Theorems on decomposition of operators in L1 and their generalization to vector lattices
The Rosenthal theorem on the decomposition for operators in L1 is generalized to vector lattices and to regular operators on vector lattices. The most general version turns out to be relatively simple, but this approach sheds new light on some known facts that are not directly related to the Rosenth...
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| Date: | 2006 |
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| Main Authors: | , , , , , |
| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2006
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3432 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509523652902912 |
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| author | Maslyuchenko, O. V. Mykhailyuk, V. V. Popov, M. M. Маслюченко, О. В. Михайлюк, В. В. Попов, М. М. |
| author_facet | Maslyuchenko, O. V. Mykhailyuk, V. V. Popov, M. M. Маслюченко, О. В. Михайлюк, В. В. Попов, М. М. |
| author_sort | Maslyuchenko, O. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:54:30Z |
| description | The Rosenthal theorem on the decomposition for operators in L1 is generalized to vector lattices and to regular operators on vector lattices. The most general version turns out to be relatively simple, but this approach sheds new light on some known facts that are not directly related to the Rosenthal theorem. For example, we establish that the set of narrow operators in L1 is a projective component, which yields the known fact that a sum of narrow operators in L1 is a narrow operator. In addition to the Rosenthal theorem, we obtain other decompositions of the space of operators in L1, in particular the Liu decomposition. |
| first_indexed | 2026-03-24T02:42:28Z |
| format | Article |
| fulltext |
UDK 517.982
O.�V.�Maslgçenko, V.�V.�Myxajlgk, M.�M.�Popov (Çernivec. nac. un-t)
TEOREMY PRO ROZKLAD OPERATORIV V L1
TA }X UZAHAL|NENNQ NA VEKTORNI ÌRATKY
We generalize the Rosenthal decomposition theorem for operators in L1 to vector lattices and to regular
operators in the vector lattices. The most general version is quite simple, but this approach brings a new
emphasis to some known facts which are not related to the Rosenthal theorem immediately. For
example, we establish that the set of all narrow operators in L1 is a projection component. This yields a
known fact that a sum of narrow operators in L1 is a narrow operator. Besides the Rosenthal theorem,
we obtain another decompositions of the space of all operators in L1 , in particular, the Liu
decomposition.
Uzahal\neno teoremu Rozentalq pro rozklad operatoriv u L1 na vektorni ©ratky ta na rehulqr-
ni operatory u vektornyx ©ratkax. Najbil\ß zahal\nyj variant vyqvlq[t\sq vidnosno prostym,
odnak cej pidxid dozvolq[ po-novomu dyvytys\ na deqki vidomi fakty, ne/pov’qzani bezposeredn\o
z teoremog Rozentalq. Napryklad, vstanovleno, wo mnoΩyna vuz\kyx operatoriv u L1 [ proek-
cijnog komponentog, zvidky vyplyva[ vidomyj fakt, wo suma vuz\kyx operatoriv u L1 [ vuz\-
kym operatorom. Krim teoremy Rozentalq oderΩano inßi rozklady prostoru operatoriv u L1 ,
zokrema rozklad Liu.
1. Poperedni vidomosti. Budemo vykorystovuvaty standartnu terminolohig
teori] klasyçnyx banaxovyx prostoriv [1, 2] ta vektornyx ©ratok i dodatnyx ope-
ratoriv [3]. Çerez L ( X , Y ) poznaçymo mnoΩynu vsix linijnyx obmeΩenyx ope-
ratoriv, qki digt\ iz banaxovoho prostoru X u banaxiv prostir Y , symvol L ( X )
[ skoroçennqm dlq L ( X , X ) . Slovo „pidprostir” u banaxovomu prostori oznaça[
zamknenyj pidprostir.
Çerez Σ poznaçymo mnoΩynu vsix vymirnyx za Lebehom pidmnoΩyn [0, 1],
çerez µ — miru Lebeha na Σ , ç e r e z χ ( A ) — xarakterystyçnu funkcig
mnoΩyny A. Krim c\oho, budemo vykorystovuvaty taki skoroçennq: Σ
+ = { A ∈
∈ Σ : µ ( A ) > 0 }, Σ A = { B ∈ Σ : B ⊆ A }, Σ A
+ = { B ∈ Σ A : µ ( B ) > 0 }.
Sformulg[mo teoremu Kaltona pro zobraΩennq linijnyx obmeΩenyx opera-
toriv, qki digt\ iz L1 u L1 [4].
Teorema 1 (teorema Kaltona pro zobraΩennq). Dlq dovil\noho T ∈ L ( L1 )
isnu[ slabko∗-vymirna funkciq µ t z vidrizka [ 0, 1 ] u mnoΩynu M [ 0, 1 ] usix
rehulqrnyx borelivs\kyx mir na [ 0, 1 ] taka, wo dlq bud\-qkoho x ∈ L1
T x ( t ) = x d t( ) ( )τ µ τ∫ (1)
majΩe skriz\. Navpaky, koΩna slabko∗ -vymirna funkciq µ t : [ 0, 1 ] → M [ 0, 1 ]
vyznaça[ deqkyj operator T ∈ L ( L1 ) za dopomohog (1).
Qkwo miru µt zapysaty u vyhlqdi sumy atomno] µt
a = a tn t
n
n
( ) ( )δσ
=
∞
∑
1
(de δτ —
mira Diraka) i neperervno] (tobto bezatomno]) µt
c çastyn, to otryma[mo take
zobraΩennq operatora T :
T x ( t ) = a t x tn n
n
( ) ( )( )σ
=
∞
∑
1
+ x d t( ) ( )τ ν τ∫ (2)
(dyv.[5]).
Rozental\ v [6] formulg[ i vykorystovu[ pry dovedenni osnovnoho rezul\ta-
tu teoremu, qku vin nazyva[ pereformulgvannqm teoremy Kaltona pro zobra-
© O./V./MASLGÇENKO, V./V./MYXAJLGK, M./M./POPOV, 2006
26 ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
TEOREMY PRO ROZKLAD OPERATORIV V L1 TA }X UZAHAL|NENNQ … 27
Ωennq. Vona, bez sumnivu, [ bil\ß prozorog ta zruçnog dlq zastosuvan\, niΩ
teorema/1.
