On the defect of nondenseness of continuous imbeddings in the scale of Hilbert spaces
We obtain a formula for the determination of a defect under a continuous imbedding of subspaces in the scale of Hilbert spaces.
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2008
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| author | Bozhok, R. V. Божок, Р. В. |
| author_facet | Bozhok, R. V. Божок, Р. В. |
| author_sort | Bozhok, R. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:47:45Z |
| description | We obtain a formula for the determination of a defect under a continuous imbedding of subspaces in the scale of Hilbert spaces. |
| first_indexed | 2026-03-24T02:37:52Z |
| format | Article |
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K O R O T K I P O V I D O M L E N N Q
UDK 517.9
R. V. BoΩok (In-t matematyky NAN Ukra]ny, Ky]v)
PRO DEFEKT NEWIL|NOSTI NEPERERVNYX VKLADEN|
U ÍKALI HIL|BERTOVYX PROSTORIV
∗∗∗∗
The formula is obtained for the determination of a defect under the continuous imbedding of subspaces
in the scale of Hilbert spaces.
Ustanovlena formula dlq opredelenyq defekta pry neprer¥vnom vloΩenyy podprostranstv v
ßkale hyl\bertov¥x prostranstv.
Dlq pary separabel\nyx hil\bertovyx prostoriv H ta K pyßemo H � K , qk-
wo K [ vlasnog pidmnoΩynog H, tobto H ⊃ K , i pry c\omu K vklada[t\sq
v H wil\no i neperervno.
Nexaj H � K . Prypustymo, wo K rozkladeno v sumu ortohonal\nyx pid-
prostoriv, K = M � N . Todi moΩe statysq, wo pidprostir M vklada[t\sq v
H znovu wil\no, tobto moΩna pysaty H � M (neobxidna i dostatnq umova dlq
c\oho vidoma, dyv. nyΩçe spivvidnoßennq (3)). Ale prypustymo, wo ce ne tak,
tobto pidprostir K ne vklada[t\sq wil\no v H . Todi moΩna vykorystaty
nastupne oznaçennq. Defektom pidprostoru M v H nazyva[t\sq rozmirnist\
pidprostoru M ⊥ = N � M , zapysu[mo
def ( )M H⊂ : = dim M ⊥
.
Zadaça polqha[ v znaxodΩenni c\oho çysla v terminax trijky K H K∗ � � ,
de K
∗
poznaça[ sprqΩenyj do K prostir vidnosno H .
Dlq zruçnosti podal\ßyx pobudov perepoznaçymo H na H 0 , K na H + , a
K ∗
na H − i bez obmeΩennq zahal\nosti prypustymo, wo trijka
H− � H0 � H+ (1)
utvorg[ zvyçajne osnawennq hil\bertovoho prostoru H 0 u sensi monohrafij
[1, 2]. Poznaçymo çerez D− + + −→, : H H zvyçajnyj operator unitarnoho izo-
morfizmu.
Teorema"1. Prypustymo, wo pozytyvnyj prostir H + rozkladeno v orto-
honal\nu sumu pidprostoriv: H + = M � N + . Todi
def ( )M H+ ⊂ 0 = dim( )N H− ∩ 0 , (2)
de N − = D− + +, N . Pry c\omu
H 0 � M + ⇔ N H− ∩ 0 = { }0 . (3)
Dovedennq. Ekvivalentnist\ N H− ∩ 0 = { }0 ta M H+ � 0 dovedeno v
teoremi:A1 z [3]. Spivvidnoßennq (2) [ naslidkom rivnosti
M +
⊥,0 ≡ H M0 � + = N H− ∩ 0 . (4)
∗
Çastkovo pidtrymano DFG 436 UKR (proekty 113/67 ta 113/78).
