On the strong matrix Hamburger moment problem
We obtain necessary and sufficient conditions for the solvability of the strong matrix Hamburger moment problem. We describe all solutions of the moment problem by using the fundamental results of A. V. Shtraus on generalized resolvents of symmetric operators.
Saved in:
| Date: | 2010 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | Russian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2010
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/2881 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508871752155136 |
|---|---|
| author | Zagorodnyuk, S. M. Загороднюк, С. М. Загороднюк, С. М. |
| author_facet | Zagorodnyuk, S. M. Загороднюк, С. М. Загороднюк, С. М. |
| author_sort | Zagorodnyuk, S. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:39:35Z |
| description | We obtain necessary and sufficient conditions for the solvability of the strong matrix Hamburger moment problem. We describe all solutions of the moment problem by using the fundamental results of A. V. Shtraus on generalized resolvents of symmetric operators. |
| first_indexed | 2026-03-24T02:32:06Z |
| format | Article |
| fulltext |
UDK 517.948
S. M. Zahorodngk (Xar\kov. nac. un-t)
O SYL|NOJ MATRYÇNOJ PROBLEME
MOMENTOV HAMBURHERA
We obtain necessary and sufficient conditions for the sovability of the strong matrix Hamburger moment
problem. We decribe all solutions of the moment problem by using fundamental results of A. V. Shtraus
on generalized resolvents of symmetric operators.
Otrymano neobxidni ta dostatni umovy toho, wo syl\na matryçna problema momentiv Hamburhera
ma[ rozv’qzok. Opysano vsi rozv’qzky problemy momentiv. Pry c\omu vykorystano fundamen-
tal\ni rezul\taty A. V. Ítrausa pro uzahal\neni rezol\venty symetryçnyx operatoriv.
1. Vvedenye. Cel\g dannoj rabot¥ qvlqetsq poluçenye uslovyj razreßymos-
ty syl\noj matryçnoj problem¥ momentov Hamburhera y opysanye ee reßenyj.
Napomnym, çto syl\naq matryçnaq problema momentov Hamburhera sostoyt v
naxoΩdenyy neprer¥vnoj sleva neub¥vagwej matryc¥-funkcyy M x( ) =
= ( ), ,( )m xk l k l
N
=
−
0
1
na R , M( )−∞ = 0, takoj, çto
x dM xn ( )
R
∫ = Sn , n ∈Z , (1)
hde { }Sn n∈Z — zadannaq posledovatel\nost\ πrmytov¥x kompleksn¥x
( )N N× -matryc (momentov), N ∈N .
Syl\naq (skalqrnaq) problema momentov Hamburhera vperv¥e poqvylas\ v
naçale vos\mydesqt¥x hodov proßloho veka v svqzy s yzuçenyem neprer¥vn¥x
drobej v rabotax W. B. Jones, O. Njåstad, W. J. Thron [1, 2]. V çastnosty, b¥ly
ustanovlen¥ uslovyq razreßymosty πtoj zadaçy [2]. V 1996 hodu v rabote [3]
b¥ly opysan¥ reßenyq skalqrnoj syl\noj problem¥ momentov Hamburhera pry
nekotorom uslovyy rehulqrnosty (sm. takΩe obzor [4]).
Syl\naq matryçnaq problema momentov Hamburhera vperv¥e voznykla v 2006
hodu v rabote K. K. Symonova [5]. Rassmotrym sledugwye bloçn¥e matryc¥,
sostavlenn¥e yz momentov:
Γn = ( ) ,Si j i j n
n
+ = − =
S S S
S S S
S S S
n n
n n
n n
− −
−
… …
… …
2 0
0
0 2
� � �
… …
� � �
, n ∈ +Z . (2)
Pry uslovyy strohoj poloΩytel\nosty matryc Γn K. K. Symonov rassmatry-
vaet y yzuçaet matryçn¥e mnohoçlen¥ Lorana. V prostranstve πtyx mnohoçle-
nov operator umnoΩenyq na nezavysymug peremennug poroΩdaet symmetryçes-
kyj operator. Pry uslovyy vpolne neopredelennosty (yndeks defekta ope-
ratora raven ( , )N N ) K.CK.CSymonov¥m ustanovlena parametryzacyq reßenyj
syl\noj matryçnoj problem¥ momentov Hamburhera. TakΩe v rabote [5] usta-
novlen¥ (pry pozytyvnosty matryc Γn ) neobxodym¥e y dostatoçn¥e uslovyq
edynstvennosty reßenyq syl\noj matryçnoj problem¥ momentov Hamburhera.
V dannoj rabote m¥ ustanovym, çto uslovyq
Γn ≥ 0, n = 0, 1, 2, … , (3)
qvlqgtsq neobxodym¥my y dostatoçn¥my uslovyqmy razreßymosty problem¥
momentov (1). V sluçae, kohda πty uslovyq v¥polnen¥, moΩno vvesty nekoto-
© S. M. ZAHORODNGK, 2010
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 4 471
472 S. M. ZAHORODNGK
r¥j operator sdvyha v abstraktnom hyl\bertovom prostranstve. ∏tot operator
symmetryçen, obratym y eho samosoprqΩenn¥e rasßyrenyq poroΩdagt vse re-
ßenyq problem¥ momentov. Takoj podxod, pozvolqgwyj rassmatryvat\ odno-
vremenno y v¥roΩdenn¥e sluçay zadaçy, b¥l pryveden v [6] dlq sluçaq
klassyçeskoj problem¥ momentov Hamburhera. Dlq opysanyq vsex reßenyj vos-
pol\zuemsq fundamental\n¥my rezul\tatamy A. V. Ítrausa ob obobwenn¥x re-
zol\ventax symmetryçeskyx operatorov [7].
