Nu~k;
yn = h; k = (k1; k2) 2 Z
2
+ 0 � k1 � n1 � 1; 0 � k2 � n2;
(29)
at the same time ~k = k � n, when 0 � k1 � n1 � 1, 0 � k2 � n2, and �nally
vk(n) = v̂k + vk�n (30)
and v̂k = Ku~k + yk, where yk is a solution of the Cauchy problem (29).
The vector-function u~k, that is obtained as a result of the shift by �n�, auto-
matically satis�es the consistency equation (5), since, according to the construc-
tion, uk is a solution of the Cauchy problem (16). And it follows from the equation
(6) that vk(n) (30) continues uniquely into the whole domain Z
2
+ as a solution of
the equation (18), that is always possible in the context of the suppositions of the
Statement 1.
The following facts justify that U(n) (28) is de�ned if n 2 Ẑ
2
+ (27): �rst,
fT1; T2g 2 C (T1) (3); second, the choice of the metric (23), and third, the con-
struction of the space D+( ~N; ~�) that is generated by the Cauchy problem (18)
with the initial data from the semiaxis Z+.
Thus, the operator-function U(n) (28) maps the space HN;� (26) into itself
for all n 2 Ẑ
2
+ (27).
Theorem 5. Suppose dim ~E < 1 and the suppositions of Lemma 1 are
taking place, then the following conservation law is true for the vector-function
f(n) = U(n)f (28):
kh(n)k2 + h~�vk(n)i
2
L
n̂
0
= khk2 + h�uki
2
~L�1
�n
(31)
200 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
On isometric dilations of commutative systems of linear operators
for all n 2 Ẑ
2
+ (27) and for all nondecreasing broken lines L̂n̂0 that connect points
O = (0; 0) and n̂ = (n1 � 1; n2) 2 Z
2
+, where
~L�1
�n̂
is a broken line that is obtained
from Ln0 by the shift (25) by �n�, at the same time the corresponding �-forms in
(31) have the appearance of (23) and (24). The operator-function U(n) (28) is a
semigroup, U(n) � U(m) = U(n+m), for all n, m 2 Ẑ
2
+ (27).
P r o o f. The equality (31) easily follows from the isometric correspondence
of the operators V1, V2 (1) in accordance with 1) and 2) (2). The fact that the
operator-function U(n) (28) is a semigroup when n 2 Ẑ
2
+ (27) follows from the
elementary calculations taking into account the continuation of the function vk(n)
(30) into the domain Z
2
+ by the equation (18).
It follows from (31) that it is natural to de�ne in the space HN;� (26) the
inde�nite, generally speaking, metric
hfi2
�
= h�uki
2
L
�1
�1
+ khk2 + h~�vki
2
L
1
0
; (32)
where L10 and L�1
�1
are nondecreasing broken lines in Z2
+ and in Ẑ2
�
(15) connect-
ing point O = (0; 0) with 1 = (1;1) and point �1 = (�1;�1) with (�1; 0)
respectively, straight segments of these broken lines are parallel to the axes OX
and OY .
Consider the subspace K from Z
2
+ that contains O = (0; 0) and is an addition
semigroup. T (n) denotes the semigroup of linear operators over K,
T (n) = T
n1
1 T
n2
2 ; n = (n1; n2) 2 K; (33)
assuming that the commutative system of linear operators fT1; T2g belongs to the
class C (T1) (3).
De�nition 3. [4] Semigroup of operators U(n); U(n)U(m) = U(n+m); 8n,
m 2 K, that is given in the Hilbert space H such that
H � H; PHU(n)jH = T (n); n 2 K; (34)
where PH is an orthoprojector on H, is called the dilation of a discrete operator
semigroup T (n) (33) that is acting in the Hilbert space H. If for every n 2 K
the operator-function U(n) is an isometric or unitary operator in H then U(n) is
called isometric or unitary dilation T (n).
Consider the family of one-parameter semigroup G+(p) in Z
2
+,
G+(p) =
n
np : p 2 Ẑ
2
+; n 2 Z+
o
; (35)
Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2 201
V.A. Zolotarev
besides the point p = (p1; p2) 2 Ẑ
2
+ is such that numbers p1 and p2 are coprime.
