On conditions under which the sum of self-adjoint operators with given integer spectra is a scalar operator
We describe the set ∑M1 ,...,Mn и of parameters γ for which there exists a decomposition of the operator γIH into a sum of n self-adjoint operators with the spectra belonging to the sets M1 ,...,Mn . The description of this set is performed for Mi = {0,1,...
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| Datum: | 2008 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2008
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/3169 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509213689643008 |
|---|---|
| author | Hrushevoi, R. V. Грушевой, Р. В. |
| author_facet | Hrushevoi, R. V. Грушевой, Р. В. |
| author_sort | Hrushevoi, R. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:47:27Z |
| description | We describe the set ∑M1 ,...,Mn и of parameters γ for which there exists a decomposition of
the operator γIH into a sum of n self-adjoint operators with the spectra belonging to the sets M1 ,...,Mn .
The description of this set is performed for Mi = {0,1, ...,ki } in the case of n ≥ 4 and in some cases for n = 3. |
| first_indexed | 2026-03-24T02:37:32Z |
| format | Article |
| fulltext |
UDK 517. 98
R. V. Hrußevoj (In-t matematyky NAN Ukra]ny, Ky]v)
KOLY SUMA SAMOSPRQÛENYX OPERATORIV
IZ ZADANYMY CILOÇYSEL|NYMY SPEKTRAMY
{ SKALQRNYM OPERATOROM
We describe the set M Mn1 , ,…∑ of parameters γ for which there exists a decomposition of the operator
γ IH into a sum of n self-adjoint operators with the spectra belonging to the sets M1, … , Mn . The
description of this set is performed for Mi = 0 1, , ,…{ }ki in the case of n ≥ 4 and in some cases for
n = 3.
Opysano mnoΩestvo M Mn1 , ,…∑ parametrov γ, dlq kotor¥x suwestvuet razloΩenye operatora
γ IH v summu n samosoprqΩenn¥x operatorov so spektramy yz mnoΩestv M1, … , Mn , dlq
Mi = 0 1, , ,…{ }ki v sluçae n ≥ 4 y nekotor¥e sluçay pry n = 3.
1. Vstup. Vyvçennq simej obmeΩenyx samosprqΩenyx operatoriv A
1
, A
2
, … , An
u hil\bertovomu prostori H iz spektramy σ( )A1 � M 1, σ( )A2 � M 2,1…
… , σ( )An � Mn
, wo pov’qzani linijnym spivvidnoßennqm Aii
n
=∑ 1
= γ I ,
de
γ ∈R , I — odynyçnyj operator v H , [ vaΩlyvog zadaçeg, wo vynyka[ u
zv’qzku z riznymy zadaçamy matematyky: deformovanymy preproektyvnymy
alhebramy [1], lokal\no skalqrnymy zobraΩennqmy hrafiv [2], problemog
Xorna ta ]] variaciqmy [3] ta in.
Vyqvylos\, wo skladnist\ opysu takyx simej operatoriv istotno zaleΩyt\
vid kil\kosti operatoriv A
1
, A
2
, … , An ta kil\kosti toçok u spektri σ( )Ai ope-
ratora Ai , i = 1, 2, … , n. Çasto navit\ opys mnoΩyny parametriv σ( )Ai , i = 1,
2, … , n, ta γ, pry qkyx vzahali isnu[ hil\bertiv prostir i nabir operatoriv u
n\omu z vidpovidnymy spektramy, wo pov’qzani navedenym vywe spivvidnoßen-
nqm, vyqvlq[t\sq dosyt\ skladnog zadaçeg.
Tak, qkwo Mi = 0 1,{ } , i = 1, 2, … , n, to jdet\sq pro vyvçennq simej orto-
proektoriv Pi, i = 1, 2, … , n, takyx, wo P1 + P2 + 1… + Pn = γ I . Taki sim’]
ortoproektoriv doslidΩeno v [4]; tam, zokrema, navedeno opys mnoΩyn
n∑ = ∃{γ hil\bertiv prostir H i nabir ortoproektoriv P1, P2, … , Pn
v n\omu takyx, wo P1 + P2 + 1… + Pn = γ I} .
Opysu mnoΩyn
M M n1, ,…∑ = ∃{γ hil\bertiv prostir H i nabir operatoriv A
1
, A
2
, … , An
v n\omu takyx, wo σ( )Ai � Mi, i = 1, 2, … , n, ta A
1
+ A2 + 1… + An = γ I}
v konkretnyx vypadkax prysvqçeno bahato robit (dyv., napryklad, [3, 5 – 9]).
U cij roboti navedeno opys mnoΩyny
M M n1, ,…∑ dlq Mi = 0 1, , ,…{ }ki .