Vvedem neobxodym¥e oboznaçenyq. Kak ob¥çno, çerez R, C, N, Z, Z+ obo-
znaçaem mnoΩestva vewestvenn¥x, kompleksn¥x, natural\n¥x, cel¥x y
neotrycatel\n¥x cel¥x çysel sootvetstvenno. Prostranstvo n-mern¥x
kompleksn¥x vektorov a = ( ), , ,a a an0 1 1… − oboznaçym çerez Cn , n ∈N ; C+ =
= { }: Imz z∈ >C 0 . Esly a n∈C , to a∗ oboznaçaet kompleksno soprqΩen-
n¥j vektor. Posredstvom PL oboznaçaem prostranstvo kompleksn¥x mnoho-
çlenov Lorana, t. e. funkcyj vyda αk
k
k a
b
x
=∑ , a, b ∈Z : a b≤ , αk ∈C . Çe-
rez PL d, oboznaçaem prostranstvo kompleksn¥x mnohoçlenov Lorana vyda
αk
k
k d
d
x
= −∑ , αk ∈C . Pust\ M x( ) qvlqetsq neprer¥vnoj sleva neub¥vag-
wej matrycej-funkcyej M x( ) = ( ), ,( )m xk l k l
N
=
−
0
1
na R, M( )−∞ = 0, y τM x( ) : =
: = m xk kk
N
, ( )
=
−∑ 0
1
; Ψ( )x = ( / ), ,( )dm x dk l M k l
Nτ =
−
0
1 . Çerez L M2( ) oboznaçaem
mnoΩestvo (klassov πkvyvalentnosty) vektornoznaçn¥x funkcyj f n: R C→ ,
f = ( ), , ,f f fN0 1 1… − , takyx, çto (sm., naprymer, [8])
f
L M2
2
( )
: = f x x f x d xM( ) ( ) ( ) ( )Ψ ∗∫ τ
R
< ∞ .
Prostranstvo L M2( ) qvlqetsq hyl\bertov¥m so skalqrn¥m proyzvedenyem
( ),
( )
f g
L M2 : = f x x g x d xM( ) ( ) ( ) ( )Ψ ∗∫ τ
R
, f, g L M∈ 2( ) .
Esly H — hyl\bertovo prostranstvo, to ( , )⋅ ⋅ H y ⋅
H
oznaçagt skalqr-
noe proyzvedenye y normu v H sootvetstvenno. Yndeks¥ mohut opuskat\sq v
oçevydn¥x sluçaqx.
Dlq lynejnoho operatora A v H oboznaçaem çerez D A( ) eho oblast\ opre-
delenyq, çerez R A( ) eho oblast\ znaçenyj, çerez Ker A eho qdro y çerez A∗
soprqΩenn¥j operator, esly on suwestvuet. Esly A obratym, to A−1
obozna-
çaet obratn¥j operator. A oznaçaet zam¥kanye operatora, esly operator do-
puskaet zam¥kanye. Esly A ohranyçen, to A oboznaçaet eho normu. Dlq
proyzvol\noho nabora πlementov { }xn n A∈ yz H oboznaçaem çerez Lin{ }xn n A∈
y span{ }xn n A∈ lynejnug oboloçku y zamknutug lynejnug oboloçku (v
metryke H ) sootvetstvenno ( A — proyzvol\noe mnoΩestvo yndeksov). Dlq
mnoΩestva M H⊆ oboznaçaem çerez M zam¥kanye mnoΩestva M po norme
H. EH oboznaçaet edynyçn¥j operator v H, t. e. EH x = x , x H∈ . Esly H1 —
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 4
O SYL|NOJ MATRYÇNOJ PROBLEME MOMENTOV HAMBURHERA 473
podprostranstvo v H, to PH1
= PH
H
1
qvlqetsq operatorom ortohonal\noho
proektyrovanyq na H1 v H.
2. Razreßymost\ y opysanye reßenyj. Rassmotrym syl\nug matryçnug
problemu momentov Hamburhera (1). Uslovyq (3) qvlqgtsq neobxodym¥my us-
lovyqmy razreßymosty zadaçy (1). Dejstvytel\no, predpoloΩym, çto zadaça
ymeet reßenye M x( ) . Voz\mem proyzvol\nug funkcyg vyda a x( ) =
= ( )( ), ( ), , ( )a x a x a xN0 1 1… − , hde
a xj ( ) = A xj k
k
k n
n
,
=−
∑ , Aj k, ∈C , n ∈ +Z .
∏ta funkcyq prynadleΩyt L M2( ) y
0 ≤ a x dM x a x( ) ( ) ( )∗∫
R
=
k l n
n
k k N k
k lA A A x dM x
,
, , ,( ), , , ( )
=−
−
+∑ ∫ …0 1 1
R
*
* ( ), , ,, , ,A A Al l N l0 1 1… −
∗ =
k l n
n
k k N k k lA A A S
,
, , ,( ), , ,
=−
− +∑ …0 1 1 *
* ( ), , ,, , ,A A Al l N l0 1 1… −
∗ = A AnΓ
∗ ,
hde
A = ( , , , , ,, , , , ,A A A A An n N n n n0 1 1 0 1 1 1− − − − − + − +… , …
… , A A A AN n n n N n− − + −… …1 1 0 1 1, , , ,, , , , , ) ;
zdes\ m¥ vospol\zovalys\ pravylamy umnoΩenyq bloçn¥x matryc.