© R. V. BOÛOK, 2008
704 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
PRO DEFEKT NEWIL|NOSTI NEPERERVNYX VKLADEN| … 705
PokaΩemo, wo M N H+
⊥
−⊂, ( )0
0∩ . Nexaj vektor g ∈H 0 naleΩyt\ do
M +
⊥,0
. Todi
0 = ( ),g M + 0 = 〈 〉+ − +g, ,M = ( ), ,I g+ − + +M ,
de 〈 〉⋅ ⋅ − +, , poznaça[ dual\nyj skalqrnyj dobutok miΩ prostoramy H − ta
H + , a I D+ − − +
−
− += →, ,: :1 H H . Ce oznaça[, wo I g+ − +∈, N . OtΩe, g ∈ −N ,
tomu wo N + = I+ − −, N . Takym çynom, g ∈ −N H∩ 0 .
Dovedemo obernene vklgçennq ( ) ,N H M− +
⊥⊂∩ 0
0 . Poznaçymo çerez
D0 0, :+ + →H H i D− −→, :0 0H H
zvyçajni operatory unitarnoho izomorfizmu. Nexaj g ∈ −N H∩ 0 . Analohiçno,
I g+ += ∈, :0 0ϕ N H∩ , de
N N0 0= − −I , i
I D+ +
−
+= →, ,: :0 0
1
0H H ,
I D0 0
1
0, ,: :− −
−
−= →H H .
Z c\oho vyplyva[, wo
0 = ( ),ϕ M 0 0 = 〈 〉+ −ϕ, ,M 0 = ( ), ,,D I0 0 0 0+ −ϕ M = ( ), ,D0 0+ +ϕ M ,
de M M N0 0 0: ,= ⊥+ +D v H 0 (dokladniße dyv. [4]). Takym çynom, oskil\ky
ϕ = I g+ +∈,0 0N H∩ ⇔ g ∈ −N H∩ 0,
to
0 = ( ), ,D0 0+ +ϕ M = ( ), , ,D I g0 0 0+ + +M = ( ),g M + 0,
a ce oznaça[, wo g ∈ +
⊥M ,0
. Tobto, qkwo g ∈ −N H∩ 0 , to g ∈ +
⊥M ,0
, a otΩe,
M N H+
⊥
−⊇,0
0∩ .
Teoremu dovedeno.
Pryklad. Nexaj A = A
∗ ≥ 1 — samosprqΩenyj operator v H 0, a H + =
= D ( )A v normi ⋅ + = A⋅ 0 . Zburennq A zadano systemog abstraktnyx hra-
nyçnyx umov
ω ϕ ϕ ω ωi i ii n( ) , , , , , ,,= 〈 〉 = = … < ∞ ∈{ }+ − −0 1 2 H ∀ ∈ϕ D ( )A .
Rozhlqnemo operator Ȧ : = A A�D ( ˙) , de D ( ˙)A = { }( ) : ( )ϕ ω ϕ∈ =D A i 0 . Çy
moΩlyvo vybraty ωi ∈ −H tak, wob Ȧ buv wil\no vyznaçenym symetryçnym
operatorom? Vidpovid\ vyplyva[ z teoremy:1. Oblast\ vyznaçennq D ( ˙)A �
� H 0 todi i lyße todi, koly def ( )( ˙)D HA ⊂ 0 = 0, a ce ekvivalentno umovi
N H− ∩ 0 = { }0 , de N − : = span{ }ωi . Prypustymo, wo Ȧ zadanyj ne wil\no.
Todi vaΩlyvo znaty defekt newil\nosti oblasti D ( ˙)A v H 0. Zrozumilo, wo
moΩna otrymaty pidprostir, ortohonal\nyj do D ( ˙)A , dovil\no] rozmirnosti.
Cq rozmirnist\ zaleΩyt\ vid vyboru N − :
D ( ˙)A ⊥ = def ( )( ˙)D HA ⊂ 0 = dim( )N H− ∩ 0 .