In particular, if p1 = (1; 0) then it is obvious that G+(p) = Z+. Narrow now the
semigroup T (n) (33) on G+(p) (35), i.e., for the given p = (p1; p2) 2 Ẑ
2
+ consider
the one-parameter semigroup Tn(p) = (T p11 T
p2
2 )
n
from n 2 Z+, which looks like
Tn (p1) = T n1 when p = p1 = (1; 0). Choose now �xed broken line L
p
0 with linear
segments that are parallel to the axes OX and OY , which connects points O and
p 2 Ẑ
2
+; and then make its group shift in Z
2
+,
L10 (p) = fn+ kp : n 2 L
p
0; k 2 Z+g (36)
and similarly shift L
p
0 in ~Z2
�
,
L�1
�1
(p) = fn+ k (p1 + 1; p2) : n 2 L10 ; k 2 Z�g : (37)
In accordance with (32), specify the quadratic form inHN;� (26) that is associated
with the semigroup G+(p) (35),
hfi2
�;p
= h�uki
2
L
�1
�1
(p)
+ khk2 + h~�vki
2
L
1
0
(p) : (38)
The next statement follows from the Theorem 5.
Theorem 6. Suppose fT1; T2g 2 C (T1) (3), dim ~E <1 and the suppositions
of the Lemma 1 are true, then for every p 2 Ẑ
2
+ (27) the operator semigroup
Tn(p) = T (np) that is narrowed on G+(p) (35) has the isometric (in metrichfi2
�;p
(38)) dilation Un(p) = U(np) (28) that acts in the Hilbert space HN;� (26).
N o t e 4. Using the semigroup property of dilation Un(p) (28) by parameter
n 2 Z+ and isometric property of Un(p) in metric (38), we obtain that
Un(p)h;Um(p)h
0
�
�;p
=
Tn�m(p)h; h
0
�
; (39)
when n � m (n,m 2 Z+) and for all h, h0 2 H. Thus, the subspace
span
n
Un(p)H : n 2 Z+; p 2 Ẑ
2
+
o
in HN;� (26) is de�ned by the initial commutative operator system fT1; T2g 2
C (T1) (3).
V. Similarly to the stated in the Paragraph III method of continuation of
subspaces D+ and D� (14) from the semiaxes Z+ and Z� by the second variable
�n2�, consider the dual situation corresponding to equations (9) and (10). Denote
by D+
�
~N�; ~��
�
the Hilbert space generated by solutions ~vn of Cauchy problem
(
@2~vn =
�
~N�@1 + ~��
�
~vn; n = (n1; n2) 2 Z
2
+;
~vnjn2=0 = vn1 2 l2
Z+
( ~E):
(40)
202 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
On isometric dilations of commutative systems of linear operators
the norm in which is induced by the norm of the initial data(
@2~vn =
�
~N�@1 + ~��
�
~vn; n = (n1; n2) 2 Z
2
+;
~vnjn2=0 = vn1 2 l2
Z+
( ~E):
(40)
Continuing the function vn1 2 l2
Z+
( ~E) by null on the left semiaxis, as in the case
of (18), it is easy to establish that the solution of the Cauchy problem (40) exists
in the cone K+ (21).
Continue now every function un1 2 l2
Z�
(E) into the domain ~Z2
�
(15) using the
Cauchy problem�
@2~un = (N�@1 + ��) ~un; n = (n1; n2) 2 ~Z2
�
;
~unjn2=0 = un1 2 l2
Z�
(E):
(41)
As a result, we obtain the Hilbert space D� (N�;��) generated by ~un, solutions
of (41), besides k~unk = kun1kl2
Z
�
(E). Constructing the solutions ~un of the Cauchy
problem (41), we have the implicit scheme of layer-to-layer calculation of n2 !
n2 � 1 solutions ~un. Using now the Lemma 1 and Note 2, we can formulate an
analogue of the Statement 1.
Statement 2. Let dimE <1 and the suppositions of Lemma 1 be true, then
the solution ~un of the Cauchy problem (41) exists and is unique in the domain ~Z2
�
(15) for all initial data un1 2 l2
Z�
(E).
Note that, as in the case of the problem (40), solutions of the Cauchy problem
(41) have wider domain of existence and uniqueness, namely, K� (17).
N o t e 5. The su�cient condition for the simultaneous existence of solutions
of Cauchy problems (18) and (41), in view of the reversibility of operators K and
K�, according to the Lemma 1, is following: all the requirements of the Lemma 1
are met and dimE = dim ~E <1.
Hence we come to the Hilbert space
HN�;�� = D� (N�;��)�H �D+
�
~N�; ~��
�
; (42)
the metric in which is induced by the norm of the initial space H = D��H�D+
(12). Note the dual features of the spaces HN;� (26) and HN�;�� (32), which
consist in that that di�erential operators of Cauchy problems (16) and (41) and
operators (18) and (40) also, are adjoint with each other correspondingly in the
metric l2.
