On Initial Data of a Simple Conservative Scattering System That Can Be Transferred to Zero by a Sequence of Inputs from l ²
We describe the lineal of initial data of a simple conservative scattering system which can be transferred to zero by a sequence from l ². The proof is based on the known connection between the Lax - Phillips scattering theory and the theory of unitary operator nodes developed by B. Szokefalvi - Na...
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2005
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509742511685632 |
|---|---|
| author | Nudel'man, M. A. Нудельман, М. А. Нудельман, М. А. |
| author_facet | Nudel'man, M. A. Нудельман, М. А. Нудельман, М. А. |
| author_sort | Nudel'man, M. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:00:05Z |
| description | We describe the lineal of initial data of a simple conservative scattering system which can be transferred to zero by a sequence from l ².
The proof is based on the known connection between the Lax - Phillips scattering theory and the theory of unitary operator nodes developed by B. Szokefalvi - Nagy, C. Foias, and M. S. Brodskii.
|
| first_indexed | 2026-03-24T02:45:56Z |
| format | Article |
| fulltext |
UDK 517.9
M.�A.�Nudel\man (Yntehr. bank. ynform. system¥, Odessa)
O NAÇAL|NÁX DANNÁX PROSTOJ
KONSERVATYVNOJ SYSTEMÁ RASSEQNYQ,
KOTORÁE MOHUT BÁT| PEREVEDENÁ V NOL|
POSLEDOVATEL|NOST|G VXODOV YZ l
2
We describe the lineal of initial data of a simple conservative scattering system which can be transferred
to zero by a sequence from l
2. The proof is based on the known connection between the Lax – Phillips
scattering theory and the theory of unitary operator nodes developed by B. Szökefalvi-Nagy, C. Foias,
and M. S. Brodskii.
Opysano lineal poçatkovyx danyx prosto] konservatyvno] systemy rozsigvannq, qki moΩna pe-
revesty v nul\ poslidovnistg vxodiv z l
2. Dovedennq ©runtu[t\sq na vidomomu zv’qzku miΩ teo-
ri[g rozsigvannq Laksa – Fillipsa ta teori[g unitarnyx vuzliv B.3Sekefal\vi-Nadq, Ç.3Foqßa
ta M.3S.3Brods\koho.
1. Predvarytel\n¥e svedenyq y formulyrovka rezul\tata. Rassmotrym
lynejnug systemu s dyskretn¥m vremenem λ vyda
xk +1 = A xk + B ξ k ,
(1)
σk = C xk + D ξ k ,
k = 0, 1, 2, … ,
hde xk ∈ X , ξ k ∈ U , σk ∈ V ; prostranstvo sostoqnyj X , prostranstvo vxodov U
y prostranstvo v¥xodov V — separabel\n¥e hyl\bertov¥ prostranstva (moΩet
b¥t\, koneçnomern¥e); A , B , C , D — ohranyçenn¥e lynejn¥e operator¥.
V sootvetstvyy s prynqtoj termynolohyej prostranstvo upravlenyj system¥
λ est\ podprostranstvo Xc
λ prostranstva X , kotoroe opredelqetsq formuloj
Xc
λ = V
k
kA BU
=
∞
0
(zdes\ znak V oboznaçaet zam¥kanye lynejnoj oboloçky), y prostranstvo na-
blgdenyj system¥ λ est\ podprostranstvo Xo
λ prostranstva X , kotoroe
opredelqetsq formuloj
Xo
λ = V
k
kA C V
=
∞
∗ ∗
0
.
Systema λ naz¥vaetsq prostoj, esly X = Xc
λ V Xo
λ .
Esly naçal\noe dannoe x0 system¥ λ ravno 0, to dejstvyem koneçnoj po-
sledovatel\nosty vxodov ξ0, ξ1, … , ξ m πta systema perexodyt v sostoqnye
xm +1 = A Bk
m k
k
m
ξ −
=
∑
0
(2)
y, takym obrazom, prostranstvo upravlenyj est\ mnoΩestvo πlementov pro-
stranstva X , kotor¥e mohut b¥t\ skol\ uhodno blyzko dostyΩym¥ yz nulevoho
naçal\noho uslovyq koneçnoj posledovatel\nost\g vxodov; moΩno analohyçno
ynterpretyrovat\ prostranstvo nablgdenyj Xo
λ , esly zamenyt\ systemu λ so-
prqΩennoj systemoj λ∗, kotoraq opredelqetsq ravenstvamy
xk +1 = A∗
xk + C∗
ξ k ,
(3)
σk = B∗
xk + D∗
ξ k ,
© M.3A.3NUDEL|MAN, 2005
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4 559
560 M.3A.3NUDEL|MAN
hde xk ∈ X , ξ k ∈ V , σk ∈ U.
