On the behavior of orbits of uniformly stable semigroups at infinity
For uniformly stable bounded analytic $C_0$-semigroups $\{T(t)\} t ≥ 0$ of linear operators in a Banach space $B$, we study the behavior of their orbits $T (t)x, x ∈ B$, at infinity. We also analyze the relationship between the order of approaching the orbit $T (t)x$ to zero as $t → ∞$ and the degre...
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| Дата: | 2006 |
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| Автори: | , , , |
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| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2006
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3443 |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509534095671296 |
|---|---|
| author | Gorbachuk, V. I. Gorbachuk, M. L. Горбачук, В. І. Горбачук, М. Л. |
| author_facet | Gorbachuk, V. I. Gorbachuk, M. L. Горбачук, В. І. Горбачук, М. Л. |
| author_sort | Gorbachuk, V. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:54:47Z |
| description | For uniformly stable bounded analytic $C_0$-semigroups $\{T(t)\} t ≥ 0$ of linear operators in a Banach space $B$, we study the behavior of their orbits $T (t)x, x ∈ B$, at infinity. We also analyze the relationship between the order of approaching the orbit $T (t)x$ to zero as $t → ∞$ and the degree of smoothness of the vector $x$ with respect to the operator $A^{-1}$ inverse to the generator A of the semigroup $\{T(t)\}_{t \geq 0}$. In particular, it is shown that, for this semigroup, there exist orbits approaching zero at infinity not slower than $e^{-at^{\alpha}}$, where $a > 0,\; 0 < \alpha < \pi/(2 (\pi - 0 )),\; \theta$ is the angle of analyticity of $\{T(t)\}_{t \geq 0}$, and the collection of these orbits is dense in the set of all orbits. |
| first_indexed | 2026-03-24T02:42:38Z |
| format | Article |
| fulltext |
UDK 517.9
V.�I.�Horbaçuk, M.�L.�Horbaçuk (In-t matematyky NAN Ukra]ny, Ky]v)
PRO POVEDINKU NA NESKINÇENNOSTI ORBIT
RIVNOMIRNO STIJKYX PIVHRUP
*
For uniformly stable bounded analytic C0-semigroups { }( )T t t≥0 of linear operators in a Banach space
�, the behavior at infinity of their orbits T ( t ) x , x ∈ �, is studied. The dependence of order of the
tending to zero of an orbit T ( t ) x as t → ∞ on the degree of smoothness of a vector x with respect to
the operator A
–1
inverse of the generator A of the semigroup { }( )T t t≥0 is investigated. In particular,
it is shown that there exist orbits of such semigroup which tend to zero at infinity not slower than e at− α
,
where a > 0, 0 < α < π / (2 ( π – θ )) , θ is an analyticity angle of { }( )T t t≥0 , and the collection of these
orbits is dense in the set of all orbits.
Vyvça[t\sq povedinka na neskinçennosti orbit T ( t ) x , x ∈ � , rivnomirno stijkyx obmeΩenyx
analityçnyx C 0 -pivhrup { }( )T t t≥0 linijnyx operatoriv u banaxovomu prostori � . Doslid-
Ωu[t\sq zaleΩnist\ miΩ porqdkom prqmuvannq orbity T ( t ) x do 0 pry t → ∞ i stepenem hlad-
kosti vektora x vidnosno operatora A
–1
, obernenoho do heneratora A pivhrupy { }( )T t t≥0 .
Zokrema pokazano, wo dlq tako] pivhrupy isnugt\ orbity, wo prqmugt\ do 0 na ∞ ne1povil\ni-
ße, niΩ e at− α
, de a > 0, 0 < α < π / (2 ( π – θ )) , θ — kut analityçnosti { }( )T t t≥0 , i sukupnist\
cyx orbit [ wil\nog u mnoΩyni vsix orbit.
1. Nexaj { }( )T t t≥0 — C0 -pivhrupa obmeΩenyx linijnyx operatoriv u kompleks-
nomu banaxovomu prostori � z normog || ⋅ || , A — ]] henerator. Zhidno z [1],
{ }( )T t t≥0 nazyva[t\sq rivnomirno stijkog, qkwo
∀x ∈ � : lim ( )
t
T t x
→∞
|| || = 0,
i rivnomirno eksponencial\no stijkog, qkwo isnugt\ stali c > 0 ta ω > 0 taki,
wo
∀t ≥ 0 : || T ( t ) || ≤ ce t−ω .
Vidomo (dyv., napryklad, [2]), wo obydva ci ponqttq ekvivalentni pry dim � <
< ∞ . Ce, vzahali kaΩuçy, ne tak, qkwo dim � = ∞ .
Z pryncypu rivnomirno] obmeΩenosti vyplyva[, wo dlq rivnomirno stijko]
{ }( )T t t≥0 zavΩdy isnu[ stala c > 0 taka, wo
∀t ≥ 0 : || T ( t ) || ≤ c .
Ne zmenßugçy zahal\nosti, pry rozhlqdi pytan\ stijkosti moΩna vvaΩaty c = 1,
tobto { }( )T t t≥0 — pivhrupog styskiv. U podal\ßomu perevaΩno budemo maty
spravu same z takymy pivhrupamy.
Poznaçymo çerez D ( ⋅ ) , ρ ( ⋅ ) , σ ( ⋅ ) , σc ( ⋅ ) oblast\ vyznaçennq, rezol\vent-
nu mnoΩynu, spektr i neperervnyj spektr operatora. Nastupne tverdΩennq
*
Vykonano za pidtrymky U. S. CRDF i Urqdu Ukra]ny (proekt UM 1-2090), a takoΩ DFG 436
UKR 113/78/0-1 ta 113/79.
© V.1I.1HORBAÇUK, M.1L.1HORBAÇUK, 2006
148 ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
PRO POVEDINKU NA NESKINÇENNOSTI ORBIT RIVNOMIRNO STIJKYX PIVHRUP 149
xarakteryzu[ rivnomirnu ta rivnomirno eksponencial\nu stijkosti pivhrupy v
terminax spektra ]] heneratora (dyv. [3, 4]).