De�ne now in the space HN�;�� (42) the operator-function
+
U (n) for n 2 Ẑ
2
+
(27), which acts on ~f =
�
~uk; ~h; ~vk
�
2 HN�;�� in the following way:
+
U (n) ~f = ~f(n) =
�
~uk(n); ~h(n); ~v(n)
�
; (43)
Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2 203
V.A. Zolotarev
where ~vk(n) = P
D+( ~N�;~��)~vk+n (PD+( ~N�;~��) is an orthoprojector onD+
�
~N�; ~��
�
);
~h(n) = ~y(�1;0), besides ~yk (k 2 ~Z2
�
) satis�es the Cauchy problem8<
:
@1~yk = T �1 ~yk + �~v~k;
@2~yk = T �2 ~yk + � ~N�~v~k;
~y(�n1;�n2) = h; k = (k1; k2) 2 ~Z2
�
(�n1 � k1 � �1; �n2 � k2 � 0);
(44)
besides ~k = k + n ø (�n1 � k1 � �1; �n2 � k2 � 0); and �nally
~uk(n) = ûk + ~uk+n; (45)
and ûk = K�~v~k +��~yk, where ~yk is a solution of the system (44).
As in the case of the mapping U(n) (28), the function ~v~k is obtained after the
shift by ��n� and automatically satis�es the consistency condition for (10) and
the function ~uk(n) (45) has natural continuation into the whole domain ~Z2
�
(15)
on account of the equation (41).
Similarly to (22), de�ne the operator-function
�� =
�
I; � = (�1; 0);
� ; � = (0;�1):
(46)
Denote by L�1
m
the nondecreasing broken line in ~Z2
�
(15) with linear segments that
are parallel to the axes OX and OY which connects points m = (m1;m2) 2 ~Z2
�
and (�1; 0). Choose now all the points fQsg
�1
M
(M =m1 +m2) on L
�1
m
that are
numerated in the nonascending order (of one of the coordinates Qs) beginning
with the point (�1; 0) and �nishing with m = (m1;m2) 2 ~Z2
�
. De�ne in the
space D� (N�;��) the quadratic form
h� ~uki
2
L
�1
m
=
�1X
s=M
�Qs�Qs+1 ~uQs ; ~uQs
�
; (47)
where Q0 = (0; 0). For the broken line Ln0 in Z
2
+, n = (n1; n2) 2 Z
2
+, of the
similar type with points fPkg
N
0 (N = n1+n2) on L
n
0 which are also chosen in the
nonascending order, de�ne the quadratic form for the functions ~vk 2 D+
�
~N�; ~��
�
h~� ~vki
2
L
n
0
=
NX
k=0
~�P
k
�P
k+1
~vP
k
; ~vP
k
�
; (48)
where PN � PN+1 = (�1; 0) and ~�� is de�ned similarly to �� (48). Denote by
~Lm0 the broken line in Z
2
+ obtained from the curve L�1
m
from ~Z2
�
using the shift
by �m�
~Lm0 =
�
Pk = (l1; l2) 2 Z
2
+ : (l1 +m1; l2 +m2) = Qs 2 L�1
m
; (49)
204 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
On isometric dilations of commutative systems of linear operators
where m = (m1;m2) 2 ~Z2
�
. Similarly to the Theorem 5, the following statement
takes place.
Theorem 7. Suppose dimE < 1 and the requirements of the Lemma 1 are
met, then for the vector-function ~f(n) =
+
U (n) ~f (43) the equality
k~h(n)k2 + h� ~uk(n)i
2
L
�1
�n
= khk2 + h~�~vki~L�n
0
(50)
takes place for all n 2 Ẑ
2
+ (27) and for all broken lines L�1
�n
connecting points
�n = (�n1;�n2) 2 ~Z2
�
and (�1; 0) where ~L�n0 is a curve in Z
2
+ obtained from
L�1
�n
using the shift (49) by ��n� and corresponding � -forms in (50) have the
appearance of (47) and (48). The operator-function
+
U (n) (43) has the semigroup
property,
+
U (n)
+
U (m) =
+
U (n+m) for all n, m 2 Ẑ
2
+ (27).
As in the case of the theorem 5, the proof is reduced to the use of the isometric
property of
+
V 1 and
+
V 2 (1) in view of 1) and 2) (2). The check of the semigroup
property of the operator-function
+
U (n) (43) is quite simple as in the proof of the
Theorem 5.