Teorema 2 (versiq Rozentalq teoremy Kaltona pro zobraΩennq). Dovil\nyj
operator T ∈ L ( L1 ) [dynym çynom rozklada[t\sq v sumu T = Tp a + Tc , de Tp a ∈
∈ L ( L1 ) — çysto atomnyj i Tc ∈ L ( L1 ) — çysto neperervnyj operatory.
Oznaçennq çysto atomnoho i çysto neperervnoho operatoriv my navedemo
nyΩçe (dyv. p./3).
Rozental\ v [6] zauvaΩyv, wo joho teoremu moΩna formal\no oderΩaty qk
naslidok teoremy Kaltona, niçoho ne/zaznaçyvßy, pravda, pro zvorotnyj zv’qzok
miΩ cymy rezul\tatamy. Ale vin poobicqv u majbutn\omu opublikuvaty ]] bezpo-
seredn[ dovedennq, ne/vykorystovugçy teoremu/1. Naskil\ky nam vidomo,
dovedennq teoremy/2 dosi ne/opublikovano.
U p./2 navedeno neobxidni vidomosti z teori] vektornyx ©ratok. My dovodymo
versig Rozentalq teoremy Kaltona pro zobraΩennq v zahal\nomu vypadku vek-
tornyx ©ratok (teorema/3). Dovedennq [ prostym, xoça otrymannq podal\ßyx
analohiv teoremy 3 dlq vypadku prostoru rehulqrnyx operatoriv na vektornyx
©ratkax (p./3), a takoΩ zhadano] teoremy Rozentalq dlq operatoriv u L1 (p./4) po-
trebu[ deqkyx dodatkovyx zusyl\. Naspravdi my da[mo svo[, na naß pohlqd,
bil\ß pryrodne oznaçennq çysto atomnoho operatora. Pry c\omu dovedennq
versi] Rozentalq v p./4 zvodyt\sq lyße do dovedennq ekvivalentnosti naßoho
oznaçennq çysto atomnoho operatora z oznaçennqm, danym Rozentalem u [6].
Usvidomlennq toho faktu, wo mnoΩyna vsix çysto neperervnyx operatoriv (vona
Ω — mnoΩyna vsix vuz\kyx operatoriv) [ komponentog (ce ne/vyplyva[ ni z
teoremy Kaltona, ni z ]] versi] Rozentalq), qke sta[ tryvial\nym zavdqky novomu
pidxodu, da[ nove dovedennq toho, wo suma vuz\kyx operatoriv u L1 [ vuz\kym
operatorom. U/p./5 teoremu/4 zastosovano do znaxodΩennq inßyx cikavyx roz-
kladiv prostoru L ( L1 ) , zokrema vidomoho rozkladu Liu.
2. Rozklad vektornyx ©ratok. NyΩçe my dotrymu[mos\ [3]. Çastkovo
vporqdkovanyj linijnyj prostir E nad polem dijsnyx çysel R nazyva[t\sq
vektornog ©ratkog, qkwo:
i) dlq dovil\nyx x , y , z ∈ E z umovy x ≤ y vyplyva[ x + z ≤ y + z ;
ii) dlq dovil\nyx x , y ∈ E ta λ ∈ [ 0 , + ∞ ) z umovy x ≤ y vyplyva[ λ x ≤ λ y ;
iii) dlq dovil\nyx x , y ∈ E isnugt\ toçna nyΩnq x Ÿ y i toçna verxnq x ⁄ y
meΩi elementiv x ta y .
Vektorna ©ratka nazyva[t\sq porqdkovo povnog, qkwo koΩna porqdkovo
obmeΩena mnoΩyna ma[ toçnu verxng meΩu. Qk i v oznaçenni vektorno] ©ratky
dostatn\o vymahaty dlq iii) isnuvannq xoça b odni[] z toçnyx meΩ, tak i v po-
rqdkovo povnij ©ratci moΩna pokazaty, wo porqdkovo obmeΩena mnoΩyna ma[ i
toçnu nyΩng meΩu. PidmnoΩyna F vektorno] ©ratky E nazyva[t\sq porqd-
kovo zamknenog, qkwo dlq dovil\no] pidmnoΩyny G ⊆ F z isnuvannq y = sup G ∈
∈ E ( abo y = inf G ∈ E ) vyplyva[, wo y ∈ F.
Nexaj E — vektorna ©ratka. Element x ∈ E nazyva[t\sq dodatnym, qkwo
x ≥ 0, a mnoΩyna vsix dodatnyx elementiv z E poznaça[t\sq çerez E
+. Dlq
koΩnoho elementa x ∈ E dodatna, vid’[mna çastyny ta modul\ vyznaçagt\sq
takym çynom: x+ = x ⁄ 0, x– = ( – x ) ⁄ 0, | x | = x ⁄ ( – x ) . Dva elementy x , y ∈ E na-
zyvagt\sq dyz’gnktnymy (abo ortohonal\nymy), qkwo | x | Ÿ | y | = 0, i cej fakt
poznaça[t\sq tak: x ⊥ y. KaΩut\, wo pidmnoΩyny A , B ⊆ E [ dyz’gnktnymy,
qkwo x ⊥ y dlq dovil\nyx x ∈ A ta y ∈ B . Dlq dovil\no] mnoΩyny A ⊆ E çe-
rez A⊥ poznaçatymemo mnoΩynu A⊥ = { x ∈ E : A ta { x } — dyz’gnktni }.
Oznaçennq 1. Nexaj E — porqdkovo povna vektorna ©ratka. Na E bude-
mo rozhlqdaty porqdkovu zbiΩnist\, a same, naprqmlenist\ ( )xs s S∈ ⊆ E budemo
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
28 O./V./MASLGÇENKO, V./V./MYXAJLGK, M./M./POPOV
vvaΩaty zbiΩnog do elementa x ∈ E ( poznaça[mo x = lim
s S sx
∈
) , qkwo isnu[ s0 ∈
S take, wo
x = inf sup
s s t s
tx
≥ ≥0
= sup inf
s s t s tx
≥ ≥
0
. (3)
ZauvaΩymo, wo formula (3) zalyßa[t\sq pravyl\nog, qkwo s0 zaminyty na
dovil\ne s1 ≥ s0 .