Budem predpolahat\ dalee, çto uslovyq (3) qvlqgtsq v¥polnenn¥my. Pust\
Si = ( ); , ,si k l k l
N
=
−
0
1 , si k l; , ∈C , i ∈Z . Rassmotrym beskoneçnug bloçnug matrycu
Γ = ( ) ,Si j i j+ = −∞
∞ =
� � �
� �
� � �
… … … …
… …
… … … …
… …
… … …
− −
−
S S S
S S S
S S
n n
n n
n
2 0
0
0 SS n2 …
� � �
, (4)
hde v¥delenn¥j πlement sootvetstvuet i = j = 0. Vzqv v kaçestve nulevoj
stroky y nulevoho stolbca tu stroku y tot stolbec, v kotor¥x naxodytsq lev¥j
verxnyj πlement v¥delennoj matryc¥ S0 , provedem numeracyg v vozrasta-
gwem porqdke strok y stolbcov Γ. ∏lement (kompleksnoe çyslo), stoqwyj v
k-j stroke y l-m stolbce, oboznaçym çerez γ k l, , − ∞ < k , l < +∞ . Pry πtom bu-
dut v¥polnen¥ ravenstva
γ rN j tN n+ +, = sr t j n+ ; , , r, t ∈Z , 0 ≤ j , n N≤ − 1. (5)
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 4
474 S. M. ZAHORODNGK
Takym obrazom, bloçnug matrycu Γ moΩno takΩe rassmatryvat\ kak skalqr-
nug beskoneçnug matrycu Γ = ( ), ,γ k l k l = −∞
∞ . Uslovye (3) ravnosyl\no tomu, çto
( ), ,γ k l k l r
r
= − ≥ 0, r ∈ +Z . (6)
Spravedlyva sledugwaq teorema (sm., naprymer, [9, c. 361 – 363]).
Teorema+1. Pust\ zadana nekotoraq beskoneçnaq kompleksnaq matryca Γ =
= ( ), ,γ k l k l = − ∞
∞ , γ k l, ∈C . Esly v¥polneno sootnoßenye (6), to najdutsq hyl\-
bertovo prostranstvo H y nabor πlementov { }xn n∈Z v H takye, çto
( ),x xn m H = γ n m, , n, m ∈Z . (7)
Pry πtom span{ }xn n∈Z = H.
Dokazatel\stvo. Rassmotrym prostranstvo V beskoneçn¥x kompleksn¥x
posledovatel\nostej
( )un n∈Z = … …( )−, , , ,u u u1 0 1 , un ∈C .
Rassmotrym πlement¥ x j πtoho prostranstva, u kotor¥x j-q komponenta ravna
edynyce, a ostal\n¥e ravn¥ nulg ( )j ∈Z . Zametym, çto πlement¥ { }xn n∈Z
lynejno nezavysym¥ y V = Lin{ }xn n∈Z . Opredelym funkcyonal
[ , ]x y = γ n m n m
n m
a b,
, ∈
∑
Z
(8)
dlq x, y ∈ V,
x = a xn n
n∈
∑
Z
, y = b xm m
m∈
∑
Z
, a bn m, ∈C .
Prostranstvo V s funkcyonalom [ , ]⋅ ⋅ qvlqetsq kvazyhyl\bertov¥m. Provodq
faktoryzacyg y popolnqq eho [10], poluçaem hyl\bertovo prostranstvo H y
nabor πlementov { }xn n∈Z (m¥ soxranyly dlq klassa πkvyvalentnosty, poroΩ-
dennoho πlementom xn , oboznaçenye xn ) v H takye, çto v¥polneno (7). Esly
span { }xn n∈Z ≠ H, to trebuem¥m prostranstvom budet span { }xn n∈Z .
Teorema dokazana.
Yz ravenstv (5) sleduet,C çto
γ a N b± , = γ a b N, ± , a b, ∈Z . (9)
Dejstvytel\no, dlq proyzvol\n¥x a rN j= + , b tN n= + , 0 ≤ j , n N≤ − 1,
r t, ∈Z , spravedlyvo
γ a N b± , = γ ( ) ,r N j tN n± + +1 = sr t j n+ ±1; , = γ rN j t N n+ ± +,( )1 = γ a b N, ± .
Oboznaçym L : = Lin{ }xn n∈Z . V¥berem proyzvol\n¥j πlement x L∈ . Po-
skol\ku πlement¥ { }xn n∈Z ne obqzatel\no lynejno nezavysym¥, πlement¥ ly-
nejnoj oboloçky L moΩno predstavlqt\ razlyçn¥my sposobamy v vyde lynej-
noj kombynacyy πlementov xn . Pust\
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 4
O SYL|NOJ MATRYÇNOJ PROBLEME MOMENTOV HAMBURHERA 475
x = αk k
k
x
= − ∞
∞
∑ (10)
y
x = βk k
k
x
= − ∞
∞
∑ , (11)
hde α βk k, ∈C — dva proyzvol\n¥x predstavlenyq πlementa x . Zdes\ lyß\
koneçnoe çyslo koπffycyentov α βk k, otlyçn¥ ot nulq. ∏to predpolahaetsq
y v dal\nejßem pry v¥bore πlementov yz lynejn¥x oboloçek. Yspol\zuq (7),
(9), moΩem zapysat\
αk k N
k
lx x±
= − ∞
∞
∑
, = αk
k
k N lx x
= − ∞
∞
±∑ ( , ) = α γk k N l
k
±
= − ∞
∞
∑ , =
= α γk k l N
k
, ±
= − ∞
∞
∑ = αk
k
k l Nx x
= − ∞
∞
±∑ ( , ) =
= αk k
k
l Nx x
= − ∞
∞
±∑
, = ( , )x xl N± , l ∈Z .
Analohyçn¥m obrazom zaklgçaem, çto
βk k N
k
lx x±
= − ∞
∞
∑
, = ( , )x xl N± , l ∈Z ,
y, znaçyt,
αk k N
k
lx x±
=
∞
∑
0
, = βk k N
k
lx x±
=
∞
∑
0
, , l ∈Z .
