On a criterion for the uniform boundedness of a C0-semigroup of operators in a Hilbert space
Let $T(t),\quad t ≥ 0$, be a $C_0$-semigroup of linear operators acting in a Hilbert space $H$ with norm $‖·‖$. We prove that $T(t)$ is uniformly bounded, i.e., $‖T(t)‖ ≤ M, \quad t ≥ 0$, if and only if the following condition is satisfied: $$\sup_{t > 0} \frac1t ∫_0^t∥(T(s)+T^{∗}(s))x ∥^2ds...
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| Дата: | 2007 |
|---|---|
| Автори: | , , , , , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2007
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3350 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509424949395456 |
|---|---|
| author | Wróbel, I. Gomilko, A. M. Zemanek, J. Врубель, И. Гомилко, А. М. Земанек, Я. Врубель, И. Гомилко, А. М. Земанек, Я. |
| author_facet | Wróbel, I. Gomilko, A. M. Zemanek, J. Врубель, И. Гомилко, А. М. Земанек, Я. Врубель, И. Гомилко, А. М. Земанек, Я. |
| author_sort | Wróbel, I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:51:58Z |
| description | Let $T(t),\quad t ≥ 0$, be a $C_0$-semigroup of linear operators acting in a Hilbert space $H$ with norm $‖·‖$. We prove that $T(t)$ is uniformly bounded, i.e., $‖T(t)‖ ≤ M, \quad t ≥ 0$, if and only if the following condition is satisfied:
$$\sup_{t > 0} \frac1t ∫_0^t∥(T(s)+T^{∗}(s))x ∥^2ds < ∞$$
forall $x ∈ H$, where $T^{*}$ is the adjoint operator. |
| first_indexed | 2026-03-24T02:40:53Z |
| format | Article |
| fulltext |
K�O�R�O�T�K�I���P�O�V�I�D�O�M�L�E�N�N�Q
UDK 517.9
A. M. Homylko (Kyev. nac. torh.- πkon. un-t),
Y. Vrubel\ (Varßav. texnol. un-t, Pol\ßa),
Q. Zemanek (Yn-t matematyky Pol\skoj akademyy nauk, Varßava, Pol\ßa)
O KRYTERYY RAVNOMERNOJ OHRANYÇENNOSTY
C0-POLUHRUPPÁ OPERATOROV
V HYL|BERTOVOM PROSTRANSTVE
Let T ( t ) , t ≥ 0, be a C0
-
semigroup of linear operators acting in the Hilbert space H with norm ⋅ .
It is proved that T ( t ) is uniformly bounded, i.e., T t M( ) ≤ , t ≥ 0, if and only if the condition
sup ( ( ) ( ))
t
t
t
T s T s x ds
>
∗∫ + < ∞
0 0
21
holds for all x ∈ H ,
where T∗
is the adjoint operator.
Nexaj T ( t ) , t ≥ 0, [ C0
-
pivhrupog linijnyx operatoriv, wo di[ u hil\bertovomu prostori H z
normog ⋅ . Dovedeno, wo T ( t ) [ rivnomirno obmeΩenog, tobto T t M( ) ≤ , t ≥ 0, todi i
til\ky todi, koly vykonu[t\sq umova
sup ( ( ) ( ))
t
t
t
T s T s x ds
>
∗∫ + < ∞
0 0
21
dlq vsix x ∈ H ,
de T∗
— sprqΩenyj operator.
1. Vvedenye. Pust\ H — hyl\bertovo prostranstvo so skalqrn¥m proyzvede-
nyem ( ⋅ , ⋅ ) y normoj ⋅ , E = E ( H ) — mnoΩestvo lynejn¥x plotno oprede-
lenn¥x zamknut¥x operatorov, dejstvugwyx v H , a L = L ( H ) — alhebra ly-
nejn¥x ohranyçenn¥x operatorov, dejstvugwyx v H. Çerez σ ( A ) oboznaçym
spektr operatora A ∈ E , I — edynyçn¥j operator y R ( A, λ ) = ( )A I− −λ 1, λ 4∉
∉ σ ( A ) , — rezol\venta operatora A . Esly operator A prynadleΩyt E, to A
∗
— eho soprqΩenn¥j operator.