TverdΩennq 1. Dlq toho wob pivhrupa { }( )T t t≥0 z heneratorom A bula
rivnomirno stijkog, neobxidno, wob
0 ∈ σc ( A ) ∪ ρ ( A ) . (1)
Qkwo { }( )T t t≥0 rivnomirno eksponencial\no stijka, to
0 ∈ ρ ( A ) . (2)
U vypadku, koly { }( )T t t≥0 [ obmeΩenog analityçnog, umovy (1) i, vidpovidno,
(2) [ takoΩ dostatnimy.
Nahada[mo, wo pivhrupa { }( )T t t≥0 nazyva[t\sq obmeΩenog analityçnog z
kutom analityçnosti θ ∈ ( 0, π / 2 ] , qkwo vona dopuska[ prodovΩennq do opera-
tor-funkci], analityçno] v sektori
Σ ( θ ) = { z ∈ C : | arg z | < θ },
syl\no neperervno] v nuli na bud\-qkomu promeni vseredyni c\oho sektora, i dlq
dovil\noho θ ′ < θ isnu[ stala c ′θ > 0 taka, wo
∀z ∈ Σ ( θ ′ ) : || T ( z ) || ≤ c ′θ .
Iz tverdΩennq 1 vyplyva[ take tverdΩennq.
TverdΩennq 2. Dlq toho wob rivnomirno stijka pivhrupa { }( )T t t≥0 ne bula
rivnomirno eksponencial\no stijkog, dostatn\o, wob
0 ∈ σc ( A ) . (3)
Dlq obmeΩeno] analityçno] { }( )T t t≥0 umova (3) [ takoΩ neobxidnog.
Osnovna meta statti — doslidyty povedinku na neskinçennosti orbit T ( t ) x ,
x ∈ � , rivnomirno, ale ne1rivnomirno eksponencial\no stijko] pivhrupy { }( )T t t≥0
v zaleΩnosti vid vlastyvostej vektora x , zokrema z’qsuvaty, çy [ sered nyx
taki, wo spadagt\ na neskinçennosti eksponencial\no, i qkwo ce tak, to naskil\-
ky bahatog [ mnoΩyna cyx orbit. Pro te, wo (na vidminu vid vypadku rivnomirno
eksponencial\no] stijkosti) spadannq na neskinçennosti orbit rivnomirno stij-
ko] pivhrupy ne kontrolg[t\sq Ωodnog funkci[g, svidçyt\ nastupne tverd-
Ωennq z [5].
TverdΩennq 3. Nexaj { }( )T t t≥0 — C0 -pivhrupa v � , a γ ( t ) — neperervna
na [ 0 , ∞ ) funkciq, γ ( t ) > 0 i γ ( t ) → 0 pry t → ∞ . Qkwo dlq koΩnoho x ∈ �
isnu[ stala c = c ( x ) > 0 taka, wo
∀t ∈ [ 0 , ∞ ) : || T ( t ) x || ≤ c γ ( t ) ,
to { }( )T t t≥0 [ rivnomirno eksponencial\no stijkog.
TverdΩennq 3 pokazu[, wo ne vsi orbity rivnomirno, ale ne rivnomirno ekspo-
nencial\no stijko] pivhrupy { }( )T t t≥0 spadagt\ na neskinçennosti eksponenci-
al\no. Spravdi, nexaj dlq dovil\noho x ∈ �
∃ c = c ( x ) > 0, ∃ ωx > 0 : || T ( t ) x || ≤ ce x t−ω .
Todi
∀t ∈ [ 0 , ∞ ) : || T ( t ) x || ≤ c
t1
1
1 +
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
150 V.1I.1HORBAÇUK, M.1L.1HORBAÇUK
de 0 < c1 = c t e
t
txsup ( )
[ , )∈ ∞
−+
0
1 ω . Pokladagçy v tverdΩenni13 γ ( t ) = 1
1+ t
, pryxo-
dymo do vysnovku, wo { }( )T t t≥0 [ rivnomirno eksponencial\no stijkog, a ce supe-
reçyt\ prypuwenng.
Iz tverdΩennq 3 vyplyva[ taka teorema.
Teorema 1. Nexaj { }( )T t t≥0 — dovil\na C0 -pivhrupa v � . Qkwo dlq koΩ-
noho x ∈ � isnu[ px > 0 take, wo
T t x dtpx( )
0
∞
∫ < ∞ , (4)
to { }( )T t t≥0 [ rivnomirno eksponencial\no stijkog.
Dovedennq. Tak samo, qk i pry dovedenni teoremy 4.1 z [6], ustanovlg[mo,
wo spivvidnoßennq (4) zumovlg[ zbiΩnist\ T ( t ) x → 0 ( t → ∞ ) dlq bud\-qkoho
x ∈ � , a tomu (za pryncypom rivnomirno] obmeΩenosti) j rivnomirnu obmeΩe-
nist\ pivhrupy { }( )T t t≥0 : || T ( t ) || ≤ c . Perexodqçy (v razi potreby) u prostori
� do normy || x ||1 = sup ( )
[ , )t
T t x
∈ ∞
|| ||
0
, ekvivalentno] poçatkovij || ||⋅ , moΩemo vva-
Ωaty c = 1, a otΩe, funkciq || T ( t ) x || [ monotonno spadnog na [ 0, ∞ ) . Iz
zbiΩnosti intehrala v (4) vyplyva[ ocinka
|| T ( x ) || ≤ c tx
px( ) /1 1+ − ( 0 < cx = const ) .
Pokladagçy γ ( t ) = 1
2ln( )+ t
, distanemo
|| T ( x ) || ≤ c1 ( x ) γ ( t ) , c1 ( x ) = c
t
t
x
t
px
sup
ln( )
( )[ , )
/∈ ∞
+
+0
1
2
1
.
Za tverdΩennqm13 { }( )T t t≥0 [ rivnomirno eksponencial\no stijkog, wo j po-
tribno bulo dovesty.