Nastoqwaq rabota posvqwena reßenyg sledugweho voprosa: kakov¥ na-
çal\n¥e dann¥e x0 = a , dlq kotor¥x suwestvuet takaq posledovatel\nost\
vxodov { } =
∞ξk k 0 ⊂ U, çto || ||
=
∞
∑ ξk
k
2
0
< + ∞ y lim
k kx
→∞
|| || = 0. ∏tot vopros yssle-
duetsq dlq sluçaq, kohda λ — prostaq konservatyvnaq systema rasseqnyq.
Napomnym sootvetstvugwee opredelenye [1]: systema λ vyda (1) naz¥vaetsq
konservatyvnoj systemoj rasseqnyq, esly ravenstva (1) vlekut ravenstvo
|| || || || || || || ||+ − = −x xk k k k1
2 2 2 2ξ σ
y analohyçnoe ravenstvo dlq system¥ λ∗
vyda (3).
Lehko vydet\, çto πto opredelenye ravnosyl\no tomu faktu, çto bloçn¥j
operator
A B
C D
,
kotor¥j dejstvuet yz X � U v X � V, yzometryçen y soprqΩenn¥j k nemu
operator takΩe yzometryçen. V druhoj termynolohyy [2] πto oznaçaet, çto
prostranstva X , U , V y operator¥ A , B , C , D obrazugt unytarn¥j uzel.
Xoroßo yzvestna svqz\ meΩdu teoryej unytarn¥x operatorn¥x uzlov y teo-
ryej rasseqnyq Laksa – Fyllypsa (opysanye πtoj svqzy sm., naprymer, vo vvede-
nyy k stat\e [3]). Central\n¥m obæektom teoryy Laksa – Fyllypsa qvlqetsq (v
dannom sluçae dyskretnaq) unytarnaq hruppa operatorov { } =−∞
∞W k
k , hde W —
unytarn¥j operator, dejstvugwyj v nekotorom separabel\nom hyl\bertovom
prostranstve � . V πtom prostranstve v¥delqgtsq dva podprostranstva � + y
�– , kotor¥e ymegt sledugwye svojstva:
1) W �+ ⊂ �+ ;
2)
∩
k
kW
=
∞
+
0
� = { 0 } ;
3) W∗
�– ⊂ �– ;
4)
∩
k
kW
=
∞
∗
−
0
� = { 0 } .
KaΩdomu unytarnomu uzlu (yly, çto ravnosyl\no, kaΩdoj konservatyvnoj
systeme rasseqnyq vyda (1)) sootvetstvuet takaq unytarnaq hruppa Laksa – Fyl-
lypsa, dlq kotoroj v¥polnen¥ sledugwye sootnoßenyq:
�+ ⊥ �– ,
� = �+ � X � �– ,
(4)
�+ � W �+ = V,
�– � W∗
�– = U.
Yspol\zovav texnyku, razrabotannug v teoryy Laksa – Fyllypsa, m¥ doka-
Ωem sledugwee utverΩdenye.
Pust\ M — mnoΩestvo tex naçal\n¥x dann¥x x0 = a ∈ X system¥ λ vyda
(1), dlq kotor¥x suwestvuet takaq posledovatel\nost\ vxodov { } =
∞ξk k 0 , çto
|| ||
=
∞
∑ ξ 2
0k
< + ∞ y lim
k kx
→∞
|| || = 0.
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
O NAÇAL|NÁX DANNÁX PROSTOJ KONSERVATYVNOJ SYSTEMÁ RASSEQNYQ … 561
Oboznaçym çerez
�
Xo
λ lynejnug oboloçku L ( A∗k
C∗
V ) , k = 0, 1, 2, … (takym
obrazom, Xo
λ =
�
Xo
λ , hde çerta oboznaçaet zam¥kanye).
Pust\ Θ ( z ) = D + z C ( I – z A )–1B — peredatoçnaq funkcyq konservatyvnoj
system¥ rasseqnyj λ (y v to Ωe vremq xarakterystyçeskaq funkcyq sootvet-
stvugweho unytarnoho uzla y matryca rasseqnyq sootvetstvugwej unytarnoj
hrupp¥ Laksa – Fyllypsa; sm. [1, 2, 4]). Yzvestno, çto πta funkcyq prynadle-
Ωyt klassu Íura sΩymagwyx analytyçeskyx operator-funkcyj, kotor¥e
opredelen¥ v otkr¥tom edynyçnom kruhe. Pust\ ∆ ( ξ ) = (I – Θ(ξ)∗Θ(ξ))1/2, hde
Θ ( ξ ) ( | ξ | = 1) — hranyçn¥e znaçenyq funkcyy Θ ( z ) .
