On the invertibility of the operator d/dt + A in certain functional spaces
We prove that the operator $\cfrac{d}{dt} + A$ constructed on the basis of a sectorial operator $A$ with spectrum in the right half-plane of $ℂ$ is continuously invertible in the Sobolev spaces $W_p^1 (ℝ, D_{α}),\; α ≥ 0$. Here, $D_{α}$ is the domain of definition of the operator $A^{α}$ and the nor...
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| Sprache: | Ukrainisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2007
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509442693398528 |
|---|---|
| author | Gorodnii, M. F. Городній, М. Ф. |
| author_facet | Gorodnii, M. F. Городній, М. Ф. |
| author_sort | Gorodnii, M. F. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:52:34Z |
| description | We prove that the operator $\cfrac{d}{dt} + A$ constructed on the basis of a sectorial operator $A$ with spectrum in the right half-plane of $ℂ$ is continuously invertible in the Sobolev spaces $W_p^1 (ℝ, D_{α}),\; α ≥ 0$. Here, $D_{α}$ is the domain of definition of the operator $A^{α}$ and the norm in $D_{α}$ is the norm of the graph of $A^{α}$. |
| first_indexed | 2026-03-24T02:41:10Z |
| format | Article |
| fulltext |
UDK 517.98
M. F. Horodnij (Ky]v. nac. un-t im. T. Íevçenka)
PRO OBOROTNIST| OPERATORA d dt A// +
V DEQKYX FUNKCIONAL|NYX PROSTORAX
We prove that the operator d dt A/ + , which is constructed on the basis of a sectorial operator A with
the spectrum in the right half-plane of C, is continuously invertible in the Sobolev spaces W Dp
1( , )R α ,
α ≥ 0. Here, Dα is the domain of definition of the operator Aα and the norm in Dα is presented by
the norm of the graph of Aα .
Dokazano, çto operator d dt A/ + , postroenn¥j s pomow\g sektoryal\noho operatora A so
spektrom v pravoj poluploskosty C , qvlqetsq neprer¥vno obratym¥m v prostranstvax Sobo-
leva W Dp
1( , )R α , α ≥ 0. Zdes\ Dα — oblast\ opredelenyq operatora Aα , norma v Dα —
norma hrafyka operatora Aα .
1. Vstup. Nexaj B — kompleksnyj banaxiv prostir iz normog ⋅ ta nul\ovym
elementom
�
0; L ( B ) — banaxiv prostir usix linijnyx obmeΩenyx operatoriv, wo
digt\ z B v B ; I — odynyçnyj, O — nul\ovyj operatory v B ; R ( T ) , σ ( T ) ,
R λ ( T ) : = ( )λ I T− −1
poznaçagt\ vidpovidno obraz, spektr ta rezol\ventnu mno-
Ωynu operatora T. Nexaj A D A B B: ( )⊂ → — sektorial\nyj operator, e As− ,
s > 0, — eksponenta vid A [1, c. 26 – 28], e IA− =0 : .
Zafiksu[mo p ∈ [ 1, + ∞ ) . Poznaçymo çerez Lp = L Bp( , )R banaxiv prostir
usix vymirnyx za Boxnerom, intehrovnyx z p -m stepenem funkcij iz normog
f p : = f p
p
R∫( )1/
(dyv., napryklad, [2, c. 102]). Poklademo takoΩ
l Bp( ) : = x x n n B x x np
p
n
p
: { ( ) : } : ( )
/
= ∈ ⊂ =
< +∞
∈
∑Z
Z
1
.
l Bp p( ), ⋅( ) — banaxiv prostir iz pokoordynatnymy dodavannqm elementiv i
mnoΩennqm na kompleksne çyslo.
Operatoru A postavymo u vidpovidnist\ linijnyj operator LA : D A( )L ⊂
⊂ Lp → Lp , qkyj vyznaça[t\sq za takym pravylom. Funkciq x Lp∈ naleΩyt\
D A( )L todi i til\ky todi, koly znajdet\sq taka funkciq f Lp∈ , wo dlq vsix
t0 ≤ t iz R spravdΩugt\sq rivnosti
x ( t ) = e x t e f s dsA t t A t s
t
t
− − − −+ ∫( ) ( )( ) ( )0
0
0 . (1)
Pry c\omu poklada[mo LA x = f.