Teorema 1 [ uzahal\nennqm teoremy 4.1 z [6] (tam px = p ne1zaleΩyt\ vid x ) ,
otrymano] v riznyx okremyx vypadkax riznymy avtoramy (dyv. [7 – 10]).
2. U c\omu punkti rozhlqdagt\sq rivnomirno stijki styskal\ni pivhrupy
{ }( )T t t≥0 i z’qsovu[t\sq, qki same vlastyvosti vektora x vyznaçagt\ porqdok
prqmuvannq orbity T ( t ) x do nulq pry t → ∞ .
Teorema 2. Nexaj { }( )T t t≥0 — rivnomirno stijka C0 -pivhrupa v � z hene-
ratorom A . Todi
x ∈ D ( A
–1
) ⇔ ∃ T s x ds( )
0
∞
∫ (5)
i
A
–1
x = – T s x ds( )
0
∞
∫ . (6)
Qkwo, krim toho, { }( )T t t≥0 obmeΩena analityçna, to
x ∈ D ( A
–
(n
+1)
) ⇔ ∃ s T s x dsk ( )
0
∞
∫ , k = 0, 1, … , n ,
i
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
PRO POVEDINKU NA NESKINÇENNOSTI ORBIT RIVNOMIRNO STIJKYX PIVHRUP 151
A
–
(n
+1)
x = ( )
!
( )
( )− + ∞
∫1 1
0
n
n
n
s T s x ds . (7)
Dovedennq. Prypustymo, wo x ∈ D ( A
–1
) . Todi dlq dovil\noho t > 0
T s x ds
t
( )
0
∫ = T s A A x ds
t
( ) −∫ 1
0
= T ( t ) A
–1
x – A
–1
x .
Pry t → ∞ ostannij vyraz zbiha[t\sq do – A
–1
x . OtΩe, intehral T s x ds( )
0
∞
∫
isnu[ i rivnist\ (6) vykonu[t\sq.
Navpaky, nexaj isnu[ T s x ds( )
0
∞
∫ . Oskil\ky yt = T s x ds
t
( )
0
∫ ∈ D ( A ) , to ma[mo
sim’g vektoriv
yt ∈ D ( A ) : yt → T s x ds( )
0
∞
∫ pry t → ∞ .
Beruçy do uvahy, wo
A yt = T ( t ) x – x → – x pry t → ∞ ,
i zamknenist\ operatora A , robymo vysnovok, wo
T s x ds( )
0
∞
∫ ∈ D ( A ) i A T s x ds( )
0
∞
∫ = – x ,
tobto
x ∈ D ( A
–1
) i A
–1
x = – T s x ds( )
0
∞
∫ .
Prypustymo teper, wo rozhlqduvana pivhrupa [ obmeΩenog analityçnog.
Todi dlq bud\-qkoho n ∈ N
x ∈ D ( A
–
n
) ⇒ t
n
T ( t ) x → 0 pry t → ∞ . (8)
Spravdi, oskil\ky dlq obmeΩeno] analityçno] pivhrupy { }( )T t t≥0 vykonu[t\sq
ocinka || A T ( t ) || ≤ c / t , 0 < c = const, to
|| T ( t ) x || = || T ( t ) A
n
A
–
n
x || = T t
n
A A x
n
n n
+
+
−
1
1
=
= T t
n
A T t
n
A x
n
n
+
+
−
1 1
≤ c
n
t
T t
n
A xn
n
n+
+
−1
1
,
zvidky
t
n
|| T ( t ) x || ≤ c n T t
n
A xn n n( )+
+
−1
1
→ 0 pry t → ∞ .
Podal\ße dovedennq provedemo metodom matematyçno] indukci].
Prypustymo, wo
x ∈ D ( A
–
n
) ⇔ ∃ s T s x dsk ( )
0
∞
∫ , k = 0, 1, … , n – 1 .
Oskil\ky dlq y ∈ D ( A )
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
152 V.1I.1HORBAÇUK, M.1L.1HORBAÇUK
( sn
T ( s ) y ) ′ = n sn
–1
T ( s ) y + sn
T ( s ) A y ,
to
tn
T ( t ) y = s T s y ds n s T s y ds s T s A y dsn
t
n
t
n
t
( ) ( ) ( )( )′ = +∫ ∫ ∫−
0
1
0 0
,
a otΩe,
s T s A y dsn
t
( )
0
∫ = tn T ( t ) y – n s T s y dsn
t
−∫ 1
0
( ) . (9)
Nexaj teper x ∈ D ( A
–
(n
+1)
) , tobto x = A zn+1 , z ∈ D( )An+1 . Poklademo v
(9) y = A
n
z . Oskil\ky y ∈ D ( A
–
n
) , to isnu[ s T s y dsn−
∞
∫ 1
0
( ) . Vraxovugçy (8) i
(9), pryxodymo do vysnovku, wo isnu[
s T s Ay ds s T s x dsn n( ) ( )
0 0
∞ ∞
∫ ∫= .
Beruçy do uvahy rivnist\
s T s x ds s T s A A x ds s T s A x dsn n n n n n n( ) ( ) ( )( ) ( ) ( )
0
1 1
0
1 1
0
∞
+ − +
∞
+ − +
∞
∫ ∫ ∫= = ,
za dopomohog intehruvannq çastynamy ostann\oho intehrala n raziv oderΩu[mo
s T s x ds n T s A x ds n A xn n n n n( ) ( ) ! ( ) ( ) !( ) ( )
0
1
0
1 11 1
∞
− +
∞
+ − +∫ ∫= − ′ = − ,
zvidky
A x
n
s T s x dsn
n
n− +
+ ∞
= − ∫( ) ( )
!
( )1
1
0
1 .