Poskol\ku L = H, poluçaem
αk k N
k
x ±
= − ∞
∞
∑ = βk k N
k
x ±
= − ∞
∞
∑ . (12)
PoloΩym
Ax = αk k N
k
x +
=
∞
∑
0
, Bx = αk k N
k
x −
=
∞
∑
0
, (13)
x L∈ , x = αk k
k
x
= − ∞
∞
∑ , αk ∈C . (14)
Sootnoßenyq (10) – (12) pokaz¥vagt, çto znaçenyq operatorov A y B na
vektore x yz L ne zavysqt ot v¥bora predstavlenyq πtoho vektora. Znaçyt,
operator¥ A y B korrektno zadan¥. V çastnosty, spravedlyv¥ ravenstva
Axk = xk N+ , Bxk = xk N− , k ∈Z .
Operator A qvlqetsq obratym¥m y A−1 = B . V¥berem proyzvol\n¥e πle-
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 4
476 S. M. ZAHORODNGK
ment¥ x y L, ∈ , x = αk kk
x
= − ∞
∞∑ , y = γ n nn
x
= − ∞
∞∑ , y zapyßem
( , )Ax y = α γk k N
k
n n
n
x x+
= − ∞
∞
= − ∞
∞
∑ ∑
, = α γk n k N n
k n
x x( , )
,
+
= − ∞
∞
∑ =
= α γk n k n N
k n
x x( , )
,
+
= − ∞
∞
∑ = α γk k
k
n n N
n
x x
= − ∞
∞
+
= − ∞
∞
∑ ∑
, = ( , )x Ay .
Takym obrazom, operator A qvlqetsq symmetryçeskym. Analohyçn¥m obrazom
proverqetsq symmetryçnost\ B.
Pust\
�A A⊇ — proyzvol\noe samosoprqΩennoe rasßyrenye operatora A v
hyl\bertovom prostranstve
�H H⊇ . Pry πtom moΩno sçytat\, çto Ker �A =
= { }0 . V protyvnom sluçae, poskol\ku Ker � �A R A⊥ ( ) , R A L( )� ⊇ , zaklgçaem,
çto Ker �A H⊥ . Poπtomu operator
�A , suΩenn¥j na
� �H A� Ker , takΩe budet
samosoprqΩenn¥m rasßyrenyem operatora A s nulev¥m qdrom.
Pust\ { }�Eλ λ∈R — neprer¥vnoe sleva ortohonal\noe razloΩenye edynyc¥
operatora
�A . V¥berem proyzvol\noe çyslo a ∈Z ,
a = rN + j, r ∈ Z , 0 ≤ j ≤ N – 1.
Yspol\zuq opredelenye operatora A, po yndukcyy lehko proveryt\, çto
xa = xrN j+ = A xr
j . (15)
Na osnovanyy (5), (7) y (15) zapys¥vaem
sr t j n+ ; , = γ rN j tN n+ +; = ( ),x xrN j tN n H+ + = ( ),A x A xr
j
t
n H = ( ),� � �A x A xr
j
t
n H ,
hde r t, ∈Z , 0 ≤ j , n N≤ − 1. Zdes\ m¥ vospol\zovalys\ tem, çto AL L⊆ y,
znaçyt,
�A Ar r⊇ , r ∈Z .
Yzvestno, çto dlq racyonal\n¥x funkcyj, nuly znamenatelq kotor¥x ne
leΩat v toçeçnom spektre operatora, neposredstvennoe zadanye πtyx funkcyj
ot operatora sohlasovano s zadanyem funkcyj ot operatora spektral\n¥m
yntehralom (sm. [11, c. 141]). Znaçyt, moΩno utverΩdat\, çto
�A xr
j = λ λ
r
jdE x�
R
∫ , r ∈Z .
(Zdes\ m¥ vospol\zovalys\ tem, çto λ = 0 ne qvlqetsq sobstvenn¥m çyslom
operatora
�A .) Znaçyt,
sr t j n+ ; , = λ λλ λ
r
j
t
n
H
dE x dE x� �
�R R
∫ ∫
, = λ λ
r t
j n Hd E x x+∫ ( ),� �
R
,
hde poslednee sootnoßenye sleduet yz [12, c. 222].
Sledovatel\no, poluçaem ravenstvo
sr t j n+ ; , = λ λ
r t
H
H
j n
H
d P E x x+ ( )∫ � � ,
R
.
Zapys¥vaq poslednee sootnoßenye v matryçnom vyde, pryxodym k ravenstvu
Sr t+ = λ λr t d M+∫ � ( )
R
, r t, ∈Z , (16)
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 4
O SYL|NOJ MATRYÇNOJ PROBLEME MOMENTOV HAMBURHERA 477
hde
�M ( )λ : = P E x xH
H
j n
H j n
N� �
λ ,
,
( )( )
=
−
0
1
. Polahaq v sootnoßenyy (16) t = 0, po-
luçaem, çto matryca-funkcyq
�M ( )λ qvlqetsq nekotor¥m reßenyem problem¥
momentov (1) (yz svojstv ortohonal\noho razloΩenyq edynyc¥ lehko sleduet,
çto
�M ( )λ qvlqetsq neprer¥vnoj sleva neub¥vagwej matrycej-funkcyej y
�M ( )−∞ = 0 ).
Takym obrazom, m¥ dokazaly sledugwug teoremu.
Teorema+2. Pust\ zadana syl\naq matryçnaq problema momentov Hambur-
hera (1) s nekotor¥m naborom momentov { }Sn n∈Z . Problema momentov yme-
et reßenye v tom y tol\ko v tom sluçae, kohda v¥polnen¥ uslovyq (3).
ProdolΩym teper\ yzuçenye problem¥ momentov (1), kak y ranee, predpola-
haq uslovyq (3) v¥polnenn¥my.