Pry n = 2 opys [ oçevydnym M M1 2,∑( = M
1
+ M2) , pry n ≥ 4 joho otrymano v
povnomu obsqzi (p. 4), u vypadku n = 3 doslidΩeno deqki special\ni çastkovi vy-
padky (p. 3).
© R. V. HRUÍEVOJ, 2008
470 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
KOLY SUMA SAMOSPRQÛENYX OPERATORIV IZ ZADANYMY … 471
2. Elementarni vlastyvosti mnoΩyn
M M n1 , ,……∑∑ . KoΩnomu naboru mno-
Ωyn M
1
, … , Mn moΩna odnoznaçno postavyty u vidpovidnist\ zirçastyj hraf
Γ1= Tk k n1, ,… z n hilkamy po ki = Mi – 1 verßyn u i-j hilci.
Dali budemo pysaty k = ki∑ i mnoΩynu
M M n1, ,…∑ poznaçatymemo çerez
Γ∑ .
Korysnog [ nastupna lema (analohy qko] dovedeno v robotax [4, 5] ), wo opy-
su[ vlastyvosti Γ∑ .
Lema 1. MnoΩyny Γ∑ magt\ taki vlastyvosti:
1a) Γ∑ � 0, k[ ];
1b) Γ∑ � 0 1 2 3, , , , ,…{ }k ;
2) Γ∑ symetryçna vidnosno k
2
, tobto γ ∈∑Γ ⇒ k – γ ∈∑Γ ;
3) Γ∑ ∩ 0 2,[ ) =
n∑ ∩ 0 2,[ );
4) qkwo Γ1 — zv’qznyj pidhraf Γ2 z tym Ωe korenem, to Γ1
∑ �
� Γ2
∑ .
Dovedennq. Vykorystovugçy spektral\nyj rozklad operatoriv A
1
, A
2
, …
… , An
, dlq koΩnoho γ z Γ∑ otrymu[mo
jP Ii j
j
k
i
n j
==
∑∑ =
11
γ , (1)
de Pi j — ortoproektory v hil\bertovomu prostori H.
1a. Oskil\ky v livij çastyni (1) sto]t\ nevid’[mnyj operator, to γ ≥ 0 . Vid-
nqvßy v (1) vid obox çastyn operator k I, otryma[mo ( – )jP k Ii j ij
k
i
n j
== ∑∑ 11
=
= ( – )γ k I , de livoruç mistyt\sq nedodatnyj operator, tomu γ ≤ k .
1b. Rozhlqdagçy H = C, a operatory Ai = li , li ∈ 0 1 2, , , ,…{ }ki , moΩna
otrymaty γ = l dlq vsix l = 0, 1, 2, … , k.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
472 R. V. HRUÍEVOJ
2. Nexaj γ ∈∑Γ , todi, za vyznaçennqm, isnugt\ hil\bertiv prostir H i na-
bir operatoriv Ai , σ( )Ai � 0 1 2, , , ,…{ }ki i Aii
n
=∑ 1
= γ I . Todi v tomu Ω pros-
tori dlq operatoriv Âi = k Ii – Ai ma[mo Âii
n
=∑ 1
= ( – )k Iγ , pryçomu σ( ˆ )Ai =
=1 σ( )Ai , zvidky k – γ ∈∑Γ .
3. Prypustymo, wo γ ∈[ )0 2, i odyn iz ortoproektoriv Pml , l ≥ 2, ne doriv-
ng[ nulg, todi z (1) otrymu[mo jP lPi j mlj
k
i
n j –== ∑∑ 11
= γ I – lPml . Tut u livij
çastyni rivnosti mistyt\sq nevid’[mnyj operator, a u pravij — ni. OtΩe, taka
sytuaciq nemoΩlyva.
4. Te, wo Γ1 — pidhraf Γ2 , oznaça[, wo u vyrazi (1), wo vidpovida[ Γ1, „ne
vystaça[” deqkyx dodankiv, qki [ v analohiçnomu vyrazi dlq Γ2 . Ale poklavßy
vidpovidni ortoproektory rivnymy nulg, otryma[mo identyçni vyrazy, a otΩe, i
potribne vklgçennq.
Lemu dovedeno.
3. Opys mnoΩyn ΓΓ∑∑ u vypadku tr\ox operatoriv. U vypadku tr\ox
operatoriv zadaça opysu mnoΩyny Γ∑ vyqvylas\ najvaΩçog. Tut [ try pryn-
cypovo rizni vypadky, qki zruçno opysuvaty v terminax hrafiv:
Γ — odna z diahram Dynkina
Γ — odna z çotyr\ox evklidovyx diahram
Γ ne [ Ωodnym iz vkazanyx vywe hrafiv.
Rozhlqnemo ci vypadky okremo.
3.1. ΓΓΓΓ — diahrama Dynkina. Opys mnoΩyny Γ∑ , qkwo Γ — diahrama
Dynkina, v inßyx terminax otrymano v roboti [3]. Dlq povnoty roboty navedemo
cej opys.