Navpaky, prypustymo, wo z isnuvannq usix intehraliv s T s x dsk ( )
0
∞
∫ , k = 0, 1, …
… , n – 1 , vyplyva[ vklgçennq x ∈ D ( A
–
n
) , i dovedemo, wo isnuvannq intehrala
s T s x dsn ( )
0
∞
∫ zumovlg[ naleΩnist\ x do D ( A
–
(n
+1)
) . Intehrugçy rivnist\
( sk
T ( s ) x ) ′ = k sk
–1
T ( s ) x + sk
T ′ ( s ) x
po vidrizku [ 1 / t , t ] ( t > 0 dovil\ne), oderΩu[mo
tk T ( t ) x – 1 1
t
T
t
x
k
= k s T s x ds A s T s x dsk
t
t
k
t
t
−∫ ∫+1
1 1
( ) ( )
/ /
.
Ale, qk vydno z (8), pry x ∈ D ( A
–
n
) , tn T ( t ) x → 0 pry t → ∞ . Oskil\ky inteh-
raly s T s x dsk ( )
0
∞
∫ , k = 0, 1, … , n , isnugt\ i
lim ( ) ( )
/
t
n
t
t
nA s T s x ds n s T s x ds
→∞
−
∞
∫ ∫= −
1
1
0
,
to vnaslidok zamknenosti operatora A
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
PRO POVEDINKU NA NESKINÇENNOSTI ORBIT RIVNOMIRNO STIJKYX PIVHRUP 153
y = s T s x dsn ( )
0
∞
∫ ∈ D ( A ) i A y = − −
∞
∫n s T s x dsn 1
0
( ) ,
a otΩe,
y = − − −
∞
∫n s T s A x dsn 1 1
0
( ) .
Todi za prypuwennqm A
–1
x ∈ D ( A
–
n
) , tobto x ∈ D ( A
–
(n
+1)
) , i, qk pokazano
vywe, dlq n\oho vykonu[t\sq (7).
Teoremu dovedeno.
ZauvaΩennq 1. Qkwo A — henerator rivnomirno stijko] C0 -pivhrupy
{ }( )T t t≥0 , to vnaslidok (6) rozv’qzok stacionarnoho rivnqnnq
A z = x ( z = A
–1
x ) (10)
moΩna podaty çerez rozv’qzok nestacionarno] zadaçi Koßi
dy t
dt
( )
= A y ( t ) ,
y ( 0 ) = x
qk
z = −
∞
∫ y t dt( )
0
.
Qkwo 0 ∈ σc ( A ) , to zadaça (10) [ nekorektnog. Tomu pry x ∉ D ( A ) cq zadaça
ne1ma[ zvyçajnoho rozv’qzku i za teoremog12 intehral y t dt( )
0
∞
∫ rozbiha[t\sq. Qk-
wo cej intehral pevnym çynom rehulqryzuvaty, to rehulqryzacig moΩna vvaΩa-
ty rehulqryzovanym uzahal\nenym rozv’qzkom stacionarnoho rivnqnnq (10).
Nexaj teper A — henerator rivnomirno stijko] obmeΩeno] analityçno] piv-
hrupy { }( )T t t≥0 . Todi iz spivvidnoßennq (8) vyplyva[, wo
x ∈ D ( A
–
n
) ⇒ || T ( t ) x || = o
t n
1
, (11)
a z (7) —
|| T ( t ) x || = O
t n
1
+
ε ⇒ x ∈ D ( A
–
n
) . (12)
Dlq dovil\noho zamknenoho operatora B v � poznaçymo çerez C
∞
( B ) mno-
Ωynu vsix joho neskinçenno dyferencijovnyx vektoriv:
C
∞
( B ) =
∩
n
nB
∈ = …N0 0 1 2{ , , , }
( )D .
Vzahali kaΩuçy, C B∞( ) ≠ � . Ale qkwo ρ ( B ) ≠ ∅ (wo zavΩdy vykonu[t\sq
dlq heneratoriv C0 -pivhrup), to C B∞( ) = � . Spivvidnoßennq (11) i (12) pryvo-
dqt\ do nastupnoho tverdΩennq.
Teorema 3. Nexaj { }( )T t t≥0 — rivnomirno stijka obmeΩena analityçna piv-
hrupa z heneratorom A . Todi
x ∈ C
∞
( A
–1
) ⇔ ∀n ∈ N0 : lim ( )
t
nt T t x
→∞
|| || = 0.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
154 V.1I.1HORBAÇUK, M.1L.1HORBAÇUK
Oskil\ky u vypadku, koly A — henerator rivnomirno, ale ne rivnomirno eks-
ponencial\no stijko] obmeΩeno] analityçno] pivhrupy { }( )T t t≥0 z kutom anali-
tyçnosti θ , operator A
–1 takoΩ heneru[ obmeΩenu analityçnu pivhrupu z tym
samym kutom θ (dyv. [11]), to mnoΩyna C
∞
( A
–1
) [ wil\nog v � . Takym
çynom, qkwo { }( )T t t≥0 — rivnomirno stijka obmeΩena analityçna pivhrupa, to
zavΩdy isnu[ wil\na v � mnoΩyna vektoriv x , a same, C
∞
( A
–1
) (i lyße vona),
dlq qkyx vidpovidni orbity T ( t ) x prqmugt\ do 0 na neskinçennosti ßvydße za
bud\-qkyj stepin\ 1 / t . Aby11z’qsuvaty, çy [ u rivnomirno stijko] obmeΩeno]
analityçno] pivhrupy orbity, spadannq qkyx na neskinçennosti ßvydße za
stepeneve, i u razi naqvnosti takyx — opysaty ]x, nahada[mo vyznaçennq deqkyx
pidprostoriv iz C
∞
( A
–1
) .
3. Dlq dovil\noho zamknenoho operatora B zi wil\nog oblastg vyznaçennq
v � i bud\-qkoho çysla β ≥ 0 poklademo
�{β} ( B ) =
∪
α β
α
>0
� ( )B , �(β) ( B ) =
∩
α β
α
>0
� ( )B ,
de
�β
α( )B = { x ∈ C
∞
( B ) | ∃ c = c ( x ) > 0 : ∀n ∈ N0 , || B
k
x || ≤ c αn
nn
β
}
— banaxiv prostir wodo normy
|| || || ||=
∈
x
B x
nB
n
n
n n�β
α
α β( )
sup
N0
.