V stat\e [1] (sm. takΩe [2]) b¥lo pokazano, çto esly dlq C0
-
poluhrupp¥
operatorov T ( t ) , t ≥ 0, dlq lgboho vektora x ∈ H v¥polnqetsq ocenka
sup ( ) ( )
t
t
t
T s x T s x ds
>
∗∫ +[ ]
0 0
2 21 < ∞ , (1)
to poluhruppa T ( t ) qvlqetsq ravnomerno ohranyçennoj. V dannoj stat\e doka-
zano, çto uslovye (1) moΩno oslabyt\, a ymenno, dlq toho çtob¥ poluhruppa
© A. M. HOMYLKO, Y. VRUBEL|, Q. ZEMANEK, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 853
854 A. M. HOMYLKO, Y. VRUBEL|, Q. ZEMANEK
T ( t ) b¥la ravnomerno ohranyçennoj, vmesto ocenky (1) dostatoçno potrebovat\
v¥polnenyq ocenky
sup ( ( ) ( ))
t
t
t
T s T s x ds
>
∗∫ +
0 0
21 < ∞ .
∏tot rezul\tat qvlqetsq neprer¥vn¥m analohom sootvetstvugweho utverΩde-
nyq dlq dyskretnoj poluhrupp¥ operatorov T
n, n = 0, 1, … . A ymenno, v rabo-
te [3] b¥lo pokazano, çto operator T ∈ L ( H ) qvlqetsq stepenn¥m ohranyçen-
n¥m, t. e. T n ≤ M, n ∈ N, tohda y tol\ko tohda, kohda dlq lgboho vektora
x4∈ H spravedlyva ocenka
sup ( )
k j
k
j j
k
A A x
∈ =
−
∗∑ +
N
1
0
1
2
< ∞ .
2. Predvarytel\n¥e svedenyq y rezul\tat¥. Napomnym neobxodym¥e
dlq dal\nejßeho yzloΩenyq svedenyq yz teoryy poluhrupp operatorov [4, 5].
Semejstvo T ( t ) , t ≥ 0, lynejn¥x ohranyçenn¥x operatorov, dejstvugwyx v H ,
obrazuet C0
-
poluhruppu operatorov, esly T I( )0 = , T t t T t T t( ) ( ) ( )1 2 1 2+ = ,
t t1 2 0, ≥ , y
T t x x( ) − → 0, t → 0, ∀ x ∈ H .
Henerator A (proyzvodqwyj operator) C0
-
poluhrupp¥ T ( t ) opredelqetsq kak
syl\n¥j predel
Ax = lim
( )
t
T t x x
t→
−
0
, x ∈ D ( A ) ,
y qvlqetsq plotno zadann¥m zamknut¥m operatorom. Pry πtom dlq sootvetst-
vugwej poluhrupp¥ budem yspol\zovat\ oboznaçenye T ( t ) = et A , t ≥ 0. Typom
C0
-
poluhrupp¥ T ( t ) = et A
naz¥vaetsq çyslo
ω0 ( A ) = ω0 = lim
ln ( )
t
T t
t→∞
,
y poluhruppa T ( t ) naz¥vaetsq ravnomerno ohranyçennoj, esly najdetsq takaq
postoqnnaq M ≥ 0, çto
T t( ) ≤ M, t ≥ 0. (2)
Pry πtom esly et A
qvlqetsq C0
-
poluhruppoj, to soprqΩenn¥j operator A
∗
takΩe poroΩdaet C0
-
poluhruppu operatorov.
Esly typ poluhrupp¥ et A
udovletvorqet neravenstvu ω0 ( A ) ≤ 0 (v çastno-
sty, esly poluhruppa qvlqetsq ravnomerno ohranyçennoj), to spektr σ ( A ) ras-
poloΩen v poluploskosty Re λ ≤ 0 y dlq rezol\vent R ( A, λ ) , R ( A
∗, λ ) , hene-
ratorov poluhrupp spravedlyv¥ predstavlenyq
R ( A, λ ) = –
0
∞
−∫ e e dtt t Aλ , R ( A
∗, λ ) = –
0
∞
−∫
∗
e e dtt t Aλ , Re λ > 0. (3)
Otsgda y yz ravenstva Parsevalq dlq preobrazovanyq Fur\e v hyl\bertovom
prostranstve sledugt yzvestnoe ravenstvo
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
O KRYTERYY RAVNOMERNOJ OHRANYÇENNOSTY C0-POLUHRUPPÁ … 855
0
2
∞
−∫ e e x dtt t Aν = 1
2
2
π
λ λ
ν
ν
− ∞
+ ∞
∫
i
i
R A x d( , ) , ν > 0, (4)
y analohyçnoe ravenstvo s zamenoj operatora A na soprqΩenn¥j operator A
∗
.