Pust\  — proyzvol\noe samosoprqΩennoe rasßyrenye postroennoho v¥ße
operatora A v nekotorom hyl\bertovom prostranstve H� . Oboznaçym çerez
R Az ( )�
rezol\ventu A� , a çerez { }E�λ λ∈R eho ortohonal\noe neprer¥vnoe sleva
razloΩenye edynyc¥. Napomnym, çto operatornoznaçnaq funkcyq Rz =
= P R AH
H
z
� �( ) naz¥vaetsq obobwennoj rezol\ventoj A, z ∈C R\ . Funkcyq
Eλ = P EH
H� �
λ , λ ∈R , qvlqetsq spektral\noj funkcyej symmetryçeskoho ope-
ratora A. MeΩdu obobwenn¥my rezol\ventamy y spektral\n¥my funkcyqmy
suwestvuet vzaymno odnoznaçnoe sootvetstvye sohlasno sootnoßenyg [9]
( ),Rz Hf g =
1
λ λ−∫ z
d f g H( ),E
R
, f g H, ∈ , z ∈C R\ . (17)
Formula (16) pokaz¥vaet, çto proyzvol\naq spektral\naq funkcyq operatora
A poroΩdaet reßenye problem¥ momentov (1) (spektral\naq funkcyq poroΩ-
daetsq nekotor¥m samosoprqΩenn¥m rasßyrenyem, kotoroe moΩno sçytat\ ob-
ratym¥m). PokaΩem, çto vse reßenyq problem¥ momentov poroΩdagtsq spekt-
ral\n¥my funkcyqmy podobn¥m obrazom.
Pust\ M x�( ) = ( ), ,( )m xk l k l
N�
=
−
0
1
qvlqetsq proyzvol\n¥m reßenyem problem¥
momentov (1). Rassmotrym prostranstvo L M2( )�
, y pust\ Q — operator umno-
Ωenyq na nezavysymug peremennug v L M2( )�
. Operator Q qvlqetsq samoso-
prqΩenn¥m y eho ortohonal\noe razloΩenye edynyc¥ ymeet vyd (sm. [8])
E Eb a− = E a b( )[ , ) : h x( ) → χ[ , ) ( ) ( )a b x h x , (18)
hde χ[ , ) ( )a b x — xarakterystyçeskaq funkcyq yntervala [ , )a b , – ∞ ≤ a < b ≤
≤ + ∞ . PoloΩym
�
ek = ( ), , ,, , ,e e ek k k N0 1 1… − , ek j, = δk j, , 0 1≤ ≤ −j N , dlq
k = 0, 1, … , N – 1,
MnoΩestvo (klassov πkvyvalentnosty) funkcyj f L M∈ 2( )�
takyx, çto (so-
otvetstvugwyj klass vklgçaet) f = ( ), , ,f f fN0 1 1… − , f L∈P , oboznaçaem çe-
rez PL M2( )�
y naz¥vaem mnoΩestvom vektorn¥x mnohoçlenov Lorana v L M2( )�
.
Polahaem L ML
2 ( )� = PL M2( )� .
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 4
478 S. M. ZAHORODNGK
Dlq proyzvol\noho vektornoho mnohoçlena Lorana f = ( ), , ,f f fN0 1 1… − ,
f j L∈P , suwestvuet edynstvennoe predstavlenye vyda
f x( ) =
k
N
k j
j
k
j
x e
=
−
= − ∞
∞
∑ ∑
0
1
α ,
�
, (19)
hde lyß\ koneçnoe çyslo koπffycyentov αk j, otlyçno ot nulq. ∏to budet
predpolahat\sq y v dal\nejßem v podobn¥x sluçaqx. Pust\ proyzvol\n¥j dru-
hoj vektorn¥j mnohoçlen Lorana g ymeet vyd
g x( ) =
l
N
l r
r
l
r
x e
=
−
= − ∞
∞
∑ ∑
0
1
β ,
�
. (20)
Tohda moΩno zapysat\
( ),
( )
f g
L M2 � =
k l
N
k j l r
j r
k
j r
lx e d M x e
,
, ,
,
( )
=
−
+
= − ∞
∞
∑ ∫∑
0
1
α β
R
� � �∗∗ =
=
k l
N
k j l r
j r
k l
j r
x dm x
,
, , ,
,
( )
=
−
+
= − ∞
∞
∑ ∫∑
0
1
α β
R
� =
k l
N
k j l r j r k l
j r
s
,
, , ; ,
,=
−
+
= − ∞
∞
∑ ∑
0
1
α β . (21)
S druhoj storon¥,
j
k j jN k
k
N
r
l r rN l
l
N
x x
= − ∞
∞
+
=
−
= − ∞
∞
+
=
−
∑ ∑ ∑α β, ,,
0
1
0
11
∑
H
=
k l
N
k j l r
j r,
, ,
,=
−
= − ∞
∞
∑ ∑
0
1
α β ∗
∗ ( ),x xjN k rN l H+ + =
k l
N
k j l r
j r
jN k rN l
,
, ,
,
;
=
−
= − ∞
∞
+ +∑ ∑
0
1
α β γ =
=
k l
N
k j l r
j r
j r k ls
,
, ,
,
; ,
=
−
= − ∞
∞
+∑ ∑
0
1
α β . (22)
Yz sootnoßenyj (21), (22) sleduet, çto
( ),
( )
f g
L M2 � =
j
k j jN k
k
N
r
l r rN l
l
N
x x
= − ∞
∞
+
=
−
= − ∞
∞
+
=
−
∑ ∑ ∑α β, ,,
0
1
0
11
∑
H
. (23)
PoloΩym
V f =
j
k j jN k
k
N
x
= − ∞
∞
+
=
−
∑ ∑ α ,
0
1
dlq f x ML( ) ( )∈P2 �
(y sootvetstvugwyj klass πkvyvalentnosty vklgçaet),
f x( ) =
k
N
k j
j
kj
x e
=
−
= − ∞
∞∑ ∑0
1 α ,
�
. Esly f, g qvlqgtsq vektorn¥my mnohoçle-
namy Lorana s predstavlenyqmy (19), (20) y pry πtom f g
L M
− 2 ( )� = 0, to yz
sootnoßenyq (23) sleduet, çto
V f V g
H
− 2
= ( )( ), ( )V f g V f g H− − =
= ( ),
( )
f g f g
L M
− − 2 � = f g
L M
− 2
2
( )� = 0.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 4
O SYL|NOJ MATRYÇNOJ PROBLEME MOMENTOV HAMBURHERA 479
Takym obrazom, V qvlqetsq korrektno zadann¥m operatorom yz P2( )M� v H.