Oznaçennq 2. Dlq dovil\no] mnoΩyny J rqd x j
j J∈
∑ , skladenyj z elemen-
tiv x j ∈ E , nazyvatymemo zbiΩnym, a sim’g ( )x j j J∈ — sumovnog, qkwo
naprqmlenist\ ( )ys s J∈ <ω , ys = x j
j s∈
∑ porqdkovo zbiha[t\sq do deqkoho y0 ∈ E ,
de J <ω — naprqmlena za vklgçennqm ⊆ systema vsix skinçennyx pidmnoΩyn
s ⊆ J . Pry c\omu y0 nazyvatymemo sumog rqdu x j
j J∈
∑ i zapysuvatymemo y0 =
= x j
j J∈
∑ . Rqd x j
j J∈
∑ budemo nazyvaty absolgtno zbiΩnym, a sim’g ( )x j j J∈ —
absolgtno sumovnog, qkwo zbiha[t\sq rqd x j
j J∈
∑ .
Nam potribni deqki vlastyvosti vvedenyx ponqt\, qki, ßvydße za vse, [ vi-
domymy.
Lema 1. Nexaj E — porqdkovo povna vektorna ©ratka, xj ∈ E
+
, j ∈ J .
Todi:
i) zbiΩnist\ rqdu x j
j J∈
∑ rivnosyl\na porqdkovij obmeΩenosti mnoΩyny
x j
j t∈
∑
: t J∈
<ω , pry c\omu x j
j J∈
∑ = sup
t J
j
j t
x
∈ ∈<
∑
ω
;
ii) qkwo rqd x j
j J∈
∑ [ zbiΩnym i yj ∈ E — taki elementy, wo | yj | ≤ xj pry
j ∈ J , to rqd yj
j J∈
∑ takoΩ [ zbiΩnym, pryçomu yj
j J∈
∑ ≤ x j
j J∈
∑ .
Dovedennq. Dlq dovedennq tverdΩennq i) dostatn\o zaznaçyty, wo
oskil\ky xj ≥ 0, to dlq dovil\noho s ∈ J <ω ma[mo
sup
t s
j
j t
x
≥ ∈
∑ = sup
t J
j
j t
x
∈ ∈<
∑
ω
pry umovi obmeΩenosti obox mnoΩyn, supremumy qkyx rozhlqdagt\sq, do toho
Ω obmeΩenist\ odni[] z mnoΩyn, oçevydno, rivnosyl\na obmeΩenosti inßo].
ii) Nexaj spoçatku yj ≥ 0 pry j ∈ J . MnoΩyna y t Jj
j t∈
<∑ ∈
: ω obmeΩena
elementom x j
j J∈
∑ . Vnaslidok porqdkovo] povnoty E isnu[
y y x xj
j J t J
j
j t t J
j
j t
j
j J∈ ∈ ∈ ∈ ∈ ∈
∑ ∑ ∑ ∑= ≤ =
< <
sup sup
ω ω
.
Rozhlqnemo teper zahal\nyj vypadok | yj | ≤ xj . Poklademo
y+
= yj
j J
+
∈
∑ , y–
= yj
j J
−
∈
∑ , y = y+ – y–.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
TEOREMY PRO ROZKLAD OPERATORIV V L1 TA }X UZAHAL|NENNQ … 29
Todi dlq dovil\noho s0 ∈ J <ω
inf sup inf sup
s s t s
j
j t s s t s
j
j t
j
j t
y y y
≥ ≥ ∈ ≥ ≥
+
∈
−
∈
∑ ∑ ∑= −
0 0
≤ inf sup
s s t s
j
j s
y y
≥ ≥
+ −
∈
−
∑
0
=
= inf
s s j
j s
y y
≥
+ −
∈
−
∑
0
= y+ – y–
= y .
Analohiçno
sup inf sup inf sup inf
s s t s j
j t s s t s j
j t
j
j t s s t s j
j s
y y y y y
≥ ≥ ∈ ≥ ≥
+
∈
−
∈ ≥ ≥
+
∈
−∑ ∑ ∑ ∑= −
≥ −
0 0 0
=
= sup
s s
j
j s
y y
≥
+
∈
−∑ −
0
= y+ – y–
= y .
OtΩe, my dovely, wo
sup inf
s s t s j
j t
y
≥ ≥ ∈
∑
0
≥ y ≥ inf sup
s s t s
j
j t
y
≥ ≥ ∈
∑
0
.
Dovedemo obernenu nerivnist\
inf sup
s s t s
j
j t
y
≥ ≥ ∈
∑
0
≥ inf sup inf ( )
s s t s
j
j t s s
y y y y
≥ ≥
+
∈
−
≥
+ −∑ −
= −
0 0
= y .
OtΩe, inf sup
s s t s
j
j t
y
≥ ≥ ∈
∑
0
= y . Analohiçno sup inf
s s t s j
j t
y
≥ ≥ ∈
∑
0
= y .
Nareßti, vykorystovugçy tverdΩennq i) , oderΩu[mo
y y y y y y y yj
j J
j
j J
j
j J∈
+ − + − +
∈
−
∈
∑ ∑ ∑= = − ≤ + = + =
= y y y xj j
j J
j
j J
j
j J
+ −
∈ ∈ ∈
+( ) = ≤∑ ∑ ∑ .
Lemu dovedeno.
Bezposeredn\o z lemy 1 vyplyva[ takyj naslidok.
Naslidok 1. Absolgtno zbiΩnyj rqd x j
j J∈
∑ u porqdkovo povnij vektornij
©ratci [ zbiΩnym, pryçomu
x xj
j J
j
j J∈ ∈
∑ ∑≤ .
PidmnoΩyna A vektorno] ©ratky E nazyva[t\sq tilesnog, qkwo dlq
dovil\nyx x ∈ A ta y ∈ E z umovy | y | ≤ | x | vyplyva[, wo y ∈ A . Tilesnyj li-
nijnyj pidprostir nazyva[t\sq idealom, porqdkovo zamknenyj ideal — kompo-
nentog. Komponenta I vektorno] ©ratky E nazyva[t\sq proekcijnog kompo-
nentog, qkwo E = I ⊕ I
⊥.