Lemma. Pust\ T — lynejn¥j ohranyçenn¥j operator v prostranstve H .
Tohda dlq lgboho vektora x ∈ H y lgboho kompleksnoho çysla α , α ≠ – 1,
v¥polnqetsq neravenstvo
T x T x2 2 2
+ ∗α ≤
≤ C T T x c T T x x( ) ( ) (( ) , )( )α α α α+ + +{ }∗ ∗2
0
2 2 , (5)
hde c0( )α = min ,{ }1 α y postoqnnaq
C( )α =
2
2 1
1
2
2 1 1
1
− −
≤
− −
>
α
α
α
α
pry
pry
,
.
/
(6)
Dokazatel\stvo. Dlq lgboho vektora x ∈ H spravedlyvo ravenstvo
( )T T x+ ∗α
2
= T x T x T x x2 2 2 22+ + { }∗α αRe ( , ) ,
yz kotoroho poluçaem ocenku
T x T x2 2 2
+ ∗α ≤ ( ) Re ( , )T T x T x x+ + { }∗α α
2 22 . (7)
Rassmotrym snaçala sluçaj α ≤ 1, α ≠ – 1, tak çto c0( )α α= . Tohda,
yspol\zuq predstavlenye
2 2Re ( , )α T x x{ } = α α( , ) ( , )T x x T x x2 2+ ∗ =
= α α α α(( ) , ) ( )( , )T T x x T x Tx2 2 1+ + −∗ ∗
y neravenstvo 2 α T x T x∗ ≤ T x T x2 2 2
+ ∗α , ymeem
2 2Re ( , )α T x x{ } ≤ α α α α(( ) , )T T x x T x T x2 2 1+ + −∗ ∗ ≤
≤ α α α α(( ) , )T T x x T x T x2 2 2 2 21
2
+ + − +{ }∗ ∗ .
Otsgda y yz (7) naxodym
T x T x2 2 2
+ ∗α ≤ ( )T T x+ ∗α
2
+
+ α α α α(( ) , )T T x x T x T x2 2 2 2 21
2
+ + − +{ }∗ ∗ ,
otkuda, v svog oçered\, s uçetom toho, çto 1 2− <α , α ≠ – 1, v¥tekaet nera-
venstvo (5) pry α ≤ 1.
Poskol\ku ( )T T∗ ∗ = , v sluçae α > 1 dlq dokazatel\stva lemm¥ moΩno
vospol\zovat\sq uΩe dokazannoj ocenkoj (5) dlq α ≤ 1. A ymenno, pry
α > 1, polahaq β α= 1/ , ymeem
T x T x2 2 2
+ ∗α = α β2 2 2 2T x T x∗ +[ ] ≤
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
856 A. M. HOMYLKO, Y. VRUBEL|, Q. ZEMANEK
≤
2
2 1
2 2 2 2α
β
β β β
− −
+ + +{ }∗ ∗( ) (( ) , )T T x T T x x =
= 2
2 1 1
2 2 2
− −
+ + +{ }∗ ∗
/
( ) (( ) , )
α
α αT T x T T x x .
Lemma dokazana.
Otmetym, çto prymer proyzvol\noho samosoprqΩennoho ohranyçennoho ope-
ratora T T= ∗
pokaz¥vaet, çto neravenstvo (5) ne v¥polnqetsq pry α = – 1.
Yz lemm¥, v ee oboznaçenyqx dlq postoqnn¥x c0( )α y C( )α , neposredst-
venno poluçaem sledugwee utverΩdenye.