ZauvaΩymo, wo v (1) vykorystovu[t\sq intehral u sensi Boxnera.
Oskil\ky e s tA t s− − −∞ < ≤ < +∞{ }( ) : [ sim’[g evolgcijnyx operatoriv
dlq linijnoho dyferencial\noho rivnqnnq ′ +x t Ax t( ) ( ) =
�
0, t ∈R, to LA =
= d dt A/ + : D A( )L ⊂ Lp → Lp [ abstraktnym paraboliçnym operatorom [3,
c. 165].
SpravdΩu[t\sq nastupna teorema.
Teorema((1. Navedeni nyΩçe tverdΩennq [ ekvivalentnymy:
© M. F. HORODNIJ, 2007
1020 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
PRO OBOROTNIST| OPERATORA d dt A/ + … 1021
1. Operator LA ma[ obernenyj operator LA pL L− ∈1 ( ) .
2. Dlq dovil\noho y l Bp∈ ( ) riznyceve rivnqnnq x e xn
A
n+
−+1 = yn
, n ∈Z ,
ma[ u prostori l Bp( ) [dynyj rozv’qzok x .
3. Spektr σ ( A ) operatora A ne peretyna[t\sq z uqvnog vissg i R : =
: = { : }it t ∈R .
Dovedennq. Ekvivalentnist\ tverdΩen\ 1 i 2 vyplyva[ z teoremyFF3 robo-
ty [4] ta teoremy Banaxa pro obernenyj operator dlq zamknenoho operatora.
TverdΩennqF2 vykonu[t\sq todi i til\ky todi, koly σ( ) :{ }e z zA− ∈ =∩ C 1 =
= ∅ (dyv. [5, 6]). Zhidno z [7, c. 98] { }: ( )e A− ∈λ λ σ ⊂ σ( )e A− , a otΩe, spravd-
Ωu[t\sq implikaciq 2) ⇒ 3).
Qkwo σ( )A i∩ R = ∅, to σ ( A ) = σ σ+ −( ) ( )A A∪ , de σ+( )A , σ−( )A — zamk-
neni mnoΩyny, qki leΩat\ vidpovidno v pravij ta livij pivplowynax C, a mno-
Ωyna σ−( )A [ obmeΩenog. Tomu zhidno z [1, c. 38] prostir B rozklada[t\sq v
prqmu sumu invariantnyx vidnosno operatora A pidprostoriv B± ; zvuΩennq A±
operatora A na B± magt\ vidpovidno spektry σ±( )A ; A− — linijnyj obmeΩe-
nyj operator, A+ — sektorial\nyj operator.
Nexaj P± — proektory v B vidpovidno na pidprostory B± . Todi na pidstavi
lemyFF1 iz [4] operator LA ma[ obernenyj operator LA pL L− ∈1 ( ) , qkyj vyzna-
ça[t\sq formulog
( )( )LA f t−1 = G t s f s dsA( ) ( )−∫
R
, t ∈R, f Lp∈ , (2)
de
G tA( ) : =
e P t
e P t
At
At
−
+
−
−
≥
− <
, ,
, .
0
0
(3)
Zaznaçymo, wo koly σ−( )A = ∅, to P+ = I, P− = O.
Takym çynom, iz tverdΩennqFF3FFvyplyva[ tverdΩennqFF1.
TeoremuFF1FFdovedeno.
Meta ci[] statti — dovesty, wo koly vykonu[t\sq tverdΩennqFF3, to opera-
tory, qki vyznaçagt\sq analohiçno do LA v deqkyx sobol[vs\kyx prostorax B-
znaçnyx funkcij, zadanyx na R, teΩ [ neperervno oborotnymy. Pro zastosu-
vannq takyx rezul\tativ do doslidΩennq linijnyx paraboliçnyx dyferencial\-
nyx operatoriv dyv. [3] (hl.FX), [4].