Dlq dovil\no] vektorno] ©ratky E ta dovil\no] pidmnoΩyny A ⊆ E çerez
Band ( A ) poznaçymo najmenßu komponentu v E , qka mistyt\ A (oçevydno, wo
peretyn dovil\no] kil\kosti komponent takoΩ [ komponentog, i tomu takyj
ob’[kt [ korektno vyznaçenym). Nahada[mo vidome tverdΩennq.
Lema 2 [3, s. 62]. Nexaj E — porqdkovo povna vektorna ©ratka, A ⊆ E —
dovil\na pidmnoΩyna. Todi A⊥ — komponenta, pryçomu E = Band ( A ) ⊕ A⊥.
Zokrema, koΩna komponenta [ proekcijnog komponentog.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
30 O./V./MASLGÇENKO, V./V./MYXAJLGK, M./M./POPOV
Nexaj A — pidmnoΩyna porqdkovo povno] vektorno] ©ratky E . Poznaçymo
çerez Abs ( A ) mnoΩynu vsix sum absolgtno zbiΩnyx rqdiv x j
j J∈
∑ z elementiv
xj ∈ A .
Teorema 3 (uzahal\nennq teoremy Kaltona – Rozentalq na vektorni ©ratky).
Nexaj A — tilesna pidmnoΩyna porqdkovo povno] vektorno] ©ratky E . Todi:
i) A⊥
= { x ∈ E : ( ∀x ∈ A ) ( ( 0 ≤ y ≤ | x | ) ⇒ ( y = 0) ) };
ii) Band ( A ) = Abs ( A ) .
Zokrema, E = Abs ( A ) ⊕ A⊥ — rozklad na proekcijni komponenty.
Dovedennq. i) Poznaçymo çerez C pravu çastynu rivnosti i) . Nexaj x ∈
∈ A⊥, y ∈ A ta 0 ≤ y ≤ | x | . Todi 0 = | x | Ÿ y = y . OtΩe, x ∈ C. Nexaj teper
x ∈ C ta y ∈ A . Oskil\ky A [ tilesnog i 0 ≤ | x | Ÿ | y | ≤ | x | , to | x | Ÿ | y | ∈ A . Za
oznaçennqm mnoΩyny C ma[mo | x | Ÿ | y | = 0, tomu x ∈ A⊥.
ii) Za lemog/2
E = Band ( A ) ⊕ A⊥. (4)
Dovedemo teper, wo
E = Abs ( A ) ⊕ A⊥. (5)
Dovedemo isnuvannq rozkladu. Nexaj x ∈ E . Oskil\ky x = x+ – x– [3, s. 51], to
dostatn\o rozhlqnuty vypadok x ≥ 0.
ZauvaΩymo, wo u dovil\nij sumovnij sim’] bud\-qkyj ]] nenul\ovyj element
moΩe povtorgvatysq ne/bil\ß niΩ skinçennu kil\kist\ raziv. Nexaj ωα — kar-
dynal potuΩnosti | |
=
∞
∑ E n
n 0
= | E | . Rozhlqnemo sukupnist\
A x = ( ) : , ,x j x A x xj j J j j
j J
∈
+
∈
⊆ ∈ ≤
∑ωα .
Budemo vvaΩaty, wo ( )xi i I∈ ≤ ( )yj j J∈ , qkwo I ⊆ J ta x i = yi p r y i ∈ I .
Oskil\ky nerivnist\ x j
j J∈
∑ ≤ x [ rivnosyl\nog nerivnostqm x j
j J∈
∑
0
≤ x dlq
dovil\no] skinçenno] pidmnoΩyny J 0 ⊆ J , to sim’q ( A x , ≤ ) — induktyvno
vporqdkovana, tobto koΩna ]] linijno vporqdkovana çastyna ma[ verxng meΩu.
Za/lemog Kuratovs\koho – Corna v A x isnu[ maksymal\nyj element ( )aj j J∈ .
Todi x1 = aj
j J∈
∑ ∈ Abs ( A ) , pryçomu x1 ≤ x . Vykorystavßy tverdΩennq i) ta
maksymal\nist\ ( )aj j J∈ , oderΩymo x2 = x – x1 ∈ A
⊥
. Takym çynom, x = x1 + x2 —
ßukanyj rozklad. Oskil\ky Abs ( A ) ⊆ Band ( A ) , to z (4) vyplyva[ Abs ( A ) ∩
∩ A⊥ = { 0 }. OtΩe, (5) dovedeno.
Zalyßa[t\sq vidmityty, wo z Abs ( A ) ⊆ Band ( A ) , (4) i (5) vyplyva[ rivnist\
Abs ( A ) = Band ( A ) .
Teoremu dovedeno.
3. Ìratky rehulqrnyx operatoriv ta ]x rozklad. Nexaj X , Y — vektorni
©ratky. Poznaçymo çerez L ( X , Y ) linijnyj prostir usix linijnyx operatoriv iz
X v Y, a çerez L+
( X , Y ) pidmnoΩynu L ( X , Y ) usix dodatnyx operatoriv, tobto
takyx, qki dodatni elementy z X perevodqt\ u dodatni elementy z Y. VvaΩagt\,
wo S ≤ T, qkwo operator T – S [ dodatnym. Qkwo Y — porqdkovo povna
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
TEOREMY PRO ROZKLAD OPERATORIV V L1 TA }X UZAHAL|NENNQ … 31
vektorna ©ratka, to (dyv. [3, s. 229]) vektornyj prostir rehulqrnyx operatoriv,
wo vyznaça[t\sq rivnistg L r
( X , Y ) = L ( X , Y ) – L+
( X , Y ) , [ porqdkovo povnog
vektornog ©ratkog, a rehulqrnist\ operatora T rivnosyl\na isnuvanng opera-
tora | T | . NevaΩko dovesty, wo
| T | x = sup : , ,T x x x x X ni
k
n
k
k
n
k
= =
+∑ ∑= ∈ ∈
1 1
N (6)
dlq dovil\nyx T ∈ Lr
( X , Y ) ta x ∈ X+, adΩe prava çastyna (6) vyznaça[ najmen-
ßyj dodatnyj operator, wo maΩoru[ T ta – T.