Sledstvye�1. Pust\ semejstvo operatorov T ( t ) ∈ L ( H ) , t ≥ 0, ymeet
poluhruppovoe svojstvo T t t T t T t( ) ( ) ( )1 2 1 2+ = , t1 , t2 ≥ 0. Tohda dlq lgb¥x
t ≥ 0, kompleksnoho çysla α , α ≠ – 1, y proyzvol\noho vektora x ∈ H v¥-
polnqetsq neravenstvo
T t x T t x( ) ( )2 2 2
+ ∗α ≤
≤ C T t T t x c T t T t x x( ) ( ( ) ( )) (( ( ) ( )) , )( )α α α α+ + +{ }∗ ∗2
0 2 2 .
3. Ravnomerno ohranyçenn¥e C0
-
poluhrupp¥. Osnovn¥m rezul\tatom
rabot¥ qvlqetsq sledugwaq teorema.
Teorema. Pust\ T ( t ) , t ≥ 0, — C0
-
poluhruppa v H y suwestvugt takye
çyslo α ∈ C , α ≠ 0, α ≠ – 1, y postoqnnaq M ≥ 1, çto v¥polnqetsq
neravenstvo
1 2
0
t
T s T s x ds
t
( ( ) ( ))+ ∗∫ α ≤ M x 2, t > 0, ∀ x ∈ H . (8)
Tohda T ( t ) qvlqetsq ravnomerno ohranyçennoj C0
-
poluhruppoj, a ymenno
T t( ) ≤
C
M c M
( ) ( )α
α
α+{ }0 , t ≥ 0, (9)
hde postoqnn¥e C( )α y c0( )α vzqt¥ yz lemm¥.
Dokazatel\stvo. V sylu poluhruppov¥x svojstv semejstva operatorov
T ( t ) dlq lgb¥x t > 0 y x ∈ H ymeem ravenstvo
( T ( t ) x , x ) = 1
0
t
T t x x ds
t
( ( ) , )∫ = 1
0
t
T t x T t s x ds
t
( )( ) , ( )∗ −∫ ,
y, sledovatel\no, yspol\zuq sledstvye41, poluçaem neravenstvo
( )( ) ,T t x x ≤ 1
0
t
T s x T t s x ds
t
∫ ∗ −( ) ( ) ≤
≤ 1
2
0
2 2 2
α
α
t
T s x T s x ds
t
∫ +{ }∗( ) ( ) ≤
≤
C
t
T s T s x ds c x T s T s x ds
t t
( )
( ( ) ( )) ( ( ) ( ))( )α
α
α α α
2
2 2
2
0
0
0
+ + +
∗ ∗∫ ∫ .
Otsgda na osnovanyy uslovyq (8) ymeem
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
O KRYTERYY RAVNOMERNOJ OHRANYÇENNOSTY C0-POLUHRUPPÁ … 857
( ( ) , )T t x x ≤
C
M x c x
t
T s T s x ds
t
( )
( ( ) ( ))( )α
α
α α
2
1
2
2
0
2
0
2
+ +
∗∫ ≤
≤
C
M c M x
( ) ( )α
α
α
2 0
2+{ } ,
otkuda, yspol\zuq ocenku norm¥ ohranyçennoho operatora çerez eho çyslovoj
radyus [6]
T ≤ 2 w T( ), w T( ) = sup ( , )
,x H x
T x x
∈ =1
,
poluçaem ocenku (9).
Teorema dokazana.
Zameçanye. Polahaq v teoreme α = 1, poluçaem, çto esly dlq C0
-
polu-
hrupp¥ T ( t ) v¥polnqetsq neravenstvo
1 2
0
t
T s T s x ds
t
( ( ) ( ))+ ∗∫ ≤ M x 2, t > 0, ∀ x ∈ H ,
to T ( t ) qvlqetsq ravnomerno ohranyçennoj C0
-
poluhruppoj, pryçem spraved-
lyva ocenka T t( ) ≤ M M+ , t ≥ 0.
Otmetym, çto teorema ne verna dlq znaçenyq α = 0. A ymenno, v [7] pryve-
den prymer C0
-
poluhrupp¥ T ( t ) v hyl\bertovom prostranstve L2( )R , dlq ko-
toroj v¥polnqetsq uslovye (8) s α = 0, odnako T ( t ) ne qvlqetsq ravnomerno
ohranyçennoj. S druhoj storon¥, dlq toho çtob¥ ubedyt\sq, çto otvet na vop-
ros o spravedlyvosty teorem¥ dlq znaçenyq α = – 1 takΩe qvlqetsq otryca-
tel\n¥m, dostatoçno rassmotret\ poluhruppu T ( t ) = etA, t ≥ 0.