TverdΩennq 1. Magt\ misce nastupni rivnosti mnoΩyn:
= …{ }∑An
n n0 1 2 1, , , – , – ,
= { }∑D4
0 1 3
2
2 3, , , , , = { }∑D5
0 1 3
2
2 5
2
3 4, , , , , , ,
= …{ }∑Dn
n
n n0 1 3
2
2 5
2
3
2 5
2
2 1, , , , , , ,
–
, – , – , n ≥ 5,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
KOLY SUMA SAMOSPRQÛENYX OPERATORIV IZ ZADANYMY … 473
= { }∑E6
0 1 3
2
2 7
3
5
2
8
3
3 7
2
4 5, , , , , , , , , , ,
= { }∑E 7
0 1 3
2
2 7
3
5
2
8
3
3 10
3
7
2
11
3
4 9
2
5 6, , , , , , , , , , , , , , ,
= { }∑E8
0 1 3
2
2 7
3
5
2
8
3
3 10
3
7
2
11
3
4 13
3
9
2
14
3
5 11
2
6 7, , , , , , , , , , , , , , , , , , .
3.2. ΓΓΓΓ — evklidova diahrama. V roboti [6] navedeno neqvnyj vyhlqd mno-
Ωyn Γ∑ (toçniße, bil\ß zahal\nyx mnoΩyn: operatory ne obov’qzkovo magt\
taki spektry, qki rozhlqdagt\sq v cij roboti). Ale v koΩnomu z vypadkiv dlq
toho, wob qvno vypysaty ci mnoΩyny, potribno provodyty dodatkovi obçyslen-
nq, abo analiz (qk v roboti [9] ). Vykonavßy ci obçyslennq, otryma[mo take
tverdΩennq.
TverdΩennq 2. Magt\ misce rivnosti
= ± ± ≥{ } { }∑ ˜ ,
D n n
n
4
2 1 2 2 1 2∪ ,
= ± ± ± ≥{ } { }∑ ˜ , ,
E n n n
n
6
3
1
3
2
3
3
1 3∪ ,
= { }∑ ˜ , , , , , , , , , , , , , , , , , , , , , ,
E8
0 1 3
2
2 7
3
5
2
8
3
3 10
3
7
2
11
3
4 13
3
9
2
14
3
5 16
3
11
2
17
3
6 19
2
7 8 .
Perelik toçok mnoΩyny
Ẽ 7
∑ potrebu[ bahato hromizdkyx obçyslen\, tomu
my joho ne navodymo. Ale zaznaçymo, wo mnoΩyna
Ẽ 7
∑ [ neskinçennog i ma[
[dynu hranyçnu toçku
7
2
(dyv. [9]).
ZauvaΩennq 1. Opys mnoΩyn
D̃4
∑ ,
Ẽ6
∑ ta vyhlqd vidpovidnyx opera-
toriv navedeno v roboti [7].
3.3. ΓΓΓΓ mistyt\ qk pidhraf Ẽ8, ale ne mistyt\ inßyx evklidovyx dia-
hram. Qkwo Γ ne [ ni diahramog Dynkina, ni evklidovog diahramog, to pytan-
nq opysu mnoΩyny Γ∑ zalyßa[t\sq vidkrytym, i lyße çastkovi vypadky
vdalos\ doslidyty, a same vypadky, koly hrafy mistqt\ qk pidhraf diahramu
Dynkina Ẽ8, ale ne mistqt\ Ωodnoho z hrafiv D̃4 , Ẽ6 , Ẽ7 . Ci hrafy magt\
vyhlqd
Z nymy pov’qzano sim’] operatoriv A, B, Q taki, wo σ( )A � {0, 1, 2, … , k},
σ( )B � {0, 1, 2}, a Q — ortoproektor. Opys mnoΩyn
T k2 3 1, , +
∑ da[ nastupna
teorema.
Teorema 1. Dlq k ≥ 5 vykonu[t\sq
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
474 R. V. HRUÍEVOJ
T k2 3 1, , +
∑ =
1 2 3
2
3 4 2 3, , , , , ,… +{ } = … +{ }k i i k∪ ∪
∪ j
j k
3
7 8 3 2= … +{ }, , , . (2)
Dovedennq provedemo indukci[g po k.
Baza indukci] k = 5: v c\omu vypadku otrymu[mo diahramu Dynkina Ẽ8, dlq
qko] tverdΩennq teoremy vykonu[t\sq zhidno z opysom mnoΩyny
Ẽ8
∑ , nave-
denym u tverdΩenni 2.