Rehulqrni operatory zi znaçennqmy v porqdkovo povnij vektornij ©ratci [
analohom obmeΩenyx operatoriv u normovanyx prostorax: operator bude rehu-
lqrnym todi i til\ky todi, koly vin vidobraΩa[ porqdkovo obmeΩeni mnoΩyny v
porqdkovo obmeΩeni mnoΩyny [3, s. 229]. ZauvaΩymo, wo qkwo X , Y — banaxovi
©ratky ( banaxiv prostir E , qkyj odnoçasno [ vektornog ©ratkog, nazyva[t\sq
banaxovog ©ratkog, qkwo dlq dovil\nyx x , y ∈ E z umovy | x | ≤ | y | vyplyva[
|| x || ≤ || y || ) , to L r
( X , Y ) ⊆ L ( X , Y ) . Vzahali kaΩuçy, Lr
( X , Y ) ≠ L ( X , Y ) ,
odnak, napryklad, Lr
( L1 , L1 ) = L ( L1 ) . Krim toho, L ( X , Y ) [ porqdkovo povnog
banaxovog ©ratkog pry umovi porqdkovo] povnoty Y.
Operator A ∈ Lr
( X , Y ) nazyva[t\sq atomom, qkwo A perevodyt\ dyz’gnk-
tni elementy v dyz’gnktni elementy. Zaznaçymo, wo v inßyx pracqx taki
operatory nazyvagt\sq operatoramy, wo zberihagt\ dyz’gnktnist\ [7].
Zhidno z Rozentalem (qkyj, wopravda, rozhlqdav lyße vypadok X = Y = L1 ,
dyv./[6]) , operator T ∈ Lr
( X , Y ) ma[ atomnu çastynu, qkwo 0 ≤ S ≤ | T | d l q
deqkoho nenul\ovoho atoma S ∈ Lr
( X , Y ) . U/protyleΩnomu vypadku budemo ho-
voryty, wo T bezatomnyj, a sukupnist\ usix bezatomnyx operatoriv iz L r
( X , Y )
poznaçatymemo çerez Lc ( X , Y ) .
Naße oznaçennq çysto atomnoho operatora formal\no vidriznq[t\sq vid
oryhinal\noho oznaçennq Rozentalq v [6]. Prote vono dozvolq[ ne lyße vyjty
za meΩi vypadku X = Y = L1 , ale j rozhlqdaty çysto atomni operatory u vektor-
nyx ©ratkax bez normy, vykorystovugçy ponqttq porqdkovo] zbiΩnosti. Rivno-
syl\nist\ oznaçen\ çysto atomnoho operatora dlq vypadku X = Y = L1 my dove-
demo v nastupnomu punkti.
Oznaçennq 3. Nexaj X , Y — vektorni ©ratky, pryçomu Y [ porqdkovo
povnog. Operator T ∈ Lr
( X , Y ) nazyvatymemo çysto atomnym, qkwo isnu[
absolgtno sumovna sim’q atomiv ( Tj ∈ Lr
( X , Y ) : j ∈ J ) taka, wo T = Tj
j J∈
∑ .
MnoΩynu vsix atomnyx (çysto atomnyx) operatoriv iz X v Y poznaçymo/çerez
La ( X , Y ) (vidpovidno Lpa ( X , Y )) . Zhidno z oznaçennqm, Lpa ( X , Y ) = Abs (La ( X , Y )).
Teorema 4 (uzahal\nennq teoremy Kaltona – Rozentalq na rehulqrni opera-
tory). Nexaj X , Y — vektorni ©ratky, pryçomu Y [ porqdkovo povnog. Todi:
i) mnoΩyna La ( X , Y ) [ tilesnog v Lr
( X , Y ) ;
ii) Band (La ( X , Y ) ) = Lpa ( X , Y ) ;
iii) La ( X , Y )⊥ = Lc ( X , Y ) ;
iv) mnoΩyny L p a ( X , Y ) ta L c ( X , Y ) [ proekcijnymy komponentamy, pry-
çomu Lr
( X , Y ) = Lpa ( X , Y ) ⊕ Lc ( X , Y ) .
Dovedennq. Vnaslidok teoremy 3 dosyt\ dovesty tverdΩennq i) . Nexaj
A ∈ La ( X , Y ) , A > 0, B ∈ Lr
( X , Y ) t a | B | ≤ A . Dovedemo, wo B ∈ La ( X , Y ) .
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
32 O./V./MASLGÇENKO, V./V./MYXAJLGK, M./M./POPOV
Nexaj x1 , x2 ∈ X , x1 ⊥ x2 . Oskil\ky
| B xi | ≤ | B | | xi | ≤ A | xi | , i = 1, 2,
to
0 ≤ | B x1 | Ÿ | B x2 | ≤ A | x1 | Ÿ A | x2 | = 0.
OtΩe, B x1 ⊥ B x2 .
4. Absolgtna porqdkova zbiΩnist\ operatornyx rqdiv u L1 . Domovy-
mos\, wo dlq x , y ∈ L1 nerivnist\ x ≤ y oznaça[, wo x ( t ) ≤ y ( t ) majΩe skriz\ na
[ 0, 1 ] .
Zhidno z Rozentalem [6], potoçkovo zbiΩnyj rqd operatoriv T x = T xn
n=
∞
∑
1
, x ∈
∈ L1 , T, Tn ∈ L ( L1 ) , nazyva[t\sq syl\no �1 -zbiΩnym, qkwo isnu[ K < ∞ take,
wo
T xn
n=
∞
∑
1
≤ K || x || (7)
dlq vsix x ∈ L1 .
ZauvaΩennq 1. V oznaçenni syl\no] �1 -zbiΩnosti dostatn\o vymahaty vyko-
nannq (7) lyße dlq x ∈ L1
+ .
ZauvaΩennq 2. Qkwo poslidovnist\ operatoriv T n ∈ L ( L1 ) zadovol\nq[ (7)
dlq vsix x ∈ L1
+ i deqkoho K < ∞ , to rqd Tn
n=
∞
∑
1
avtomatyçno potoçkovo zbiha-
[t\sq do deqkoho T ∈ L ( L1 ) .