Yz dokazannoj teorem¥ v kaçestve sledstvyq netrudno poluçyt\ sledugwee
utverΩdenye.
Sledstvye�2. Pust\ operator A ∈ E qvlqetsq heneratorom C0
-
polu-
hrupp¥ T ( t ) = etA, t ≥ 0, y α ∈ C , α ≠ 0, α ≠ – 1, — nekotoroe fyksyro-
vannoe kompleksnoe çyslo. Tohda πkvyvalentn¥ sledugwye utverΩdenyq :
1) poluhruppa T ( t ) qvlqetsq ravnomerno ohranyçennoj ;
2) poluhruppa T ( t ) ymeet typ ω0 ( A ) ≤ 0 y dlq kaΩdoho x ∈ H v¥polnq-
etsq ocenka
sup ( ( , ) ( , ))
ν ν
ν
ν λ α λ λ
> − ∞
+ ∞
∗∫ +
0
2
i
i
R A R A x d < ∞ ;
3) dlq lgboho vektora x ∈ H
sup ( ( ) ( ))
ν
νν α
>
−
∞
∗∫ +
0 0
2
e T t T t x dtt < ∞ ;
4) dlq lgboho vektora x ∈ H spravedlyva ocenka
sup ( ( ) ( ))
t
t
t
T s T s x ds
>
∗∫ +
0 0
21 α < ∞ .
Dokazatel\stvo. Ymplykacyq 1) ⇒ 2) sleduet yz oçevydnoho neraven-
stva
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
858 A. M. HOMYLKO, Y. VRUBEL|, Q. ZEMANEK
( ( , ) ( , ))R A R A xλ α λ+ ∗ 2
≤ 2 2 2 2
R A x R A x( , ) ( , )λ α λ+( )∗
y ravenstva Parsevalq (4) dlq operatorov A y A
∗
. Dalee, esly T ( t ) = etA, t ≥
≥ 0, qvlqetsq C0
-
poluhruppoj nepoloΩytel\noho typa ω0 ( A ) ≤ 0, to sohlas-
no (3) ymeem ravenstvo
( ( , ) ( , ))R A R A xλ α λ+ ∗ = – e e e x dtt t A t A−
∞
∫ +
∗λ α
0
( ) , Re λ > 0.
Tohda yz ravenstva Parsevalq dlq preobrazovanyq Fur\e dlq lgboho vektora
x4∈ H poluçaem sootnoßenye
e T t T t x dtt− ∗
∞
+∫ ν α( ( ) ( ))
2
0
=
= 1
2
2
π
λ α λ λ
ν
ν
− ∞
+ ∞
∗∫ +
i
i
R A R A x d( ( , ) ( , )) ∀ ν > 0,
otkuda sleduet ymplykacyq 2) ⇒ 3).
Dlq lgboj yzmerymoj neotrycatel\noj funkcyy f ( t ) , t ≥ 0, spravedlyva
ocenka (sm. [7])
e
t
f s ds
t
t
−
>
∫1
0 0
1sup ( ) ≤ sup ( )
ν
νν
>
∞
−∫
0 0
e f t dtt ≤ sup ( )
t
t
t
f s ds
>
∫
0 0
1 .
Prymenqq πtu ocenku k funkcyy f ( t ) = ( ( ) ( ))T t T t x+ ∗α
2
, poluçaem πkvyva-
lentnost\ uslovyj43) y 4).
Esly v¥polneno uslovye 4), to sohlasno teoreme Banaxa – Ítejnhauza o rav-
nomernoj ohranyçennosty [5], najdetsq takaq postoqnnaq M, çto budet v¥pol-
neno y uslovye (8). Teper\ dlq zaverßenyq dokazatel\stva sledstvyq nuΩno
soslat\sq na dokazannug teoremu, otkuda budet sledovat\ spravedlyvost\ ym-
plykacyy 4) ⇒ 1).
Sledstvye dokazano.
1. Homylko A. M. Ob uslovyqx na proyzvodqwyj operator ravnomerno ohranyçennoj C0
-
poluhrupp¥ operatorov // Funkcyon. analyz y pryl. – 1999. – 33, v¥p.44. – S.466 – 69.