Pust\ H
2
( U ) — prostranstvo Xardy, kotoroe traktuetsq kak podprostran-
stvo prostranstva L
2
( U ) na edynyçnoj okruΩnosty.
Teorema. Dlq kaΩdoj prostoj konservatyvnoj system¥ rasseqnyq λ yme-
et mesto vklgçenye
�
Xo
λ ⊂ M ⊂ Xo
λ . (5)
Ravenstvo M = Xo
λ ymeet mesto tohda y tol\ko tohda, kohda lyneal
∆ ( ξ ) H
2
( U ) zamknut v topolohyy prostranstva L
2
( U ) .
2. Dokazatel\stvo teorem¥. Kak b¥lo otmeçeno v p.31, kaΩdoj konserva-
tyvnoj systeme rasseqnyq λ kanonyçesky sootvetstvuet nekotoraq unytarnaq
hruppa Laksa – Fyllypsa { } =−∞
∞W k
k , dlq kotoroj spravedlyvo sootnoßenye
�– ⊥ �+ . Pry πtom prostota system¥ λ ravnosyl\na sootnoßenyg
V VV
k
k
k
kW W
=
+∞
∗
+ =
+∞
−
0 0
� � = � .
Yzvestno (sm., naprymer, [3]), çto πvolgcyq unytarnoj poluhrupp¥ { } =
∞W k
k 0
vosproyzvodyt dynamyku system¥ λ v sledugwem sm¥sle: podprostranstvo
�+ ⊂ � moΩet b¥t\ kanonyçesky otoΩdestvleno s prostranstvom l
2
( V ) y
podprostranstvo �– ⊂ � moΩet b¥t\ kanonyçesky otoΩdestvleno s prostran-
stvom l
2
( U ) ; esly { } =
∞ξk k 0 ⊂ l
2
( U ) — posledovatel\nost\ vxodov system¥ λ y
x0 = a ∈ X — naçal\noe dannoe,
h = col ( … , λ3, λ2, λ1, λ0 ; a ; ξ 0, ξ1, ξ2, … )
(zdes\ { } =
∞ξk k 0 ⊂ l
2
( U ) ynterpretyruetsq kak πlement �– y { } =
∞λk k 0 ⊂ l
2
( V )
— kak πlement �+ ) , to
W
k
h = col ( … , λ1, λ0, σ0, … , σk –2 , σk –1 ; xk ; ξk , ξk +1, ξk +2, … ) . (6)
Otsgda qsno, çto mnoΩestvo M soderΩyt te y tol\ko te πlement¥ a ∈ X , dlq
kotor¥x suwestvuet takoe d– ∈ �– , çto
lim ( )–k X
kP W a d
→ +∞
|| ||+ = 0 (7)
(zdes\ y dalee symvolom PL oboznaçaetsq operator ortohonal\noho proektyro-
vanyq na podprostranstvo L ) .
Zapyßem vektor W
k
( a + d– ) ∈ � v vyde
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
562 M.3A.3NUDEL|MAN
W
k
( a + d– ) = { }+
=
+∞dn n 0 � P W a dX
k ( )–+ � { }−
=
+∞dn n 0
v sootvetstvyy s ortohonal\n¥m razloΩenyem (4).
Pust\ v¥polneno predel\noe ravenstvo (7). Poskol\ku v sylu formul¥ (6)
posledovatel\nost\ { }−
=
+∞dn n 0 stremytsq k nulg v metryke prostranstva l
2
( U )
pry k → ∞ , to poluçaem
0 = lim ( )–k X
kP W a d
→ +∞
|| ||+ = lim ( )–k
k
n nW a d d
→ +∞
+
=
+∞|| { } ||+ − 0 =
= lim –k
k
n na d W d
→ +∞
∗ +
=
+∞|| { } ||+ − 0
(napomnym, çto operator W unytaren).
Takym obrazom, a + d– ∈
V
k
kW
=
+∞ ∗
+0
� y, sledovatel\no,
a ∈
P WX k
kV
=
+∞
∗
+0
� . (8)
Obratno, pust\ v¥polneno vklgçenye (8). Tohda suwestvuet vektor l takoj,
çto l ⊥ X y a + l ∈
V
k
kW
=
+∞ ∗
+0
� . V sootvetstvyy s ortohonal\n¥m razloΩenyem
(4) l predstavlqetsq v vyde d+ � d– , hde d+ ∈ � + , d– ∈ � – . Poskol\ku
V
k
kW
=
+∞ ∗
+0
� ⊃ �+ , to d+ ∈
V
k
kW
=
+∞ ∗
+0
� y, takym obrazom, a + d– ∈
V
k
kW
=
+∞ ∗
+0
� .