Vykorystovugçy ideg dovedennq teoremy:1, moΩna oderΩaty bil\ß tonki
rezul\taty pro wil\nist\ vkladennq odnoho pidprostoru v inßyj v A-ßkali
hil\bertovyx prostoriv, abo daty xarakterystyku defektu takoho vkladennq,
qkwo vono ne [ wil\nym.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
706 R. V. BOÛOK
Rozhlqnemo osnawennq hil\bertovoho prostoru, asocijovanoho z samosprq-
Ωenym operatorom A ≥ 1,
H − � H 0 � H + ,
de H+ = D( )A v normi ⋅ + = A ⋅ 0 i H− — sprqΩenyj do H+ vidnosno
H0 . Prypustymo, wo H+ rozkladeno v ortohonal\nu sumu: H+ = M N+ +�
tak, wo H0 � M+ . KoΩen osnawenyj prostir moΩna rozßyryty do lancgΩ-
ka z p’qty prostoriv (dyv. znovu [1, 2]), otΩe,
H −− � H − � H 0 � H + � H ++ , (5)
de H ++ = D( )A 2 . Slid zaznaçyty, wo H ++ [ sprqΩenym do H0 vidnosno
H+ .
Rozhlqnemo v H ++ linijnu mnoΩynu M̃+ : = M H+ ++∩ . Lehko baçyty,
wo ce zamknenyj v H ++ pidprostir. Dijsno, nexaj poslidovnist\ ϕn ∈ +M̃ .
Qkwo vona zbiΩna do ϕ ∈ ++H , to zbiΩna i v H+ vnaslidok nerivnosti ⋅ + ≤
≤ ⋅ ++ . Tomu ϕ ∈ +M , oskil\ky pidprostir M+ [ zamknenym v H+ . OtΩe,
ϕ ∈ +M̃ .
Takym çynom, prostir H ++ rozklada[t\sq v ortohonal\nu sumu:
H ++ =
˜ ˜M N+ +� .
Analohiçnyj rozklad ma[ misce i dlq H0 :
H 0 =
˜ ˜M N0 0� ,
de
M̃0 : =
D0,
˜
++ +M ≡ A 2 M̃+ ,
Ñ 0 : =
D0,
˜
++ +N ≡
A 2 Ñ +
i D0 0, :++ ++ →H H poznaça[ operator unitarnoho izomorfizmu.
Teorema"2. Nexaj pozytyvni prostory H+ = D( )A ta H++ = D( )A 2
z
A -ßkaly rozkladeno v ortohonal\ni sumy, qk opysano vywe: H+ = M N+ +� i
H++ =
˜ ˜M N+ +� , de M̃+ : = M H+ ++∩ . Pry c\omu prypuska[t\sq, wo pid-
prostir M+ [ wil\nym v H0 : H0 � M+ .
Pidprostir M̃+ bude wil\nym v M+ todi i til\ky todi, koly pidprostir
N +
cl,0
(zamykannq
N + v H0 ) ne matyme nenul\ovyx vektoriv, spil\nyx z
M+:
M̃ M+ +� ⇐⇒ N M+ +
cl,0 ∩ = { }0 . (6)
Qkwo Ω umova (6) ne vykonu[t\sq i pidprostir M̃+ ne [ wil\nym v M+ , to
joho defekt newil\nosti vyznaça[t\sq formulog
def ( ˜ )M M+ +⊂ = dim( )N M+ +
cl,0 ∩ . (7)
Pry dovedenni ci[] teoremy vykorystovu[t\sq nastupna lema.
Lema. Ma[ misce rivnist\
N +
cl,0 =
Ñ 0 ,
de
Ñ 0 = H M0
2� A ˜
+ =
D0,
˜
++ +N ≡
A 2 Ñ + .
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
PRO DEFEKT NEWIL|NOSTI NEPERERVNYX VKLADEN| … 707
Dovedennq. Z oznaçennq M̃ + lehko baçyty, wo
M̃ + = { ( ) }: ,ϕ ϕ∈ =++ + +H N 0 . (8)
Z inßoho boku, 〈 〉+ ++
˜ ˜, ,N M0 0 = 0. Zrozumilo, wo N +
cl,0 ⊂ Ñ 0. Prypu-
stymo, wo Ñ 0 = N S+
cl,0
0� . PokaΩemo, wo S0 = 0. Z ohlqdu na heometrig
ßkaly (5) ma[mo
0 = 〈 〉+ + ++N S Mcl,0
0 0� , ˜
, = ( [ ] ˜ ), ,I++ + + ++0
0
0N S Mcl, � ,
de I++,0 = D0
1
,++
−
: H 0 → H ++ . Zokrema,
0 = (
˜ ), ,,I I++ + ++ + ++0
0
0 0N S Mcl, � = 〈 〉+ ++ + ++N S Mcl,0
0 0 0, , ,
˜I � .