2. Shi D.-H., Feng D.-X. Characteristic conditions of the generation of C0
-
semigroups in a Hilbert
space // J. Math. Anal. and Appl. – 2000. – 247. – P. 356 – 376.
3. Gomilko A., Wróbel I., Zemánek J. Numerical ranges in a strip // Proc. 20th Int. Conf. Operator
Theory (Timis,oara, Romania, 2004). – Bucharest: Theta, 2006. – P. 111 – 121.
4. Kato T. Teoryq vozmuwenyj lynejn¥x operatorov. – M.: Myr, 1972. – 740 s.
5. Danford N., Ívarc DΩ. T. Lynejn¥e operator¥. Obwaq teoryq. – M.: Yzd-vo ynostr. lyt.,
1962. – 896 s.
6. Xalmoß P. Hyl\bertovo prostranstvo v zadaçax. – M.: Myr, 1970. – 352 s.
7. Van Casteren J. A. Operators similar to unitary or selfadjoint ones // Pacif. J. Math. – 1983. – 104.
– P. 241 – 255.
Poluçeno 18.04.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
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| id | umjimathkievua-article-3350 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:40:53Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/fb/396ebff0ebffb0fe85c674f00f65dafb.pdf |
| spelling | umjimathkievua-article-33502020-03-18T19:51:58Z On a criterion for the uniform boundedness of a C0-semigroup of operators in a Hilbert space О критерии равномерной ограниченности C0-полугруппы операторов в гильбертовом пространстве Wróbel, I. Gomilko, A. M. Zemanek, J. Врубель, И. Гомилко, А. М. Земанек, Я. Врубель, И. Гомилко, А. М. Земанек, Я. Let $T(t),\quad t ≥ 0$, be a $C_0$-semigroup of linear operators acting in a Hilbert space $H$ with norm $‖·‖$. We prove that $T(t)$ is uniformly bounded, i.e., $‖T(t)‖ ≤ M, \quad t ≥ 0$, if and only if the following condition is satisfied: $$\sup_{t > 0} \frac1t ∫_0^t∥(T(s)+T^{∗}(s))x ∥^2ds < ∞$$ forall $x ∈ H$, where $T^{*}$ is the adjoint operator. Нехай $T(t),\quad t ≥ 0$, є $C_0$-півгрупою лінійних операторів, що діє у гільбертовому просторі $H$ з нормою $‖·‖$. Доведено, що $T(t)$ є рівномірно обмеженою, тобто $‖T(t)‖ ≤ M, \quad t ≥ 0$, тоді і тільки тоді, коли виконується умова $$\sup_{t > 0} \frac1t ∫_0^t∥(T(s)+T^{∗}(s))x ∥^2ds < ∞$$ для всіх $x ∈ H$, де $T^{*}$ — спряжений оператор. Institute of Mathematics, NAS of Ukraine 2007-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3350 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 6 (2007); 853-858 Український математичний журнал; Том 59 № 6 (2007); 853-858 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3350/3444 https://umj.imath.kiev.ua/index.php/umj/article/view/3350/3445 Copyright (c) 2007 Wróbel I.; Gomilko A. M.; Zemanek J. |
| spellingShingle | Wróbel, I. Gomilko, A. M. Zemanek, J. Врубель, И. Гомилко, А. М. Земанек, Я. Врубель, И. Гомилко, А. М. Земанек, Я. On a criterion for the uniform boundedness of a C0-semigroup of operators in a Hilbert space |
| title | On a criterion for the uniform boundedness of a C0-semigroup of operators in a Hilbert space |
| title_alt | О критерии равномерной ограниченности C0-полугруппы операторов в гильбертовом пространстве |
| title_full | On a criterion for the uniform boundedness of a C0-semigroup of operators in a Hilbert space |
| title_fullStr | On a criterion for the uniform boundedness of a C0-semigroup of operators in a Hilbert space |
| title_full_unstemmed | On a criterion for the uniform boundedness of a C0-semigroup of operators in a Hilbert space |
| title_short | On a criterion for the uniform boundedness of a C0-semigroup of operators in a Hilbert space |
| title_sort | on a criterion for the uniform boundedness of a c0-semigroup of operators in a hilbert space |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3350 |
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