Qkwo operator B obmeΩenyj, to dlq dovil\noho β > 0 ma[mo C
∞
( B ) =
= �(β) ( B ) = �{β} ( B ) = �{0} ( B ) = � . Ce, vzahali kaΩuçy, ne1tak u vypadku neob-
meΩenoho B .
Prostory �{1} ( B ) ta �(1) ( B ) vidomi qk prostory analityçnyx [13] ta, vid-
povidno, cilyx [14] vektoriv operatora B . Prostir �{0} ( B ) [ ne wo inße, qk
prostir vektoriv eksponencial\noho typu operatora B , vvedenyx u [15].
Pry 0 < β < β′ < ∞ ma[mo lancgΩok
�{0} ( B ) ⊆ �(β) ( B ) ⊆ �{β} ( B ) ⊆ �(β′ ) ( B ) ⊆ �{β′ } ( B ) .
Pryklad 1. Nexaj
� = C ( [ a , b ] ) , – ∞ < a < b < ∞ ,
B = d
dx
, D ( B ) = C1
( [ a , b ] ) .
Todi C
∞
( B ) zbiha[t\sq z mnoΩynog C
∞
( [ a , b ] ) usix neskinçenno dyferenci-
jovnyx na [ a , b ] funkcij; �{1} ( B ) , � (1) ( B ) i � {0} ( B ) — prostory anali-
tyçnyx na [ a , b ] , cilyx i cilyx eksponencial\noho typu funkcij vidpovidno;
�{β} ( B ) i �(β) ( B ) , β > 1,— klasy Ûevre typu Rum’[ ta B\orlinha (dyv. [16]).
Pryklad 2. Nexaj
� = L2 ( R ) , B = B0 ,
B0 = − +d
dx
x
2
2
2 , D ( B0 ) = C0
∞( )R .
Todi, qk pokazano v [17],
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
PRO POVEDINKU NA NESKINÇENNOSTI ORBIT RIVNOMIRNO STIJKYX PIVHRUP 155
C
∞
( B ) = S , �{β} ( B ) = Sβ
β
/
/
2
2 , β ≥ 1,
de S — vidomyj prostir Ívarca, a Sβ
β
/
/
2
2 — vidpovidnyj prostir Hel\fanda –
Íylova.
Usi pidprostory, navedeni v prykladax, wil\ni v � . U zahal\nomu vypadku
prostir �{β} ( B ) moΩe ne1buty wil\nym u � . Napryklad, qkwo
� = L2 ( [ 0 , 1 ] ) , B y ( t ) = – y ′ ( t ) , D ( B ) = { y ∈ W2
1 0 1( )[ , ] : y ( 0 ) = 0 } ,
to �{1} ( B ) = { 0 } . NevaΩko baçyty, wo cej operator B heneru[ C0 -pivhrupu
styskiv. Prote pry β > 1 dlq bud\-qkoho operatora B , kotryj heneru[ C0 -
pivhrupu,
�{ }β ( )B = � .
Pytannq wil\nosti v � prostoriv �{β} ( B ) dlq riznyx klasiv operatoriv B
u zv’qzku z riznymy zadaçamy rozhlqdalos\ bahat\ma matematykamy ( dyv.,
napryklad, [18 – 21]) . Navedemo dekil\ka kryteri]v, wo vykorystovugt\sq u
podal\ßomu.
TverdΩennq 4. Magt\ misce nastupni oznaky wil\nosti:
1. Nexaj B — normal\nyj operator u hil\bertovomu prostori � . Todi
�{0} ( B ) = { x ∈ � | x = E∆ y , ∀y ∈ � , ∆ — dovil\nyj kompakt z R 2
} ( E∆ —
spektral\na mira operatora B ), a otΩe, �{0} ( B ) = � .
2. Qkwo B — samosprqΩenyj operator u prostori Pontrqhina Πκ , κ < ∞ ,
to �{0} ( B ) wil\ne v Πκ .
3. Qkwo B — henerator obmeΩeno] analityçno] C0 -pivhrupy u prostori �
z kutom analityçnosti θ ∈ ( 0, π / 2 ] , to pry β > 1 – 2θ
π
� ( )( )β B = � . Wo
stosu[t\sq � (0) ( B ) , to isnugt\ pivhrupy z θ = π / 2 , dlq qkyx �{0} ( B ) =
= { 0 } . Ale qkwo prypustyty, wo B dodatkovo zadovol\nq[ umovu
ln ln ( )M s ds
0
1
∫ < ∞ , M ( s ) =
sup ( )
| |≥
|| ||
�z s
zR B (13)
( Rz ( B ) — rezol\venta operatora B ), to �{ }0 ( )B = � .
4. Oçevydno, wo usi orbity T ( t ) x rivnomirno eksponencial\no stijko] piv-
hrupy { }( )T t t≥0 spadagt\ do nulq na neskinçennosti eksponencial\no. Posta[
pytannq pro ]xng povedinku u vypadku, koly { }( )T t t≥0 [ rivnomirno, ale ne1riv-
nomirno eksponencial\no stijkog, tobto koly 0 ∈ σc ( A ) ( A — henerator
{ }( )T t t≥0 ) , zokrema, çy [ sered nyx taki, wo prqmugt\ na neskinçennosti do
nulq eksponencial\no. Vidpovid\ da[ nastupna teorema.
Teorema�4. Nexaj { }( )T t t≥0 — rivnomirno stijka obmeΩena analityçna piv-
hrupa styskiv. Todi pry 0 < α ≤ 1
∃ a > 0 : lim ( )
t
ate T t x
→∞
|| ||
α
= 0 ⇔ x ∈ �{β} ( A
–1
) , (14)
∀a > 0 : lim ( )
t
ate T t x
→∞
|| ||
α
= 0 ⇔ x ∈ �(β) ( A
–1
) , (15)
de α i β pov’qzani spivvidnoßennqm β = ( 1 – α ) / α .