Oskil\ky N + � N +
cl,0 , to 0 = ( ˜ ), ,N S M+ ++ + +I 0 0 � . Ale I++ ++∈,0 0S H . A
ce pryvodyt\ do supereçnosti z tym faktom, wo vsi vektory z H ++ , ortohonal\-
ni do N + , naleΩat\ M̃+ (dyv. (8)). Takym çynom, I++,0 0S = 0, a otΩe, i
S0 = 0. Ce j oznaça[, wo N +
cl,0 = Ñ 0.
Dovedennq teoremy. Dlq dovedennq neobxidno vvesty we odnu ßkalu pro-
storiv. Oskil\ky H 0 � M + , to cg paru moΩna rozhlqnuty qk peredosnawe-
nyj prostir (dyv. [1, 2]), qkyj [dynym çynom rozßyrg[t\sq do osnawennq hil\-
bertovoho prostoru H 0 . Vvedemo poznaçennq dlq novoho osnawenoho prostoru:
�
H − � H 0 �
�
H + ,
de
�
H + ≡ M + z normog H + , a
�
H − — sprqΩenyj do
�
H + vidnosno H 0 .
Nexaj
�
A =
�
A∗ ≥ 1 poznaça[ samosprqΩenyj operator, asocijovanyj z cym
osnawennqm. Rozhlqnemo rozßyrene, po analohi] z (5), osnawennq H 0 :
�
H −− �
�
H − � H 0 �
�
H + �
�
H ++ . (9)
U robotax [4 – 6] pokazano, wo
�
H ++ =
PM H
+ ++ , de
PM +
— ortoproektor
v H + na M + . Pry c\omu norma v
�
H ++ vyznaça[t\sq tak: dlq koΩnoho ϕ =
=
PM +
ψ , ψ ∈ ++H ,
ϕ ++
∼ : =
ψ H ++
.
OtΩe, (9) moΩna perepysaty u vyhlqdi
�
H −− �
�
H − � H 0 � M + � PM H
+ ++ , (10)
zvidky zrozumilo, wo D ( )
�
A ≡ M + i D ( )
�
A2 =
PM H
+ ++ . Za pobudovog (dok-
ladniße dyv. [5]) operator
�
A pov’qzanyj s operatorom A takym çynom:
�
A P2
M +
ϕ = A2ϕ , ϕ ∈ H ++ ≡ D ( )A2 ,
de, nahada[mo, A — samosprqΩenyj operator, asocijovanyj zi ßkalog (5).
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
708 R. V. BOÛOK
Zrozumilo, wo M̃ + : = M H+ ++∩ [ vlasnog pidmnoΩynog prostoru
PM H
+ ++ . Tomu dlq bud\-qkoho ϕ ∈ +M̃ ma[mo
PM +
ϕ = ϕ
( ˜ ˜ )PM M M
+ + += .
OtΩe,
A
2 � M̃ + =
�
A2 � M̃ + . (11)
Oskil\ky
�
A2
— unitarnyj operator u ßkali (10),
�
A2
: PM H
+ ++ → H 0 i
spravdΩu[t\sq rivnist\ (11), robymo vysnovok, wo ortohonal\nyj rozklad
H 0 =
˜ ˜M N0 0� moΩna perenesty na rozklad prostoru
�
H ++ =
PM H
+ ++ . A
same,
PM H
+ ++ =
� �
M N++ ++� , de
�
M ++ =
�
A−2
0M̃ = M̃ + zavdqky (11). Za-
stosuvavßy teper teoremu:A1 z [3] do trijky H 0 � M + �
PM H
+ ++ , qk do os-
nawennq hil\bertovoho prostoru M + , otryma[mo (6).
Spivvidnoßennq (7) [ naslidkom rivnosti (]] dovedennq take Ω, qk i v teore-
mi:1)
( ˜ ) ,M +
⊥ + = Ñ M0 ∩ + , (12)
de (
˜ ) ,M +
⊥ +
poznaça[ ortohonal\ne dopovnennq do M̃ + u prostori M + .