Sootnoßenye (23) pokaz¥vaet, çto V qvlqetsq yzometryçeskym otobraΩe-
nyem PL M2( )�
na L . Rasprostranym po neprer¥vnosty V do yzometryçeskoho
otobraΩenyq L ML
2 ( )�
na H. V çastnosty, zametym, çto
V x ej
k
�
= x jN k+ , j ∈Z , 0 1≤ ≤ −k N .
Pust\ L M1
2( )� : = L M L ML
2 2( ) ( )� �� y U : = V E
L M
�
1
2 ( )� . Operator U qvlqetsq
yzometryçeskym otobraΩenyem L M2( )�
na H L M� 1
2( )� = : H� . PoloΩym
A� : = UQU −1.
Operator A� qvlqetsq samosoprqΩenn¥m operatorom v H� . Pust\ { }E�λ λ∈R —
eho neprer¥vnoe sleva ortohonal\noe razloΩenye edynyc¥. Zametym, çto
UQU x jN k
−
+
1 = VQV x jN k
−
+
1 = VQx ej
k
�
= V x ej
k
+1�
= x j N k( )+ +1 =
= x jN k N+ + = Ax jN k+ , j ∈Z , 0 1≤ ≤ −k N .
V sylu lynejnosty spravedlyvo
UQU x−1 = Ax , x L D A∈ = ( ) ,
y, znaçyt, A A� ⊇ . V¥berem proyzvol\n¥j πlement z ∈C R\ y zapyßem
1
λ
λ
− ( )∫ z
d E x xk j
H
� ,
ˆ
R
=
1
λ
λ
−
∫ z
d E x xk j
H
� ,
ˆR
=
=
1 1 1
2λ
λ
−
− −∫ z
dU E Ue U xk j
L M
� �
�
,
( )R
=
1
2λ λ−
∫ z
dE e ek j
L M
� �
�
,
( )R
=
=
1
2λ λ−∫ z
d E e ek j L M
( ),
( )
� �
�
R
, 0 ≤ k , j N≤ − 1.
Yspol\zuq (18), zapys¥vaem
( ),
( )
E e ek j L Mλ
� �
�2 = mk j� , ( )λ ,
y, znaçyt,
1
λ
λ
− ( )∫ z
d P E x xH
H
k j
H
� � ,
R
=
1
λ
λ
−∫ z
d mk j� , ( )
R
, 0 ≤ k , j N≤ − 1.
Yspol\zuq formulu obrawenyq Stylt\esa – Perrona (sm., naprymer, [13]), za-
klgçaem, çto
mk j� , ( )λ = P E x xH
H
k j
H
� �
λ ,( ) .
PokaΩem teper\, çto yndeks defekta A raven ( , )m n , 0 ≤ m , n N≤ . V¥berem
proyzvol\n¥j πlement u = c xk kk = − ∞
∞∑ ∈ L, ck ∈C , y çyslo z ∈C R\ . Pust\
ck = 0 pry k ≤ R− y pry k ≥ R+ , hde R− ≤ – 2, R+ ≥ N + 1. Polahaem po
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 4
480 S. M. ZAHORODNGK
opredelenyg dk = 0 pry k ≤ R− y pry k ≥ R N+ − . Polahaem dalee
dk =
1
z
d N ck k− −( ) , k = R− + 1, … , – 1,
dk N− = zd ck k+ , k = R+ − 1, R+ − 2 , … , N.
Pust\ v : = d xk kk = − ∞
∞∑ ∈ L . Tohda neposredstvenno poluçaem
( )A zE uH− −v = ( )d zd c xk N k k k
k
N
−
=
−
− −∑
0
1
,
u = u xz k k
k
N
+
=
−
∑ α
0
1
, u L∈ , αk ∈C , u Hz z∈ , (24)
hde Hz : = ( )A zE LH− = ( ) ( )A zE D AH− . V çastnosty, yz ravenstva (24) sle-
duet, çto
u = �u yz k k
k
N
+
=
−
∑ α
0
1
,
hde �uz = u P xz k H kk
N
z
+
=
−∑ α
0
1
, yk = x P xk H kz
− . Yz posledneho ravenstva sle-
duet, çto H = H yz k k
N� span { } =
−
0
1 . Sledovatel\no, defektn¥e çysla A ne
prev¥ßagt N.
Teorema+3. Pust\ zadana syl\naq matryçnaq problema momentov Hambur-
hera (1) y v¥polneno uslovye (3). Pust\ operator A postroen dlq problem¥
momentov, kak v (13). Vse reßenyq problem¥ momentov ymegt vyd
M( )λ = ( ), ,( )mk j k j
Nλ =
−
0
1 , mk j, ( )λ = ( ),Eλ x xk j H ,
hde Eλ qvlqetsq neprer¥vnoj sleva spektral\noj funkcyej operatora A .
Pry πtom sootvetstvye meΩdu vsemy neprer¥vn¥my sleva spektral\n¥my
funkcyqmy operatora A y vsemy reßenyqmy problem¥ momentov vzaymno
odnoznaçno.