Dovedennq. Prypustymo, wo x ∈ �{β} ( A
–1
) ( β ≥ 0 ) , tobto
∃ δ > 0 ∀n ∈ N : || A
–
n
x || ≤ c δ
n
nn
β, 0 < c = const . (16)
Todi
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
156 V.1I.1HORBAÇUK, M.1L.1HORBAÇUK
|| T ( t ) x || = T t
n
A A x n
t
A x c n
t
n c
t
n
n
n
n
n
n
n n
n
≤
≤
=
− − +δ δβ β 1 .
Oskil\ky liva çastyna nerivnosti ne1zaleΩyt\ vid n , to
|| T ( t ) x || ≤ c
t
n
n
n
inf
∈
+
N0
1δ β .
NevaΩko pidraxuvaty, wo funkciq δ β
t
s
s
+
1 , s ∈ [ 0 , ∞ ) , dosqha[ svoho minimu-
mu v toçci s t = 1
1
1
e
t
δ
β
+ , pryçomu st nabuva[ znaçennq n ∈ N0 pry t = tn =
= δ β( )ne +1. OtΩe,
|| T ( tn ) x || ≤ ce cee
t tn
n
− + ( ) −
+
=
1 1 1 1( ) / / ( )β δ γ
β
α
, (17)
de α = 1
1β +
, γ = ( )α δαe −1, a tomu dlq bud\-qkoho a < γ
lim ( )
n
at
ne T t xn
→∞
|| ||
α
= 0.
Nexaj teper t = tn + s , de s ∈ ( tn , tn +1 ) , tobto
s < tn +1 – tn = δ β β β( )n n e+ −( )+ + +1 1 1 1.
Oskil\ky
lim
( )
lim
n n
n n
n
n
n
→∞
+ +
→∞
+
+ − =
+
−1 1 1 1
1
1 1
1
β β
β
β
= β + 1,
to
s ≤ c1 nβ, 0 < c1 = const . (18)
Zavdqky (17) ma[mo
|| T ( t ) x || = || T ( s ) T ( tn ) x || ≤ || T ( tn ) x || ≤ ce t s− −γ α( ) , 0 < c = const .
Vraxovugçy nerivnist\ ( p – q )r ≥ pr
– qr, wo vykonu[t\sq pry dovil\nyx
p > q ≥ 0, r ∈ ( 0 , 1 ) , dista[mo
|| T ( t ) x || ≤ ce et s− −γ γα α
.
Ale, qk vyplyva[ z (18), dlq dovil\noho ε > 0
e e e e e et s ne c n en c nn− − −≤ =
+ −ε γ ε δ γ εδ γα α β α β α α α α( ) ( )( ) 1
1 1
1
→ 0 pry n → ∞ .
Tomu dlq velykyx n > N = N ( ε )
e es tnγ εα α
< .
Beruçy ε > 0 dostatn\o malym, oderΩu[mo
|| T ( t ) x || ≤ ce e ce e cet t t t tn− − − −≤ =γ ε γ ε γ εα α α α α( ) .
Takym çynom, pry velykyx t
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
PRO POVEDINKU NA NESKINÇENNOSTI ORBIT RIVNOMIRNO STIJKYX PIVHRUP 157
|| T ( t ) x || ≤ ce t− −( )γ ε α
, t ∈ [ 0 , ∞ ) , (19)
de c > 0 — deqka stala, i, otΩe, spivvidnoßennq u pravij çastyni (14) vykonu-
[t\sq pry a = γ – 2ε , de γ = ( )α δαe −1, α = 1
1β +
.
Qkwo Ω x ∈ �(β) ( A ) , to (16) vykonu[t\sq dlq dovil\noho δ > 0, a tomu
nerivnist\ (19) vykonu[t\sq dlq dovil\noho γ > 0, wo j zumovlg[ spivvidno-
ßennq (15).
Dovedemo zvorotni tverdΩennq.
Nexaj
∃ a > 0 : lim ( )
t
ate T t x
→∞
|| ||
α
= 0. (20)
Todi za teoremog 2 x ∈ C
∞
( A
–1
) , i z formuly (7) vyplyva[
|| A
–
n
x || ≤ 1
1 1
1
0
1 1
0( )!
( )
( )!n
s T s x ds
c
n
s e dsn n as
−
≤
−
−
∞
− −
∞
|| ||∫ ∫
α
=
=
c
n
s e ds
c
n
s en a s
s
n a s1 1 2
0
1
0
1 2
1 1( )! ( )!
max( / ) ( / )
−
≤
− { }− −
∞
≥
− −∫
α α
,
de 0 < c1 = const, c = c e dsa s
1
2
0
−
∞
∫ ( / ) α
. Oskil\ky maksymum funkci] s en a s− −1 2( / ) α
dosqha[t\sq v toçci s =
2 1
1
( )
/
n
a
−
α
α
, to
|| A
–
n
x || ≤ c
n
n
a
e
n
n
( )!
( )
( )/
( )/
−
−
−
− −
1
2 1
1
1
α
α
α .
Zastosovugçy formulu Stirlinha
n n e n O
n
n n! = +
− 2 1 1π ,
pryxodymo do vysnovku, wo
|| A
–
n
x || ≤ c δ
n
nn
β, 0 < c = c ( a ) = const ,
de
δ =
2 1 1( )β β
β
+ +
ae
. (21)
OtΩe, x ∈ �{β} ( A
–1
) .
Qkwo u spivvidnoßenni (20) a > 0 dovil\ne, to v (21) δ > 0 bude takoΩ
dovil\nym, a tomu x ∈ �(β) ( A
–1
) .
Teoremu dovedeno.
MnoΩyna usix orbit T ( t ) x rivnomirno stijko] pivhrupy styskiv { }( )T t t≥0 v �
utvorg[ banaxiv prostir �T vidnosno normy
|| || || ||⋅ =
∈ ∞
T x T t x
T t
( ) max ( )
[ , )� 0
= || x || .