Dalee, tak kak { }∗
+ =
+∞W k
k� 0 — rasßyrqgwaqsq posledovatel\nost\ mno-
Ωestv, t.3e.
�+ ⊂ W
∗
�+ ⊂ W
∗2
�+ ⊂ … ,
to suwestvuet posledovatel\nost\ { } =
+∞hk k 0 ⊂ �+ takaq, çto a + d– = lim
k
k
kW h
→+∞
∗ .
Esly teper\ rassmotret\ dvojnug posledovatel\nost\ { } =
+∞gkl k l, 0 ⊂ � , hde
g P W W hkl X
l k
k= ∗ , k = 0, 1, 2, … ,
to budem ymet\
lim ( )
k kl X
lg P W a d
→∞ −= + , l = 0, 1, 2, … , (9)
pryçem sxodymost\ ravnomerna otnosytel\no l , poskol\ku
|| ||− +g P W a dkl X
l( )– = || ||∗ − +P W W h P W a dX
l k
k X
l( )– ≤
≤ || ||∗ − +W W h W a dl k
k
l( )– = || ||∗ − +W h a dk
k ( )–
(poslednee ravenstvo qvlqetsq sledstvyem toho, çto operator W unytaren) .
Ymeem
lim ( )
l X
lP W a d
→ +∞ −+ = lim lim
l k X
l k
kP W W h
→ +∞ → +∞
∗ = lim lim
l k X
l k
kP W h
→ +∞ → +∞
− .
Poskol\ku v predel\nom processe (9) sxodymost\ ravnomerna otnosytel\no
l , to korrektna perestanovka predel\n¥x perexodov y m¥ poluçaem
lim ( )
l X
lP W a d
→ +∞ −+ = lim lim
k l X
l k
kP W h
→ +∞ → +∞
− .
No hk ∈ �+ pry vsex k ≥ 0, poπtomu pry l ≥ k P W hX
l k
k
− = 0, otkuda qsno, çto
poslednyj povtorn¥j predel raven nulg, çto dokaz¥vaet ravenstvo
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
O NAÇAL|NÁX DANNÁX PROSTOJ KONSERVATYVNOJ SYSTEMÁ RASSEQNYQ … 563
M = P WX k
kV
=
+∞
∗
+0
� . (10)
Yzvestno, çto esly πvolgcyq poluhrupp¥ { } =
+∞W k
k 0 opys¥vaet dynamyku sys-
tem¥ λ, to πvolgcyq poluhrupp¥ { }∗
=
+∞W k
k 0 opys¥vaet dynamyku system¥ λ∗.
Yspol\zuq formulu (2), naxodym
P WX
m∗
+� = L ( A∗k
C∗
V ) , k = 0, 1, 2, … , m (11)
(zdes\ L oboznaçaet lynejnug oboloçku). Yz ravenstv (10) y (11) sleduet dvoj-
noe vklgçenye (5).
Dlq zaverßenyq dokazatel\stva otmetym, çto v sylu πtoho dvojnoho vklg-
çenyq ravenstvo M = Xo
λ ravnosyl\no tomu faktu, çto mnoΩestvo M zamknuto.
S pomow\g prostoho heometryçeskoho rassuΩdenyq lehko dokazat\ sledug-
wee ravenstvo:
V
k
kW
=
+∞
∗
+0
� � I P
k
kW� �
�Ò −( )
=
+∞ ∗
+
−V 0
= �+ � M � �– .
Pry dokazatel\stve πtoho ravenstva udobno vospol\zovat\sq tem faktom, çto
vklgçenye
h ∈ �+ � M � �–
ravnosyl\no tomu, çto suwestvuet takoj vektor w ∈
V
k
kW
=
+∞ ∗
+0
� , çto
PX h = PX w
(sm. ravenstvo (10)).
Takym obrazom, zamknutost\ mnoΩestva M ravnosyl\na zamknutosty mno-
Ωestva
Z = I P
k
kW� �
�Ò −( )
=
+∞ ∗
+
−V 0
.