Dokazatel\stvo. Vse utverΩdenyq teorem¥, krome posledneho, b¥ly
dokazan¥ v¥ße. PokaΩem, çto razlyçn¥e neprer¥vn¥e sleva spektral\n¥e
funkcyy operatora A poroΩdagt razlyçn¥e reßenyq problem¥ momentov (1).
PredpoloΩym, çto dve razlyçn¥e neprer¥vn¥e sleva spektral\n¥e funkcyy
poroΩdagt odno y to Ωe reßenye problem¥ momentov. ∏to znaçyt, çto est\ dva
samosoprqΩenn¥x operatora A Aj ⊇ v hyl\bertov¥x prostranstvax H Hj ⊇
takye, çto
P EH
H1
1,λ ≠ P EH
H2
2,λ ,
( ), ,P E x xH
H
k j H
1
1 λ = ( ), ,P E x xH
H
k j H
2
2 λ , 0 ≤ k, j ≤ N – 1, λ ∈R ,
hde { },En λ λ∈R — ortohonal\n¥e neprer¥vn¥e sleva razloΩenyq edynyc¥
operatorov An , n = 1, 2. PoloΩym LN : = Lin{ }xk k
N
=
−
0
1 . Yspol\zuq lynej-
nost\, zapys¥vaem
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 4
O SYL|NOJ MATRYÇNOJ PROBLEME MOMENTOV HAMBURHERA 481
( ), ,P E x yH
H
H
1
1 λ = ( ), ,P E x yH
H
H
2
2 λ , x y LN, ∈ , λ ∈R . (25)
Oboznaçym çerez Rn,λ rezol\ventu An y poloΩym Rn,λ : = P RH
H
n
n
,λ , n = 1, 2.
Yz (25), (21) sleduet, çto
( ), ,R1 λ x y H = ( ), ,R2 λ x y H , x y LN, ∈ , λ ∈C R\ . (26)
V¥berem proyzvol\noe çyslo z ∈C R\ y rassmotrym prostranstvo Hz , opre-
delennoe v¥ße. Poskol\ku
R A zE xj z H, ( )− = ( ) ( )A zE A zE xj H j Hj j
− −−1 = x , x L∈ = D A( ) ,
to
R uz1, = R uz2, ∈ H , u Hz∈ , (27)
R1,z u = R2,z u , u Hz∈ , z ∈C R\ . (28)
M¥ moΩem zapysat\
( ), ,Rn z Hx u = ( ), ,Rn z Hx u
n
= ( ), ,x R un z Hn
= ( ), ,x un z HR ,
hde x LN∈ , u Hz∈ , n = 1, 2, y, znaçyt,
( ), ,R1 z Hx u = ( ), ,R2 z Hx u , x LN∈ , u Hz∈ . (29)
Sohlasno (24) proyzvol\n¥j πlement y L∈ moΩno predstavyt\ v vyde y =
= y yz + ′ , y Hz z∈ , ′ ∈y LN . Yspol\zuq (26) y (29), poluçaem
( ), ,R1 z Hx u = ( ), ,R1 z z Hx y y+ ′ = ( ), ,R2 z z Hx y y+ ′ = ( ), ,R2 z Hx y ,
hde x LN∈ , y L∈ . Poskol\ku L = H, ymeem
R1,z x = R2,z x , x LN∈ , z ∈C R\ . (30)
Dlq proyzvol\noho x L∈ , x = x xz + ′ , x Hz z∈ , ′ ∈x LN , yspol\zuq sootno-
ßenyq (28), (30), zapys¥vaem
R1,z x = R1, ( )z zx x+ ′ = R2, ( )z zx x+ ′ = R2,z x , x L∈ , z ∈C R\ ,
y
R1,z x = R2,z x , x H∈ , z ∈C R\ .
Sohlasno (17) πto oznaçaet, çto sootvetstvugwye spektral\n¥e funkcyy sov-
padagt. Poluçennoe protyvoreçye zaverßaet dokazatel\stvo teorem¥.
Napomnym nekotor¥e opredelenyq yz [7], neobxodym¥e nam v dal\nejßem.
Pust\ B — zamknut¥j symmetryçeskyj operator v hyl\bertovom prostran-
stveCCH s oblast\g opredelenyq D B( ) , D B( ) = H . Oboznaçym ∆B( )λ =
= ( ) ( )B E D BH− λ y Nλ = N Bλ ( ) = H B� ∆ ( )λ , λ ∈C R\ . Rassmotrym pro-
yzvol\n¥j ohranyçenn¥j lynejn¥j operator C, otobraΩagwyj Ni v N i− .
Dlq
g = f C+ −ψ ψ , f D B∈ ( ) , ψ ∈Ni ,
polahaem
B gC = B f iC i+ +ψ ψ .
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 4
482 S. M. ZAHORODNGK
Poskol\ku pereseçenye D A( ) , Ni y N i− sostoyt lyß\ yz nulevoho πlementa,
πto opredelenye korrektno. Operator BC naz¥vagt kvazysamosoprqΩenn¥m
rasßyrenyem operatora B, opredelqem¥m operatorom C. Yspol\zuq funda-
mental\n¥j rezul\tat A. V. Ítrausa [7] (teoremaC7) y teoremuC3, pryxodym k
sledugwemu opysanyg reßenyj problem¥ momentov.
Teorema+4. Pust\ zadana syl\naq matryçnaq problema momentov Hambur-
hera (1) y v¥polneno uslovye (3). Pust\ operator A postroen dlq problem¥
momentov, kak v (13) . Vse reßenyq problem¥ momentov ymegt vyd
M x( ) = ( ), ,( )m xk j k j
N
=
−
0
1 ,
hde mk j, udovletvorqgt sootnoßenyg
1
x
dm xk j−∫ λ , ( )
R
= ( )( ) ,( )A E x xF H k j Hλ λ− −1 , λ ∈ +C , (31)
F( )λ qvlqetsq analytyçeskoj v C+ operatornoznaçnoj funkcyej, znaçenyqmy
kotoroj qvlqgtsq sΩatyq, otobraΩagwye N Ai ( ) v N Ai− ( ) F( )λ ≤( )1 ,
a AF( )λ — kvazysamosoprqΩenn¥m rasßyrenyem A , opredelqem¥m F( )λ .