Teorema 5. Dlq dovil\no] poslidovnosti ( )Tn 1
∞ operatoriv z L ( L1 ) r q d
Tn
n∈
∑
N
[ absolgtno porqdkovo zbiΩnym todi i til\ky todi, koly rqd Tn
n=
∞
∑
1
—
syl\no �1 -zbiΩnyj. Krim toho, v razi zbiΩnosti ci rqdy magt\ odnakovu sumu.
Dovedennq. Qkwo rqd Tn
n∈
∑
N
[ absolgtno porqdkovo zbiΩnym, to S =
= Tn
n∈
∑
N
— porqdkovo zbiΩnyj. Todi S = sup
n
k
k
n
T
∈ =
∑
N 1
ta S x = T xn
n=
∞
∑
1
d l q
koΩnoho x ∈ L1
+ , zvidky ma[mo
T x T x T xk
k
n
k
k
n
k
k
n
= = =
∑ ∑ ∑≤ =
1 1 1
≤ || S x || ≤ || S || || x ||
dlq vsix x ∈ N i x ∈ L1
+ . OtΩe, (7) dovedeno.
Pered dovedennqm obernenoho zaznaçymo, wo dlq bud\-qkoho U ∈ L ( L1 )
ma[mo
U x = sup : , ,U x x x x X mi
i
m
i
i
m
i
= =
+∑ ∑= ∈ ∈
1 1
N (8)
dlq vsix x ∈ L1
+ . Spravdi, qkwo x = xi
i
m
=
∑
1
i xi = xi j
j
mi
,
=
∑
1
, de xi , xi , j ∈ L1
+ , to
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
TEOREMY PRO ROZKLAD OPERATORIV V L1 TA }X UZAHAL|NENNQ … 33
U xi
i
m
=
∑
1
≤ U xi j
j
m
i
m i
,
==
∑∑
11
. (9)
Todi z vlastyvosti rozkladnosti vektornyx ©ratok [3, s. 53] oderΩu[mo, wo
mnoΩyna
Ex = U x x x x X mi
i
m
i
i
m
i
= =
+∑ ∑= ∈ ∈
1 1
: , , N ⊆ L1
+
— naprqmlena, a otΩe, vraxovugçy (6), oderΩu[mo
U x = sup lim lim supE u u ux u E u E u Ex x x
= = =
∈ ∈ ∈
.
Nexaj teper T xn
n=
∞
∑
1
≤ K || x || dlq vsix x ∈ L1
+ i deqkoho K < ∞ . Po-
kaΩemo, wo
T xn
n=
∞
∑
1
≤ K || x || dlq vsix x ∈ L1
+ . (10)
Zafiksu[mo x ∈ L1
+ , n ∈ N i ε > 0. Vykorystovugçy (8) dlq U = Tk , k = 1, … , n ,
vlastyvist\ rozkladnosti i (9), vybyra[mo x1 , … , xm ∈ L1
+ taki, wo x = xi
i
m
=
∑
1
i
T x T x
nk i
i
m
k
=
∑ ≥ −
1
ε
dlq k = 1, … , n . Todi
T x T x T xk
k
n
k i
i
m
k
n
k i
k
n
i
m
= == ==
∑ ∑∑ ∑∑≤ + ≤ +
1 11 11
ε ε ≤
≤ K xi
i
m
=
∑ +
1
ε = K || x || + ε .
Sprqmuvavßy ε → 0 i n → ∞ , oderΩymo (10).
NevaΩko baçyty, wo isnu[ S ∈ L ( L1) , qkyj prodovΩu[ za linijnistg rivnist\
S x = lim
n k
k
n
T x
=
∑
1
dlq x ∈ L1
+ , pryçomu || S || ≤ K ta Tk
k
n
=
∑
1
≤ S . Todi isnu[
sup
n
k
k
n
T
∈ =
∑
N 1
, a otΩe, rqd Tn
n∈
∑
N
[ porqdkovo zbiΩnym, tobto rqd Tn
n∈
∑
N
— abso-
lgtno porqdkovo zbiΩnyj.
Nexaj teper rqdy [ zbiΩnymy. Dovedemo rivnist\ porqdkovo] sumy ′T =
= Tn
n∈
∑
N
i syl\no] �1 -sumy T ″ = Tn
n=
∞
∑
1
. Dlq c\oho dosyt\ pokazaty, wo vyko-
nu[t\sq rivnist\ T ′x = T xn
n=
∞
∑
1
dlq vsix x ∈ L1
+ .
Nexaj x ∈ L1
+ . Todi
′ −
=
≤
= > >
∑ ∑ ∑T T x T x T xk
k
n
k
k n
k
k n1
=
= T x T x x Tk
k n
k
k n
k
k n
( ) ≤ ≤
= +
∞
= +
∞
= +
∞
∑ ∑ ∑
1 1 1
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
34 O./V./MASLGÇENKO, V./V./MYXAJLGK, M./M./POPOV
OtΩe, pry n → ∞ otrymu[mo T ′ = T ″.
ZauvaΩymo, wo porqdkovo zbiΩna sim’q ( Ti : i ∈ I ) nenul\ovyx dodatnyx
operatoriv T i ∈ L ( L1 ) [ ne/bil\ß niΩ zliçennog. Ce moΩna lehko oderΩaty,
rozhlqnuvßy sim’g ( Ti χ [ 0, 1 ] : i ∈ I ) .
Nastupnyj naslidok vstanovlg[ ekvivalentnist\ oznaçennq çysto atomnoho
operatora, danoho u poperedn\omu punkti, z oznaçennqm Rozentalq [6].
Naslidok 2. Operator T ∈ L ( L1 ) [ çysto atomnym todi i til\ky todi,
koly isnu[ ne7bil\ß niΩ zliçenna sim’q atomiv ( Ti : i ∈ I ) , T i ∈ L ( L1 ) i kon-
stanta K < ∞ taki, wo Tx = T xi
i I∈
∑ ta T xi
i I∈
∑ ≤ K || x || dlq vsix x ∈ L1 .
Zhidno z [8] (dyv. takoΩ [9]), operator T ∈ L ( L1 , X ) nazyva[t\sq vuz\kym, qk-
wo dlq bud\-qkyx A ∈ Σ ta ε > 0 isnu[ x ∈ L1 takyj, wo x2 = χ ( A ) , x dµ
[ , ]0 1
∫ =
= 0 ta || Tx || < ε .