2. Osnovnyj rezul\tat. Nexaj σ( )A i∩ R = ∅. Skorysta[mosq poznaçen-
nqmy iz dovedennq teoremyFF1 i poklademo f± : = P f t± ( ), t ∈R. Vnaslidok (2),
(3) pry f Lp∈ [dynyj rozv’qzok x Lp∈ rivnqnnq LA x = f zobraΩu[t\sq u
vyhlqdi x ( t ) = x t x t+ −+( ) ( ), t ∈R, de
x t+( ) = e f s dsA t s
t
− −
+
−∞
+∫ ( ) ( ) , t ∈R, (4)
x t−( ) = – e f s dsA t s
t
− −
−
+∞
−∫ ( ) ( ) , t ∈R, (5)
— rozv’qzky u sensi (1) rivnqn\ LA x
± ± = f± u prostorax B± . Oskil\ky
A L B− −∈ ( ) , to do operatora LA−
moΩna zastosuvaty nastupnyj rezul\tat.
Poznaçymo çerez Wp
1 = W Bp
1( , )R prostir Sobol[va funkcij iz Lp , uza-
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
1022 M. F. HORODNIJ
hal\neni poxidni qkyx naleΩat\ Lp , i çerez ⋅ 1, p normu v Wp
1.
Teorema((2 (dyv.F[8]). Nexaj A L B∈ ( ) , σ( )A i∩ R = ∅. Todi D A( )L = Wp
1
i linijnyj operator LA p pW L: 1 → [ obmeΩenym i neperervno oborotnym,
tobto
∃ >M 0 ∀ ∈f Lp : LA p
f−1
1,
≤ M f p .
Z uraxuvannqm teoremyFF2 u podal\ßomu budemo doslidΩuvaty vlastyvosti
rozv’qzkiv rivnqnnq LA x
+ + = f+ . Dlq skoroçennq zapysu vvaΩatymemo, wo
A+ = A , tobto
σ ( A ) ⊂ { }: Rez z∈ >C 0 . (6)
Vnaslidok (6) dlq koΩnoho α ≥ 0 vyznaçenyj zamknenyj operator Aα :
Dα = D A( )α ⊂ B → B , pryçomu Dα — banaxiv prostir iz normog x α : =
: = A xα , x D∈ α [1] (§1.4). U podal\ßomu budemo vykorystovuvaty banaxovi
prostory Lp( )α : = L Dp( , )R α , normy v qkyx poznaçatymemo ⋅ α, p , a takoΩ
taki klasy funkcij:
Cb : = f B: R →
f neperervna na R
za normog ⋅ ; f f t
t
∞
∈
= < +∞
: sup ( )
R
,
Cb
1 : = f B: R →{ f ma[ neperervnu poxidnu na R
za normog ⋅ ; f f f1, :∞ ∞ ∞= + ′ < +∞} ,
Lp
1 : = f C f f Lb p∈ ′ ⊂{ }1 { , } ,
F( )α : = f D: R →{ α f finitna i
neskinçenno dyferencijovna na R za normog ⋅ }α .
Dlq f F∈ ( )α poklademo f pα, ,1 : = f fp pα α, ,+ ′ i poznaçymo çerez
Wp
1( )α zamykannq F( )α za ⋅ α, ,p 1. Zaznaçymo, wo Cb, ⋅( )∞ , Cb
1
1, ,⋅( )∞ ,
Wp p
1
1( ), , ,α α⋅( ) — banaxovi prostory, pryçomu Wp p
1
0 10( ), , ,⋅( ) = Wp p
1
1, ,⋅( ) .
Zafiksu[mo p ≥ 1, α ≥ 0 i vyznaçymo operator U : D ( U ) ⊂ Wp
1( )α → Wp
1( )α
za takym pravylom. Funkciq x Wp∈ 1( )α naleΩyt\ D ( U ) todi i til\ky todi,
koly znajdet\sq taka funkciq f Wp∈ 1( )α , wo dlq vsix t0 ≤ t iz R spravdΩu-
gt\sq rivnosti (1). Pry c\omu my poklada[mo Ux = f .
Osnovnym rezul\tatom statti [ nastupna teorema.
Teorema((3. Qkwo sektorial\nyj operator A zadovol\nq[ umovu (6), to
linijnyj operator U ma[ obernenyj operator U L Wp
− ∈1 1( )( )α .
Pry dovedenni teoremyFF3 budemo vykorystovuvaty take tverdΩennq.