Teorema14 pokazu[, wo u vypadku, koly { }( )T t t≥0 [ we j obmeΩenog analityç-
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
158 V.1I.1HORBAÇUK, M.1L.1HORBAÇUK
nog, mnoΩyna ]] orbit T ( t ) x z eksponencial\nym prqmuvannqm do nulq na
neskinçennosti [ wil\nog v � todi i til\ky todi, koly prostir �{0} ( A
–1
) vek-
toriv eksponencial\noho typu operatora A
–1 wil\nyj v � . Ale, qk dovedeno v
[20], �{0} ( A
–1
) zbiha[t\sq z mnoΩynog vektoriv x ∈ D ( A
–1
) , dlq qkyx roz-
v’qzok zadaçi Koßi
dy t
dt
( )
= A
–1
y ( t ) , t ∈ [ 0 , ∞ ) ,
(22)
y ( 0 ) = x
dopuska[ prodovΩennq do cilo] vektor-funkci] eksponencial\noho typu. Tomu z
teoremy14 vyplyva[ takyj naslidok.
Naslidok 1. Dlq toho wob zadaça Koßi (22) bula rozv’qznog v klasi cilyx
vektor-funkcij eksponencial\noho typu iz znaçennqmy v �, neobxidno i do-
statn\o, wob orbita T ( t ) x eksponencial\no prqmuvala do nulq pry t → ∞ .
Sformulg[mo u vyhlqdi teoremy osnovni rezul\taty statti, qki vyplyvagt\
iz tverdΩennq14 ta teoremy14.
Teorema 5. Magt\ misce taki tverdΩennq.
1. Qkwo A — normal\nyj operator u hil\bertovomu prostori � , spektr
qkoho zadovol\nq[ umovu
σ ( A ) ⊆ z C z∈ − ≤ − < ≤{ }arg( ) ,π θ θ π
2
0
2
i 0 ∈ σc ( A ) ,
to A heneru[ rivnomirno, ale ne6rivnomirno eksponencial\no stijku obmeΩenu
analityçnu pivhrupu { }( )T t t≥0 z kutom analityçnosti θ , dlq qko] mnoΩyna
orbit z eksponencial\nym spadannqm na neskinçennosti [ wil\nog v � . Ci or-
bity vidpovidagt\ vektoram vyhlqdu x = E∆ y , de y — dovil\nyj vektor z � ,
E∆ — spektral\na mira operatora A , ∆ ⊂ R 2 — bud\-qka vymirna mnoΩyna,
rozmiwena na dodatnij vidstani vid nulq.
2. Nexaj A — samosprqΩenyj operator u prostori Πκ , κ < ∞ , wo heneru[
rivnomirno stijku obmeΩenu analityçnu C0 -pivhrupu { }( )T t t≥0 . Todi mnoΩyna
orbit T ( t ) x , eksponencial\no spadnyx na neskinçennosti, [ wil\nog v �T .
3. Qkwo A — henerator rivnomirno stijko] obmeΩeno] analityçno] z kutom
θ ∈ ( 0, π / 2 ] pivhrupy { }( )T t t≥0 v � , to dlq koΩnoho α ∈ 0
2
,
( )
π
π θ−
mno-
Ωyna orbit T ( t ) x , dlq qkyx
|| T ( t ) x || ≤ ce at− α
( stali a , c > 0 dovil\ni) , [ wil\nog v � T . Isnugt\ obmeΩeni pivhrupy z
kutom analityçnosti π / 2 , dlq qkyx mnoΩyna eksponencial\no spadnyx orbit
sklada[t\sq lyße z nul\ovo] orbity. Ale za umovy (13) z operatorom B = A
–1
mnoΩyna takyx orbit [ wil\nog v �T .
TverdΩennq13 teoremy15 pokazu[, wo isnugt\ rivnomirno stijki obmeΩeni
analityçni pivhrupy v � , mnoΩyna eksponencial\no spadnyx orbit kotryx zbi-
ha[t\sq z { 0 } , prote mnoΩyna orbit, wo prqmugt\ do nulq na neskinçennosti
ßvydße za e at− 1 2/
z deqkym a > 0, [ wil\nog v � T . Vynyka[ pytannq: na-
skil\ky pryncypovog tut [ umova analityçnosti pivhrupy1?
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
PRO POVEDINKU NA NESKINÇENNOSTI ORBIT RIVNOMIRNO STIJKYX PIVHRUP 159
Hipoteza. Qkwo { }( )T t t≥0 — dovil\na rivnomirno stijka C0 -pivhrupa v � ,
to mnoΩyna orbit, wo povodqt\ sebe na neskinçennosti qk e at− α
( 0 < α < 1/2 ,
a > 0 ) , [ wil\nog v �T .
1. Neerven van J. M. A. M. The asymptotic behaviour of semigroups of linear operators. – Basel:
Birkhäuser, 1996. – 236 p.
2 Hantmaxer6F.6R. Teoryq matryc. – M.: Nauka, 1966. – 5761s.
3. Grabosch A., Greiner G. et al. One parameter semigroups of positive operators // Lect. Notes
Math. – 1986. – 1184. – 460 p.
4. Vasyl\ev6V.6V., Krejn6S.6H., Pyskarev6S.6Y. Poluhrupp¥ operatorov, kosynus operator-
funkcyy y lynejn¥e dyfferencyal\n¥e uravnenyq // Ytohy nauky y texnyky. Mat. analyz /
VYNYTY. – 1990. – 28. – S.187 – 201.
5. Horbaçuk6V.6M. Povedenye na beskoneçnosty reßenyj dyfferencyal\no-operatorn¥x
uravnenyj // Dokl. AN SSSR. – 1989. – 308, #11. – S.123 – 27.
6. Pazy A. Semigroups of linear operators and applications to partial differential equations. – New
York etc.: Springer, 1983. – 279 p.
7. Datko R. Extending a theorem of A. M. Liapunov to Hilbert space // J. Math. Anal. and Appl. –
1970. – 32. – P. 610 – 616.
8. Pazy A. On the applicability of Lyapunov’s theorem in Hilbert space // SIAM J. Math. Anal. –
1972. – 3. – P. 291 – 294.