S druhoj storon¥, proyzvol\noj operatornoznaçnoj funkcyy F( )λ s upo-
mqnut¥my svojstvamy sootvetstvuet, sohlasno (31), nekotoroe reßenye
syl\noj matryçnoj problem¥ momentov Hamburhera. Pry πtom sootvetstvye
meΩdu vsemy takymy operatornoznaçn¥my funkcyqmy y reßenyqmy problem¥
momentov, ustanavlyvaemoe sootnoßenyem (31), vzaymno odnoznaçno.
1. Jones W. B., Njåstad O., Thron W. J. Continued fractions and strong Hamburger moment problems
// Proc. London Math. Soc.. – 1983. – (3) 47, # 2. – P. 363 – 384.
2. Jones W. B., Thron W. J. Njåstad O. Orthogonal Laurent polynomials and the strong Hamburger
moment problem // J. Math. Anal. and Appl. – 1984. – 98, # 2. – P. 528 – 554.
3. Njåstad O. Solutions of the strong Hamburger moment problem // Ibid. – 1996. – 197. –
P. 227 – 248.
4. Jones W. B., Njåstad O. Orthogonal Laurent polynomials and strong moment theory: a survey // J.
Comput. Appl. Math. – 1999. – 105, # 1-2. – P. 51 – 91.
5. Simonov K. K. Strong matrix moment problem of Hamburger // Meth. Funct. Anal. and Top. –
2006. – 12, # 2. – P. 183 – 196.
6. Zagorodnyuk S. M. Positive definite kernels satisfying difference equations // Ibid. – 2010. – 16,
# 1. – P. 83 – 100.
7. Ítraus A. V. Obobwenn¥e rezol\vent¥ symmetryçeskyx operatorov // Yzv. AN SSSR. –
1954. – 18. – S.C51 – 86.
8. Malamud M. M., Malamud S. M. Operatorn¥e mer¥ v hyl\bertovom prostranstve // Alhebra
y analyz. – 2003. – 15, # 3. – S.C1 – 52.
9. Axyezer N. Y., Hlazman Y. M. Teoryq lynejn¥x operatorov v hyl\bertovom prostranstve. –
M.; L.: Hostexteoryzdat, 1950. – 484 s.
10. Berezanskyj G. M. RazloΩenye po sobstvenn¥m funkcyqm samosoprqΩenn¥x operatorov.
– Kyev: Nauk. dumka, 1965. – 800 s.
11. Byrman M. Í., Solomqk M. Z. Spektral\naq teoryq samosoprqΩenn¥x operatorov v
hyl\bertovom prostranstve. – L.: Yzd-vo Lenynhrad. un-ta, 1980. – 265 s.
12. Stone M. H. Linear transformations in Hilbert space and their applications to analysis. –
Providence, Rhode Island: Amer. Math. Soc. Colloq. Publ., 1932. – Vol. 15. – 622 p.
13. Axyezer N. Y. Klassyçeskaq problema momentov y nekotor¥e vopros¥ analyza, svqzann¥e s
neg. – M.: Fyzmathyz, 1961. – 312 s.
Poluçeno 07.10.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 4
|
| id | umjimathkievua-article-2881 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:32:06Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/31/4818d207150bca4b12ebfa2007889f31.pdf |
| spelling | umjimathkievua-article-28812020-03-18T19:39:35Z On the strong matrix Hamburger moment problem О сильной матричной проблеме моментов Гамбургера Zagorodnyuk, S. M. Загороднюк, С. М. Загороднюк, С. М. We obtain necessary and sufficient conditions for the solvability of the strong matrix Hamburger moment problem. We describe all solutions of the moment problem by using the fundamental results of A. V. Shtraus on generalized resolvents of symmetric operators. Отримано необхідні та достатні умови того, що сильна матрична проблема моментів Гамбургера має розв'язок. Описано всі розв'язки проблеми моментів. При цьому використано фундаментальні результати А. В. Штрауса про узагальнені резольвенти симетричних операторів. Institute of Mathematics, NAS of Ukraine 2010-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2881 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 4 (2010); 471–482 Український математичний журнал; Том 62 № 4 (2010); 471–482 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/2881/2513 https://umj.imath.kiev.ua/index.php/umj/article/view/2881/2514 Copyright (c) 2010 Zagorodnyuk S. M. |
| spellingShingle | Zagorodnyuk, S. M. Загороднюк, С. М. Загороднюк, С. М. On the strong matrix Hamburger moment problem |
| title | On the strong matrix Hamburger moment problem |
| title_alt | О сильной матричной проблеме моментов Гамбургера |
| title_full | On the strong matrix Hamburger moment problem |
| title_fullStr | On the strong matrix Hamburger moment problem |
| title_full_unstemmed | On the strong matrix Hamburger moment problem |
| title_short | On the strong matrix Hamburger moment problem |
| title_sort | on the strong matrix hamburger moment problem |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2881 |
| work_keys_str_mv | AT zagorodnyuksm onthestrongmatrixhamburgermomentproblem AT zagorodnûksm onthestrongmatrixhamburgermomentproblem AT zagorodnûksm onthestrongmatrixhamburgermomentproblem AT zagorodnyuksm osilʹnojmatričnojproblememomentovgamburgera AT zagorodnûksm osilʹnojmatričnojproblememomentovgamburgera AT zagorodnûksm osilʹnojmatričnojproblememomentovgamburgera |