Lema((1. Nexaj sektorial\nyj operator A zadovol\nq[ umovu (6) i
f Lp∈ 1 . Todi dlq [dynoho rozv’qzku x Lp∈ rivnqnnq LA x f= spravdΩugt\-
sq taki tverdΩennq:
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
PRO OBOROTNIST| OPERATORA d dt A/ + … 1023
1) x ( t ) ∈ D ( A ) ta isnu[ ′x t( ) dlq koΩnoho t ∈ R , a takoΩ
′ +x t Ax t( ) ( ) = f ( t ) , t ∈ R ; (7)
2) funkci] ′x t( ), Ax t( ), t ∈ R , naleΩat\ do mnoΩyny C Lb p∩ .
Dovedennq. Oskil\ky f Lp∈ 1 , to
∀ ∈t s, R : f t f s( ) ( )− ≤ ′ −∞f t s . (8)
Tomu dyferencial\ne rivnqnnq (7) ma[ [dynyj vidpovidnyj do f obmeΩenyj roz-
v’qzok x [9], pryçomu
x ( t ) = e f s dsA t s
t
− −
−∞
∫ ( ) ( ) , t ∈ R , (9)
′x t( ) = – Ae f s f t dsA t s
t
− −
−∞
−∫ ( )( )( ) ( ) , t ∈ R . (10)
Iz (4), (9) i oznaçennq obmeΩenoho rozv’qzku dyferencial\noho rivnqnnq (7)
vyplyva[, wo vykonu[t\sq tverdΩennqFF1.
Perevirymo pravyl\nist\ tverdΩennqFF2. Vnaslidok (6) isnugt\ taki dodatni
stali δ, c [1, c. 28], wo
∀ >t 0 : e At− ≤ ce t−δ , Ae At− ≤ c
t
e t−δ . (11)
Tomu z uraxuvannqm (8) intehral (10) zbiha[t\sq absolgtno, a otΩe, dlq koΩno-
ho t ∈ R
′x t( ) = – Ae f t u f t duAu−
+∞
− −∫ ( )( ) ( )
0
= d
du
e f t u f t duAu−
+∞
− −( )∫ ( )( ) ( )
0
+
+ e f t u duAu−
+∞
′ −∫ ( )
0
= e f t u f t e f s dsAu
u
u A t s
t
−
→ +
→+∞ − −
−∞
− − + ′∫( )( ) ( ) ( )( )
0 .
Oskil\ky f L Cp p∈ ⊂1 1 , to vnaslidok (11) e f t u f tAu− − −( )( ) ( ) → 0 pry
u → +0 abo u → +∞ . Tomu dlq koΩnoho t ∈ R
′x t( ) = e f s dsA t s
t
− −
−∞
′∫ ( ) ( ) , (12)
a otΩe, ′ ∈x Lp . TakoΩ, skorystavßys\ zobraΩennqm (12) i zastosuvavßy
lemuFF1 iz [4] do rivnqnnq LA x′ = ′f u prostori Cb zamist\ Lp
, robymo
vysnovok, wo ′ ∈x Cb .
Zalyßylos\ zauvaΩyty, wo iz (7) i vklgçennq { , }x x′ ⊂ C Lb p∩ vyplyva[,
wo Ax C Lb p∈ ∩ .
LemuFF1 dovedeno.
3. Dovedennq teoremy((3. I. Oskil\ky x ≤ A x− ⋅α
α dlq koΩnoho
x D∈ α , to Wp
1( )α ⊂ Lp
. Tomu z oznaçen\ operatoriv U ta LA, umovy (6) i
teoremyFF1 vyplyva[, wo D U D A( ) ( )⊂ L , Ux xA= L dlq koΩnoho x D U∈ ( ) , a
takoΩ U [ bi[kci[g z D U( ) v R U( ).
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
1024 M. F. HORODNIJ
II. Dovedemo, wo F R U( ) ( )α ⊂ i
∀ ∈f F( )α : U f
p
−1
1α, ,
≤
c f pδ α, ,1, (13)
de c, δ — stali z nerivnostej (11).