ZauvaΩennq 3. Proekcijna komponenta L c ( L1 ) usix bezatomnyx operatoriv
toçno zbiha[t\sq z mnoΩynog Narr ( L1 ) usix vuz\kyx operatoriv na L1 (ozna-
çennq dyv. u [9, 8]). Cej fakt dovedeno v [6] (teorema/1.5). Zokrema, z vywe-
zhadano] teoremy Rozentalq i z teoremy/4 oderΩu[mo nove dovedennq toho, wo
suma dvox vuz\kyx operatoriv [ vuz\kym operatorom (dyv. [10, 11]).
5. Inßi rozklady prostoru L (((( L1 )))) . Nexaj Y — banaxiv prostir. Nahada-
[mo, wo operator T ∈ L ( L1 , Y ) nazyva[t\sq reprezentovnym [12, s. 61] (v inßyx
pracqx — operatorom Radona – Nikodyma; dyv., napryklad, [13]), qkwo isnu[
istotno obmeΩena intehrovna za Boxnerom funkciq f ∈ : [ 0, 1 ] → Y taka, wo
Tx = f t x t d t( ) ( ) ( )
[ , ]
µ
0 1
∫
dlq koΩnoho x ∈ L1 .
Poznaçymo çerez LR sukupnist\ usix reprezentovnyx operatoriv iz L ( L1 ) ,
çerez LA sukupnist\ usix ne bil\ß niΩ odnovymirnyx operatoriv, LB — sukup-
nist\ skinçennovymirnyx operatoriv iz L ( L1 ) .
Neskladno zrozumity, wo LR — najmenßa komponenta v L ( L1 ) , qka mistyt\
usi odnovymirni (skinçennovymirni) operatory.
Oznaçennq 4. Operator T ∈ L ( L1 ) nazvemo koreprezentovnym, qkwo
ne7isnu[ reprezentovnoho operatora S ∈ L ( L1 ) , dlq qkoho 0 < | S | ≤ | T | .
Poznaçymo çerez LCR sukupnist\ usix koreprezentovnyx operatoriv. Bezpo-
serednim naslidkom teoremy/3 [ nastupne tverdΩennq.
TverdΩennq 1. L ( L1 ) = LR ⊕ LCR , pryçomu normy vidpovidnyx proektoriv
dorivnggt\ 1.
PokaΩemo, qk za dopomohog teoremy/3 sprostyty dovedennq teoremy Liu
[13] pro rozklad prostoru L ( L1 ) na çotyry komponenty vidomyx klasiv ope-
ratoriv.
Dlq formulgvannq teoremy Liu nam potribni deqki oznaçennq (qkwo danyj
termin zustriça[t\sq til\ky v [13], to my v duΩkax vkazu[mo avtora Liu).
Operator T ∈ L ( L1 ) nazyva[t\sq:
operatorom Danforda – Petisa, qkwo T perevodyt\ slabko zbiΩni posli-
dovnosti v syl\no zbiΩni (sukupnist\ usix operatoriv Danforda – Petisa z L ( L1 )
poznaçymo çerez LDP ) ;
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
TEOREMY PRO ROZKLAD OPERATORIV V L1 TA }X UZAHAL|NENNQ … 35
istotnym operatorom Danforda – Petisa (zhidno z Liu [13]), qkwo T [
operatorom Danforda – Petisa i dlq dovil\noho S ∈ L ( L1 ) z umovy 0 < | S | ≤ | T |
vyplyva[, wo S ne/[ reprezentovnym operatorom (sukupnist\ usix istotnyx
operatoriv Danforda – Petisa z L ( L1 ) poznaçymo çerez LPDP ) ;
operatorom Enflo, qkwo isnu[ pidprostir E ⊆ L1 , izomorfnyj L1 , takyj,
wo zvuΩennq T E [ izomorfnym vkladennqm ( sukupnist\ usix operatoriv iz
L ( L1 ) , qki ne/[ operatoramy Enflo, poznaçymo çerez LN � ) ;
istotnym operatorom Enflo (zhidno z Liu [13]), qkwo T [ operatorom
Enflo i dlq dovil\noho S ∈ L ( L1 ) z umovy 0 < | S | ≤ | T | vyplyva[, wo S [
operatorom Enflo (sukupnist\ usix istotnyx operatoriv Enflo z L ( L1 ) pozna-
çymo çerez LP � ) ;
operatorom Rozentalq (zhidno z Liu [13]), qkwo T ne/[ ni operatorom
Danforda – Petisa, ni operatorom Enflo;
istotnym operatorom Rozentalq (zhidno z Liu [13]), qkwo T [ operatorom
Rozentalq i dlq dovil\noho S ∈ L ( L1 ) z umovy 0 < | S | ≤ | T | vyplyva[, wo S [
operatorom Rozentalq (sukupnist\ usix istotnyx operatoriv Rozentalq z L ( L1 )
poznaçymo çerez LPR ) .
Nastupne tverdΩennq oderΩu[t\sq qk (n – 1) -kratne zastosuvannq teoremy/3.
TverdΩennq 2. Nexaj X — porqdkovo povna vektorna ©ratka, X0 , X1 , …
… , Xn — komponenty v X , pryçomu X0 = { 0 }, Xk ⊆ Xk +1 ta Xn = X . Todi X =
Y1 ⊕ X2 ⊕ … ⊕ Yn — rozklad na komponenty, de
Yk = { y ∈ Xk : ( ∀x ∈ X ) ( 0 < | x | ≤ | y | ) ⇒ ( x ∉ Yk –1 ) }.
Teorema 6 (Liu [13]). L ( L1 ) = LR ⊕ L PDP ⊕ L PR ⊕ L P � — rozklad na
proekcijni komponenty.
Dovedennq. Dlq dovedennq poklademo X1 = LR , X2 = LDP ta X3 = LN � . Za
dovedennqm toho, wo LDP ta LN � — komponenty, my vidsyla[mo çytaça do [13].