De�ne in HN�;�� (43) the quadratic form
h ~fi2
�
= h� ~uki
2
L
�1
�1
+ k~hk2 + h~�~vki
2
L
1
0
; (51)
where L�1
�1
and L10 are nondecreasing allowable broken lines in ~Z2
�
and Z2
+ (with
segments parallel to the axes OX and OY ) connecting points �1 = (�1;�1)
with (�1; 0) and (0; 0) with 1 = (1;1) respectively.
Further, consider the family of one-parameter semigroup in ~Z2
�
[ (0; 0)
G�(q) =
n
nq : q = (q1; q2) 2 ~Z2
�
;n 2 Z+
o
; (52)
where numbers q1 and q2 are coprime ideals and, moreover, (�q1 � 1;�q2) 2
Ẑ
2
+ (27). Choose �xed allowable broken line L�1~q in ~Z2
�
connecting points ~q =
(q1; q2) 2 ~Z2
�
(where (�q1 � 1;�q2) 2 Ẑ
2
+) and (�1; 0) and make its group shift
in ~Z2
�
,
L�1
�1
(q) =
n
n+ kq : n 2 L�1~q ; k 2 Z+
o
; (53)
and in Z
2
+,
L10 (q) =
�
n+ kq : n 2 L�1
q
; k 2 Z�
; (54)
respectively. Similarly to (38), de�ne the metric along G�(p) (52) in the space
HN�;��
hfi2
�;q
= h� ~uki
2
L
�1
�1
(q)
+ k~hk2 + h~�~vki
2
L
1
0
(q) ; (55)
Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2 205
V.A. Zolotarev
where broken lines L�1
�1
(q) and L10 (q) have the appearance of (53) and (54).
Theorem 8. Suppose fT1; T2g 2 C (T1) (3), dimE <1; and the conditions
of lemma 1 are met, then for all q 2 ~Z2
�
(15) such that (�q1 � 1;�q2) 2 Ẑ
2
+ (27),
the operator semigroup T �
n
(q) = T �(�nq) (33) from n 2 Z+, narrowed on G�(q)
(50), always has an isometric (in the metric h ~fi2
�;q
(55)) dilation
+
Un (jqj) =
+
U
(�nq) (43) which acts in the space HN�;�� (42).
N o t e 6. For the dual dilation
+
U (jqj) =
+
U (�nq) (43), as well as for
Un(p) = U(np) (28), the relation�
+
Un (jqj)h;
+
Um (jqj)h0
�
�;q
=
T �
n�m
(q)h; h0
�
(56)
is true when n � m and for all h, h0 2 H. Hence, the subspace
span
�
+
Un (jqj)h : h 2 H;n 2 Z+; (�q1 � 1;�q2) 2 Ẑ
2
+
�
in HN�;�� (42) is de�ned by the initial operator system fT1; T2g from C (T1) (3).
V. Note that Hilbert spaces HN;� (26) and HN�;�� (42) have the common
part, namely the space H (12) which per se de�nes them in view of the corre-
sponding Cauchy problems (16), (18) and (40), (41). Moreover, narrowings of
the dilations U (n1; 0) (28) and
+
U (n1; 0) (43) on the invariant subspace H are
unitary operators, besides U� (n1; 0) =
+
U (n1; 0) 8n1 2 Z+. It follows from the
Note 4 that the dilation U(n) (28) has the �+ minimality� property, that means
the �observability� of the system (4), and it follows from the Note 6 respectively
that dilation
+
U (n) (43) satis�es �� minimality� condition, that corresponds with
�controllability� of the open system (8), [2, 7, 9]. The next de�nition follows from
notes made earlier.
De�nition 4. Consider the operator semigroup T (n), de�ned when n 2 Ẑ
2
+
(27), that corresponds to the commutative operator system fT1; T2g from the class
C (T1) (3). Let U(n) be the isometric dilation (in terms of the De�nition 3 of
the semigroup T (n)) that acts in the space H+ and the operator-function
+
U (n),
de�ned in H�, be the isometric dilation of the adjoint semigroup T �(n). The pair
of dilations U(n) and
+
U (n) is called minimally-unitarily connected if the following
conditions are met.
1) The Hilbert space H0 = H+ \ H� is invariant with regard to the operator-
functions U (n1; 0) and
+
U (n1; 0) 8n1 2 Z+, besides restrictions of U (n1; 0) and
206 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
On isometric dilations of commutative systems of linear operators
+
U (n1; 0) on H0 are unitary operators and, moreover, U� (n1; 0) =
+
U (n1; 0) 8n1 2
Z+.