Spravdi, F Lp( )α ⊂ 1
, tomu vnaslidok rivnostej (9), (12), teoremyFF3.7.12 iz [2,
c. 97] i vlastyvostej drobovyx stepeniv operatora A dlq koΩnoho t ∈ R oder-
Ωu[mo
A U f tα( )( )−1 = e A f s dsA t s
t
− −
−∞
∫ ( ) ( )α , (14)
A U f tα( ) ( )− ′1 = e A f s dsA t s
t
− −
−∞
′∫ ( ) ( )α . (15)
Vykorystovugçy zobraΩennq (14), ocinky (11) i nerivnist\ Gnha [10, c. 318], ma-
[mo
A U f
p
α −1 ≤ ce A f s ds dtt s
t
p p
− −
−∞
∫∫
δ α( )
/
( )
R
1
≤ c A f
pδ
α ,
tobto U f
p
−1
α,
≤ c f pδ α, . Analohiçno vnaslidok (15) U f
p
− ′1
α,
≤
≤ c f pδ α′ , . Z dvox ostannix rivnostej vyplyva[, wo spravdΩu[t\sq ocinka (13).
III. Perevirymo, wo R ( U ) = Wp
1( )α i ocinka (13) [ pravyl\nog dlq koΩnoho
f Wp∈ 1( )α .
Zafiksu[mo f Wp∈ 1( )α . Zaznaçymo, wo vnaslidok (4) analohiçno do (14)
∀ ∈t R : A f tA
α( )( )L
−1
= e A f s dsA t s
t
− −
−∞
∫ ( ) ( )α . (16)
Oskil\ky mnoΩyna F( )α [ skriz\ wil\nog v Wp
1( )α , to isnu[ taka poslidov-
nist\ { } ( )f Fn ⊂ α , wo
f fn p− α, ,1 → 0, n → ∞ . (17)
Vnaslidok (17) poslidovnist\ { }fn [ fundamental\nog v Wp
1( )α , tomu z (13)
vyplyva[, wo { }U fn
−1
— fundamental\na poslidovnist\ v Wp
1( )α . OtΩe, zna-
jdet\sq taka funkciq u Wp∈ 1( )α , wo
U f un p
− −1
1α, ,
→ 0, n → ∞ . (18)
Z inßoho boku, koΩna z funkcij fn , n ≥ 1, zadovol\nq[ rivnist\ (14). Tomu z
uraxuvannqm (11), (16) i (17)
U f fn A p
− −−1 1
1
L
α, ,
→ 0, n → ∞ . (19)
Vnaslidok (18), (19) LA pf u W− = ∈1 1( )α , a otΩe, f ∈ R ( U ) .
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
PRO OBOROTNIST| OPERATORA d dt A/ + … 1025
Oskil\ky (13) spravdΩu[t\sq dlq koΩno] z funkcij fn , n ≥ 1, i LA f u− =1 ,
to z (17), (18) vyplyva[, wo (13) vykonu[t\sq i dlq funkci] f.
IV. Na pidstavi vykladenoho v pp.FFI, III linijnyj operator U [ bi[kci[g z
D ( U ) v Wp
1( )α i dlq koΩnoho f Wp∈ 1( )α spravdΩu[t\sq ocinka (13). Tomu
isnu[ operator U L Wp
− ∈1 1( )( )α .
TeoremuFF3 dovedeno.
4. Vysnovok. U danij statti dovedeno, wo operator d dt A/ + , pobudovanyj
za dopomohog sektorial\noho operatora A , wo zadovol\nq[ umovu (6), [ nepe-
rervno oborotnym u prostorax Sobol[va Wp
1( )α . Analohiçnyj rezul\tat dlq
abstraktnoho dyferencial\noho operatora d dt A t/ ( )+ u prostorax Cb ta Lp
vstanovleno v robotax [3, 4].
1. Xenry D. Heometryçeskaq teoryq polulynejn¥x parabolyçeskyx uravnenyj. – M.: Myr,
1985. – 376Fs.
2. Xylle ∏., Fyllyps R. Funkcyonal\n¥j analyz y poluhrupp¥. – M.: Yzd-vo ynostr. lyt.,
1962. – 829 s.