Teoremu dovedeno.
Cikavym z toçky zoru heometri] ßkaly hil\bertovyx prostoriv moΩe buty
takyj naslidok z teoremy:2 ta rivnist\ (5.12) z teoremy:5.6 z roboty [4].
Naslidok.
N M+ +
cl,0 ∩ = { }0 ⇐⇒ N H+ +
cl,0 ∩ = N + . (13)
Dovedennq c\oho naslidku vyplyva[ z toho faktu, wo prava çastyna (13) ta-
koΩ ekvivalentna wil\nosti M̃ + v M + (dyv. [4]).
1. Berezanskii Yu. M. Expansion in eigenfunctions of self-adjoint operators. – Providence, Rhode
Island: AMS, 1968.
2. Berezanskii Yu. M. Self-adjoint operators in spaces of function of infinitely many of variables. –
Providence, Rhode Island: AMS, 1986.
3. Albeverio S., Karwowski W., Koshmanenko V. Square power of singularly perturbed operators //
Math. Nachr. – 1995. – 173. – P. 5 – 24.
4. Albeverio S., Bozhok R., Dudkin M., Koshmanenko V. Dense subspace in scales of Hilbert spaces //
Meth. Funct. Anal. and Top. – 2005. – 11, # 2. – P. 156 – 169.
5. Bozhok R., Koshmanenko V. D. Singular perturbations of self-adjoint operators associated with
rigged Hilbert spaces // Ukr. Math. J. – 2005. – 57, # 5.
6. Koshmanenko V. Construction of singular perturbations by the method of rigged Hilbert spaces //
J. Phys. A: Math. and Gen. – 2005. – 38. – P. 4999 – 5009.
OderΩano 05.12.05,
pislq doopracgvannq — 23.05.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
|
| id | umjimathkievua-article-3188 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:37:52Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/57/82cc7a992360d3f4b6a5132ba26ff657.pdf |
| spelling | umjimathkievua-article-31882020-03-18T19:47:45Z On the defect of nondenseness of continuous imbeddings in the scale of Hilbert spaces Про дефект нещільності неперервних вкладень у шкалі гільбертових просторів Bozhok, R. V. Божок, Р. В. We obtain a formula for the determination of a defect under a continuous imbedding of subspaces in the scale of Hilbert spaces. Установлена формула для определения дефекта при непрерывном вложении подпространств в шкале гильбертовых пространств. Institute of Mathematics, NAS of Ukraine 2008-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3188 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 5 (2008); 704–708 Український математичний журнал; Том 60 № 5 (2008); 704–708 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3188/3122 https://umj.imath.kiev.ua/index.php/umj/article/view/3188/3123 Copyright (c) 2008 Bozhok R. V. |
| spellingShingle | Bozhok, R. V. Божок, Р. В. On the defect of nondenseness of continuous imbeddings in the scale of Hilbert spaces |
| title | On the defect of nondenseness of continuous imbeddings in the scale of Hilbert spaces |
| title_alt | Про дефект нещільності неперервних вкладень у шкалі гільбертових просторів |
| title_full | On the defect of nondenseness of continuous imbeddings in the scale of Hilbert spaces |
| title_fullStr | On the defect of nondenseness of continuous imbeddings in the scale of Hilbert spaces |
| title_full_unstemmed | On the defect of nondenseness of continuous imbeddings in the scale of Hilbert spaces |
| title_short | On the defect of nondenseness of continuous imbeddings in the scale of Hilbert spaces |
| title_sort | on the defect of nondenseness of continuous imbeddings in the scale of hilbert spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3188 |
| work_keys_str_mv | AT bozhokrv onthedefectofnondensenessofcontinuousimbeddingsinthescaleofhilbertspaces AT božokrv onthedefectofnondensenessofcontinuousimbeddingsinthescaleofhilbertspaces AT bozhokrv prodefektneŝílʹnostíneperervnihvkladenʹuškalígílʹbertovihprostorív AT božokrv prodefektneŝílʹnostíneperervnihvkladenʹuškalígílʹbertovihprostorív |