1. Lindenstrauss J., Tzafriri L. Classical Banach spaces. I. – Berlin etc.: Springer, 1977. – 188 p.
2. Lindenstrauss J., Tzafriri L. Classical Banach spaces. II. – Berlin etc.: Springer, 1979. – 243 p.
3. Schaefer H. H. Banach lattices and positive operators. – Berlin etc.: Springer, 1974. – 376 p.
4. Kalton N. J. The endomorphisms of Lp ( 0 ≤ p ≤ 1 ) // Indiana Univ. Math. J. – 1978. – 27, # 3. –
P. 353 – 381.
5. Godefroy G., Kalton N. J., Li D. Operators between subspaces and quotients of L1 // Ibid. – 2000.
– 49, # 1. – P. 245 – 286.
6. Rosenthal H. P. Embeddings of L1 in L1 // Contemp. Math. – 1984. – 26. – P. 335 – 349.
7. Abramovich Yu. A., Kitover A. K. Inverses of disjointness preserving operators // Mem. Amer.
Math. Soc. – 2000. – 143. – 164 p.
8. Plichko A. M., Popov M. M. Symmetric function spaces on atomless probability spaces // Dis.
Math. – 1990. – 306. – P. 1 – 85.
9. Popov7M.7M. ∏lementarnoe dokazatel\stvo otsutstvyq nenulev¥x kompaktn¥x operatorov,
opredelenn¥x na prostranstve Lp , 0 ≤ p ≤ 1 // Mat. zametky. – 1990. – 47 , #/5. –
S./154 – 155.
10. Shvydkoy R. V. Operators and integrals in Banach spaces: Dis. – Missouri - Columbia, 2001. –
125 p.
11. Kadets V. M., Popov M. M. Some stability theorems on narrow operators acting in L1 and C ( K ) //
Mat. fyzyka, analyz, heometryq. – 2003. – 10, #/1. – S./49 – 60.
12. Diestel J., Uhl J. J. Vector measures // Amer. Math. Soc. Math. Surv. and Monogr. – 1977. – 15. –
322 p.
13. Liu Z. A decomposition theorem for operators on L1 // J. Oper. Theory. – 1998. – 40, #/1. –
P. 3 – 34.
OderΩano 16.02.2005
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| id | umjimathkievua-article-3432 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:42:28Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/5c/8847913ed052f76e9f5eb99ea793cf5c.pdf |
| spelling | umjimathkievua-article-34322020-03-18T19:54:30Z Theorems on decomposition of operators in L1 and their generalization to vector lattices Теореми про розклад операторів в L1 та їх узагальнення на векторні ґратки Maslyuchenko, O. V. Mykhailyuk, V. V. Popov, M. M. Маслюченко, О. В. Михайлюк, В. В. Попов, М. М. The Rosenthal theorem on the decomposition for operators in L1 is generalized to vector lattices and to regular operators on vector lattices. The most general version turns out to be relatively simple, but this approach sheds new light on some known facts that are not directly related to the Rosenthal theorem. For example, we establish that the set of narrow operators in L1 is a projective component, which yields the known fact that a sum of narrow operators in L1 is a narrow operator. In addition to the Rosenthal theorem, we obtain other decompositions of the space of operators in L1, in particular the Liu decomposition. Узагальнено теорему Розенталя про розклад операторів у L1 на векторні ґратки та на регулярні оператори у векторних ґратках. Найбільш загальний варіант виявляється відносно простим, однак цей підхід дозволяє по-новому дивитись на деякі відомі факти, не пов'язані безпосередньо з теоремою Розенталя. Наприклад, встановлено, що множина вузьких операторів у L1 є проекційною компонентою, звідки випливає відомий факт, що сума вузьких операторів у L1 є вузьким оператором. Крім теореми Розенталя одержано інші розклади простору операторів у L1 , зокрема розклад Ліу. Institute of Mathematics, NAS of Ukraine 2006-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3432 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 1 (2006); 26-35 Український математичний журнал; Том 58 № 1 (2006); 26-35 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3432/3604 https://umj.imath.kiev.ua/index.php/umj/article/view/3432/3605 Copyright (c) 2006 Maslyuchenko O. V.; Mykhailyuk V. V.; Popov M. M. |
| spellingShingle | Maslyuchenko, O. V. Mykhailyuk, V. V. Popov, M. M. Маслюченко, О. В. Михайлюк, В. В. Попов, М. М. Theorems on decomposition of operators in L1 and their generalization to vector lattices |
| title | Theorems on decomposition of operators in L1 and their generalization to vector lattices |
| title_alt | Теореми про розклад операторів в L1 та їх узагальнення на векторні ґратки |
| title_full | Theorems on decomposition of operators in L1 and their generalization to vector lattices |
| title_fullStr | Theorems on decomposition of operators in L1 and their generalization to vector lattices |
| title_full_unstemmed | Theorems on decomposition of operators in L1 and their generalization to vector lattices |
| title_short | Theorems on decomposition of operators in L1 and their generalization to vector lattices |
| title_sort | theorems on decomposition of operators in l1 and their generalization to vector lattices |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3432 |
| work_keys_str_mv | AT maslyuchenkoov theoremsondecompositionofoperatorsinl1andtheirgeneralizationtovectorlattices AT mykhailyukvv theoremsondecompositionofoperatorsinl1andtheirgeneralizationtovectorlattices AT popovmm theoremsondecompositionofoperatorsinl1andtheirgeneralizationtovectorlattices AT maslûčenkoov theoremsondecompositionofoperatorsinl1andtheirgeneralizationtovectorlattices AT mihajlûkvv theoremsondecompositionofoperatorsinl1andtheirgeneralizationtovectorlattices AT popovmm theoremsondecompositionofoperatorsinl1andtheirgeneralizationtovectorlattices AT maslyuchenkoov teoremiprorozkladoperatorívvl1taíhuzagalʹnennânavektornígratki AT mykhailyukvv teoremiprorozkladoperatorívvl1taíhuzagalʹnennânavektornígratki AT popovmm teoremiprorozkladoperatorívvl1taíhuzagalʹnennânavektornígratki AT maslûčenkoov teoremiprorozkladoperatorívvl1taíhuzagalʹnennânavektornígratki AT mihajlûkvv teoremiprorozkladoperatorívvl1taíhuzagalʹnennânavektornígratki AT popovmm teoremiprorozkladoperatorívvl1taíhuzagalʹnennânavektornígratki |