V sylu formul¥ (2.6) § 2 hl.3VI knyhy [5] y neposredstvenno predßestvugweho
ej v [5] teksta mnoΩestvo Z zamknuto tohda y tol\ko tohda, kohda lyneal
∆ ( ξ ) H
2
( U ) zamknut v topolohyy prostranstva L2
( U ) .
Teorema dokazana.
Zameçanye. V knyhe [5] rassuΩdenye, na kotoroe m¥ ss¥laemsq, pryvodyt-
sq ne3dlq proyzvol\noho unytarnoho uzla, a dlq unytarnoho uzla, poroΩdenno-
ho prost¥m (t.3e. vpolne neunytarn¥m) sΩatyem. Odnako πto rassuΩdenye bez
yzmenenyj perenosytsq na sluçaj proyzvol\noho prostoho unytarnoho uzla.
Avtor v¥raΩaet blahodarnost\ recenzentu za suwestvennoe uluçßenye y
uprowenye dokazatel\stva.
1. Arov'D.'Z. Passyvn¥e lynejn¥e stacyonarn¥e dynamyçeskye system¥ // Syb. mat. Ωurn. –
1979. – #320. – S.3211 – 228.
2. Brodskyj'M.'S. Unytarn¥e uzl¥ y yx xarakterystyçeskye funkcyy // Uspexy mat. nauk. –
1978. – 33, #34. – S.3141 – 168.
3. Nudel\man'M.'A. Dostatoçn¥e uslovyq absolgtnoj ustojçyvosty optymal\n¥x passyvn¥x
system rasseqnyq // Alhebra y analyz. – 1994. – 6, #34. – S.3187 – 203.
4. Adamqn'V.'M., Arov'D.'Z. Ob unytarn¥x sceplenyqx poluunytarn¥x operatorov // Mat.
yssled. – 1966. – #31. – S.33 – 66.
5. Sekefal\vy-Nad\'B., Foqß'Ç. Harmonyçeskyj analyz operatorov v hyl\bertovom prostran-
stve. – M.: Myr, 1970. – 4313s.
Poluçeno 20.01.2003
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 4
|
| id | umjimathkievua-article-3621 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:45:56Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/04/0130a780094ada0dc062e5e481476704.pdf |
| spelling | umjimathkievua-article-36212020-03-18T20:00:05Z On Initial Data of a Simple Conservative Scattering System That Can Be Transferred to Zero by a Sequence of Inputs from l ² О начальных данных простой консервативной системы рассеяния, которые могут быть переведены в ноль последовательностью входов из l ² Nudel'man, M. A. Нудельман, М. А. Нудельман, М. А. We describe the lineal of initial data of a simple conservative scattering system which can be transferred to zero by a sequence from l ². The proof is based on the known connection between the Lax - Phillips scattering theory and the theory of unitary operator nodes developed by B. Szokefalvi - Nagy, C. Foias, and M. S. Brodskii. Описано лінеал початкових даних простої консервативної системи розсіювання, які можна перевести в нуль послідовністю входів з l ². Доведення Грунтується на відомому зв'язку між теорією розсіювання Лакса - Філліпса та теорією унітарних вузлів B. Секефальві - Надя, Ч. Фояша та M. С. Бродського. Institute of Mathematics, NAS of Ukraine 2005-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3621 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 4 (2005); 559–563 Український математичний журнал; Том 57 № 4 (2005); 559–563 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3621/3973 https://umj.imath.kiev.ua/index.php/umj/article/view/3621/3974 Copyright (c) 2005 Nudel'man M. A. |
| spellingShingle | Nudel'man, M. A. Нудельман, М. А. Нудельман, М. А. On Initial Data of a Simple Conservative Scattering System That Can Be Transferred to Zero by a Sequence of Inputs from l ² |
| title | On Initial Data of a Simple Conservative Scattering System That Can Be Transferred to Zero by a Sequence of Inputs from l ² |
| title_alt | О начальных данных простой консервативной системы рассеяния, которые могут быть переведены в ноль последовательностью входов из l ² |
| title_full | On Initial Data of a Simple Conservative Scattering System That Can Be Transferred to Zero by a Sequence of Inputs from l ² |
| title_fullStr | On Initial Data of a Simple Conservative Scattering System That Can Be Transferred to Zero by a Sequence of Inputs from l ² |
| title_full_unstemmed | On Initial Data of a Simple Conservative Scattering System That Can Be Transferred to Zero by a Sequence of Inputs from l ² |
| title_short | On Initial Data of a Simple Conservative Scattering System That Can Be Transferred to Zero by a Sequence of Inputs from l ² |
| title_sort | on initial data of a simple conservative scattering system that can be transferred to zero by a sequence of inputs from l ² |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3621 |
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