3. Levytan B. M., Ûykov V. V. Poçty peryodyçeskye funkcyy y dyfferencyal\n¥e uravne-
nyq. – M.: Yzd-vo Mosk. un-ta, 1978. – 204 s.
4. Baskakov A. H. Poluhrupp¥ raznostn¥x operatorov v spektral\nom analyze lynejn¥x
dyfferencyal\n¥x operatorov // Funkcyon. analyz y eho pryl. – 1996. – 30, # 3. –
S.F1 – 11.
5. Baskakov A. H., Pastuxov A. Y. Spektral\n¥j analyz operatorov vzveßennoho sdvyha s ne-
ohranyçenn¥my operatorn¥my koπffycyentamy // Syb. mat. Ωurn. – 2001. – 42, # 6. –
S.F1231 – 1243.
6. Horodnij M. F. l p-Rozv’qzky odnoho riznycevoho rivnqnnq v banaxovomu prostori // Ukr.
mat. Ωurn. – 2003. – 55, # 3. – S.F425 – 430.
7. Holdstejn DΩ. Poluhrupp¥ lynejn¥x operatorov y yx pryloΩenyq. – Kyev: Vywa ßk.,
1989. – 348 s.
8. Horodnij M. F. Lp-rozv’qzky dyferencial\noho rivnqnnq z obmeΩenym operatornym koefi-
ci[ntom // Nauk. zap. NaUkma. Fiz.-mat. nauky. – 2003. – 21. – S.F32 – 35.
9. Horodnij M. F., Çajkovs\kyj A. V. Pro nablyΩennq obmeΩenoho rozv’qzku dyferencial\-
noho rivnqnnq z neobmeΩenym operatornym koefici[ntom // Dop. NAN Ukra]ny. – 2002. –
# 6. – S.F10 – 14.
10. Stejn Y. M. Synhulqrn¥e yntehral¥ y dyfferencyal\n¥e svojstva funkcyj. – M.: Myr,
1973. – 342 s.
OderΩano 27.10.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 8
|
| id | umjimathkievua-article-3365 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:41:10Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c2/8231689af1a19c16c3276b4f0e335bc2.pdf |
| spelling | umjimathkievua-article-33652020-03-18T19:52:34Z On the invertibility of the operator d/dt + A in certain functional spaces Про оборотність оператора d/dt + A в деяких функціональних просторах Gorodnii, M. F. Городній, М. Ф. We prove that the operator $\cfrac{d}{dt} + A$ constructed on the basis of a sectorial operator $A$ with spectrum in the right half-plane of $ℂ$ is continuously invertible in the Sobolev spaces $W_p^1 (ℝ, D_{α}),\; α ≥ 0$. Here, $D_{α}$ is the domain of definition of the operator $A^{α}$ and the norm in $D_{α}$ is the norm of the graph of $A^{α}$. Доказано, что оператор $\cfrac{d}{dt} + A$, построенный с помощью секториального оператора $A$ со спектром в правой полуплоскости $ℂ$. является непрерывно обратимым в пространствах Соболева $W_p^1 (ℝ, D_{α}),\; α ≥ 0$. Здесь $D_{α}$ — область определения оператора $A^{α}$, норма в $D_{α}$ — норма графика оператора $A^{α}$. Institute of Mathematics, NAS of Ukraine 2007-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3365 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 8 (2007); 1020–1025 Український математичний журнал; Том 59 № 8 (2007); 1020–1025 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3365/3473 https://umj.imath.kiev.ua/index.php/umj/article/view/3365/3474 Copyright (c) 2007 Gorodnii M. F. |
| spellingShingle | Gorodnii, M. F. Городній, М. Ф. On the invertibility of the operator d/dt + A in certain functional spaces |
| title | On the invertibility of the operator d/dt + A in certain functional spaces |
| title_alt | Про оборотність оператора d/dt + A в деяких функціональних просторах |
| title_full | On the invertibility of the operator d/dt + A in certain functional spaces |
| title_fullStr | On the invertibility of the operator d/dt + A in certain functional spaces |
| title_full_unstemmed | On the invertibility of the operator d/dt + A in certain functional spaces |
| title_short | On the invertibility of the operator d/dt + A in certain functional spaces |
| title_sort | on the invertibility of the operator d/dt + a in certain functional spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3365 |
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