9. Qkubovyç6V.6A. Çastotnaq teorema dlq sluçaq, kohda prostranstva sostoqnyj y upravlenyj
hyl\bertov¥, y ee prymenenyq v nekotor¥x zadaçax synteza optymal\noho upravlenyq. II //
Syb. mat. Ωurn. – 1975. – 16, #115. – S.11081 – 1102.
10. Krejn6M.6H. Zameçanye ob odnoj teoreme yz stat\y V.1A.1Qkubovyça „Çastotnaq teorema
dlq sluçaq, kohda…II” // Tam Ωe. – 1977. – 18, #16. – S.11411 – 1413.
11. Laubenfels de R. Inverses of generators // Proc. Amer. Math. Soc. – 1988. – 104 , # 2. –
P. 433 – 448.
12. Gorbachuk V. I., Gorbachuk M. L. Boundary value problems for operator differential equations. –
Dordrecht etc.: Kluwer Acad. Publ., 1991. – 347 p.
13. Nelson E. Analytic vectors // Ann. Math. – 1959. – 79. – P. 572 – 615.
14. Goodman R. Analytic and entire vectors for representations of Lie groups // Trans. Amer. Math.
Soc. – 1969. – 143. – P. 55 – 76.
15. Rad¥no6Q.6V. Prostranstvo vektorov πksponencyal\noho typa // Dokl. AN BSSR. – 1983. –
27, #19. – S.1791 – 793.
16. Lions J. L., Magenes E. Problèmes aux limites non homogènes et applications. 3. – Paris: Dunod,
1970. – 328 p.
17. Kaßpirovs\kyj6O.6I. Analityçne zobraΩennq uzahal\nenyx funkcij S -typu // Dopov. AN
USSR. Ser.1A. – 1980. – #14. – S.112 – 14.
18. Horbaçuk6V.6Y., Knqzgk6A.6V. Hranyçn¥e znaçenyq reßenyj dyfferencyal\no-operatorn¥x
uravnenyj // Uspexy mat. nauk. – 1989. – 44, #13. – S.155 – 91.
19. Horbaçuk6M.6L., Horbaçuk6V.6I. Pro nablyΩennq hladkyx vektoriv zamknenoho operatora
vektoramy eksponencial\noho typu // Ukr. mat. Ωurn. – 1995. – 47, #15. – S.1616 – 628.
20. Horbaçuk6M.6L. Pro analityçni rozv’qzky dyferencial\no-operatornyx rivnqn\ // Tam Ωe. –
2000. – 52, #15. – S.1596 – 607.
21. Gorbachuk M. L., Mokrousov Yu. G. On density of some sets of infinitely differentiable vectors of
a closed operator on a Banach space // Meth. Funct. Anal. and Top. – 2002. – 8, # 1. – P. 23 – 29.
OderΩano 10.10.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
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| id | umjimathkievua-article-3443 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:42:38Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d2/1438fb22602edf5f8525ed901df6c9d2.pdf |
| spelling | umjimathkievua-article-34432020-03-18T19:54:47Z On the behavior of orbits of uniformly stable semigroups at infinity Про поведінку на нескінченності орбіт рівномірно стійких півгруп Gorbachuk, V. I. Gorbachuk, M. L. Горбачук, В. І. Горбачук, М. Л. For uniformly stable bounded analytic $C_0$-semigroups $\{T(t)\} t ≥ 0$ of linear operators in a Banach space $B$, we study the behavior of their orbits $T (t)x, x ∈ B$, at infinity. We also analyze the relationship between the order of approaching the orbit $T (t)x$ to zero as $t → ∞$ and the degree of smoothness of the vector $x$ with respect to the operator $A^{-1}$ inverse to the generator A of the semigroup $\{T(t)\}_{t \geq 0}$. In particular, it is shown that, for this semigroup, there exist orbits approaching zero at infinity not slower than $e^{-at^{\alpha}}$, where $a > 0,\; 0 < \alpha < \pi/(2 (\pi - 0 )),\; \theta$ is the angle of analyticity of $\{T(t)\}_{t \geq 0}$, and the collection of these orbits is dense in the set of all orbits. Вивчається поведінка на нескінченності орбіт $T(t)x,\;\; x \in \mathfrak{B}$, рівномірно стійких обмежених аналітичних $C_0$-півгруп $\{T(t)\}_{t \geq 0}$ лінійних операторів у банаховому просторі $\mathfrak{B}$. Досліджується залежність між порядком прямування орбіти $T(t) x$ до 0 при $t \rightarrow \infty$ степенем гладкості вектора $x$ відносно оператора $A^{-1}$ оберненого до генератора $A$ півгрупи $\{T(t)\}_{t \geq 0}$. Зокрема показано, що для такої півгрупи існують орбіти, що прямують до 0 на $\infty$ не повільніше, ніж $e^{-at^{\alpha}}$, де $a > 0,\; 0 < \alpha < \pi/(2 (\pi - 0 )),\; \theta$ — кут аналітичності $\{T(t)\}_{t \geq 0}$, і сукупність цих орбіт є щільною у множині всіх орбіт. Institute of Mathematics, NAS of Ukraine 2006-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3443 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 2 (2006); 148–159 Український математичний журнал; Том 58 № 2 (2006); 148–159 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3443/3624 https://umj.imath.kiev.ua/index.php/umj/article/view/3443/3625 Copyright (c) 2006 Gorbachuk V. I.; Gorbachuk M. L. |
| spellingShingle | Gorbachuk, V. I. Gorbachuk, M. L. Горбачук, В. І. Горбачук, М. Л. On the behavior of orbits of uniformly stable semigroups at infinity |
| title | On the behavior of orbits of uniformly stable semigroups at infinity |
| title_alt | Про поведінку на нескінченності орбіт рівномірно стійких півгруп |
| title_full | On the behavior of orbits of uniformly stable semigroups at infinity |
| title_fullStr | On the behavior of orbits of uniformly stable semigroups at infinity |
| title_full_unstemmed | On the behavior of orbits of uniformly stable semigroups at infinity |
| title_short | On the behavior of orbits of uniformly stable semigroups at infinity |
| title_sort | on the behavior of orbits of uniformly stable semigroups at infinity |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3443 |
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