2) Restriction of the semigroup U (n1; 0) on H0 is the minimal [4, 9] unitary
dilation of the semigroup T n11 when n1 2 Z+,
H0 = span fU (n1; 0) h : h 2 H;n 2 Zg :
3) The equalities
H+ = span
n
U(n)H0 : n 2 Ẑ
2
+
o
;
H� = span
�
+
U (n)H0 : n 2 Ẑ
2
+
�
are taking place.
Note that Point 3) of the De�nition 4 means that there are no adduction
subspaces in H+ and H� for the operators U(n) and
+
U (n) on which U(n) and
+
U (n) are unitary and which are not connected with the initial system fT1; T2g.
It is easy to see that minimally-unitary connected dilations U(n) in H+ and
+
U (n) in H� are de�ned up to isomorphism. As is well known [4, 9], the minimal
unitary dilation U (n1; 0) of the contraction semigroup T
n1
1 (n1 2 Z+) in H0 is
de�ned uniquely (up to isomorphism). And from the point 3) of the De�nition 4
follows that corresponding isomorphism between U(n) in H+ and U 0(n) in H0
+
(for example) could be de�ned in the following way: U(n)f ! U 0(n)f where
f 2 H0, though this correspondence not necessarily is a unitary operator. Note
that from the constructions of the dilations U(n) (28) in HN;� (26) and
+
U (n) (43)
in HN�;�� (42), it follows that the pair U(n) and
+
U (n) is de�ned �unambiguously�
by the initial operator system fT1; T2g from C (T1) (3) in accordance with (49)
and (55).
References
[1] M.S. Liv�sic, Commutative operators and solutions of systems of di�erential equa-
tions in partial derivatives that are generated by them. � Soobsh. Acad. Nauk
Gruz.SSR 91 (1978), No. 2, 281�284. (Russian)
[2] M.S. Liv�sic, N. Kravitsky, A. Markus, V. Vinnikov, Theory of commuting non-
selfadjoint operators. Math. and Appl. 332, Kluver Acad. Publ. Groups, Dordrecht,
1995.
[3] V.A. Zolotarev, Time cones and functional model on the Riemann surface. � Mat.
Sb. 181 (1990), 965�995. (Russian)
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V.A. Zolotarev
[4] B. Szekefalvi-Nagy and Ch. Foya�s, Harmony analisys of operators in the Hilbert
space. Mir, Moscow, 1970. (Russian)
[5] S. Parrott, Unitary dilations for commuting contractions. Preprint, Boston, 1969.
[6] V.A. Zolotarev, Model representations of commutative systems of linear operators.
� Funkts. Analiz i yego Prilozhen. 22 (1988), 66�68.
[7] M.S. Brodskiy, Unitary operator knots and their characteristic functions. � Uspekhi
Mat. Nauk 33 (1978), No. 4, 141�168. (Russian)
[8] V.A. Zolotarev, Isometric expansions of commutative systems of linear operators.
� Mat. �z., analiz, geom. 11 (2004), 282�301. (Russian)
[9] V.A. Zolotarev, Analitic methods of spectral representations of nonselfadjoint and
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208 Journal of Mathematical Physics, Analysis, Geometry , 2005, v. 1, No. 2
|
| id | nasplib_isofts_kiev_ua-123456789-106573 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1812-9471 |
| language | English |
| last_indexed | 2025-12-02T10:25:10Z |
| publishDate | 2005 |
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| spelling | Zolotarev, V.A. 2016-09-30T18:02:49Z 2016-09-30T18:02:49Z 2005 On isometric dilations of commutative systems of linear operators / V.A. Zolotarev // Журнал математической физики, анализа, геометрии. — 2005. — Т. 1, № 2. — С. 192-208. — Бібліогр.: 9 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106573 en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии On isometric dilations of commutative systems of linear operators Article published earlier |
| spellingShingle | On isometric dilations of commutative systems of linear operators Zolotarev, V.A. |
| title | On isometric dilations of commutative systems of linear operators |
| title_full | On isometric dilations of commutative systems of linear operators |
| title_fullStr | On isometric dilations of commutative systems of linear operators |
| title_full_unstemmed | On isometric dilations of commutative systems of linear operators |
| title_short | On isometric dilations of commutative systems of linear operators |
| title_sort | on isometric dilations of commutative systems of linear operators |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/106573 |
| work_keys_str_mv | AT zolotarevva onisometricdilationsofcommutativesystemsoflinearoperators |