Krok indukci]. Dlq dovil\noho γ ∈
+
∑T k2 3 1, ,
∩ 0
3
2
,
k +
, qk i v dovedenni
p.13 lemy11, otrymu[mo, wo u vyrazi Pki
k
=∑ 1
+ B + Q = γ I ortoproektory Pi = 0
dlq i >
k + 3
2
. Takym çynom, vraxovugçy, wo
k + 3
2
≤ k – 1 dlq k ≥ 6, ma[mo
T k2 3 1, , +
∑ ∩ 0
3
2
,
k +
�
T k2 3, ,
∑ ∩ 0
3
2
,
k +
. Z p. 4 lemy otrymu[mo obernene
vklgçennq. Zhidno z p. 3 ti[] Ω lemy mnoΩyna
T k2 3 1, , +
∑ [ symetryçnog vid-
nosno
k + 3
2
. OtΩe, za prypuwennqm indukci] otrymu[mo potribnu rivnist\, wo i
zaverßu[ dovedennq teoremy.
4. Opys mnoΩyn ΓΓ∑∑ u vypadku n operatoriv, n ≥≥≥≥ 4. U robotax [4, 5]
bulo rozhlqnuto nabory proektoriv, asocijovani z hrafamy K n1, ta K n2, vidpo-
vidno, qki magt\ takyj vyhlqd:
Opys mnoΩyny Γ∑ , koly Γ = Tk k n1, ,… , n ≥ 4, tobto Γ mistyt\ D̃4 qk
pidhraf, da[ nastupna teorema.
Teorema 2. Nexaj Γ = Tk k n1, ,… , n ≥ 4. Todi
Γ∑ = Λ Λn n
( ) ( )0 1∪ ∪
n n n
k
n n n– –
, –
– –2 24
2
4
2
∪ k n– ( )Λ 1( ) ∪ k n– ( )Λ 0( ),
(3)
de
Λ n
( )0 =
0 1 1
1
1 1
2 1
1
1 1
2 1
2 1
1
1
,
–
,
( – ) –
–
, ,
( – ) –
( – ) –
–
–
,+ + … + …
n n
n
n
n
n
�
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
KOLY SUMA SAMOSPRQÛENYX OPERATORIV IZ ZADANYMY … 475
Λ n
( )1 =
1 1 1
2
1 1
2 1
2
1 1
2 1
2 1
1
2
,
–
,
( – ) –
–
, ,
( – ) –
( – ) –
–
–
,+ + … + …
n n
n
n
n
n
�
.
Dovedennq. V roboti [4] dovedeno, wo
K n1,
∑ = Λ Λn n n( ) ( )0 1∪ ∪
n n n n n n– –
,
–2 24
2
4
2
+
∪ n n– ( )Λ 1( ) ∪ n n– ( )Λ 0( ).
Lehko pereviryty, wo pry n ≥ 4 vykonu[t\sq nerivnist\
n n n– –2 4
2
≤ 2, pry-
çomu pry n > 4 nerivnist\ [ strohog.
Zvidsy, vraxovugçy p. 3 lemy11, otrymu[mo Γ∑ ∩ 0 2,[ ) = Λ n
( )0 ∪ Λ n
( )1 ∪
∪
n n n– –
,
2 4
2
2 . Z vykorystannqm pp. 1 ta 2 lemy11 dosyt\ dovesty, wo
Γ∑ mistyt\ vidrizok 2
2
, m
. Dovedemo ce indukci[g za kil\kistg verßyn
hrafa.
Baza indukci] — krytyçni hrafy K1 5, ta T2 2 2 3, , , . Dlq perßoho vypadku v
roboti [4] pobudovano nabory z 5 ortoproektoriv, suma qkyx dorivng[ γ I dlq
koΩnoho γ ∈[ ]2 3, . Dlq T2 2 2 3, , , , wo ma[ vyhlqd
teΩ, qk v roboti [5], dlq koΩnoho γ ∈[ ]2 3, pobudu[mo 5 ortoproektoriv u pros-
tori l2 tak, wo P1 + P2 + P3 + P4 + 2Q4 = γ I , pryçomu P4 ⊥ Q4.
Rozhlqnemo spoçatku sumu dvox ortoproektoriv:
P1 =
τ τ τ
τ τ τ
1 1 1
1 1 1
1
1 1
( – )
( – ) –
, P2 =
1 0
0 0
.
}xn\og sumog bude samosprqΩenyj operator zi spektrom {x, 2 – x}, de x ∈
∈ 0 2,[ ] (qkwo τ1 = ( – )x 1 2
). Tomu operator
x
x
0
0 2 –
[ sumog dvox orto-
proektoriv.
Poklademo teper
P3 =
τ τ τ
τ τ τ
2 2 2
2 2 2
1
1 1
( – )
( – ) –
, P4 =
0 0
0 1
, Q4 =
1 0
0 0
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
476 R. V. HRUÍEVOJ
SamosprqΩenyj operator P3 + P4 + 2Q4 ma[ spektr {y, 4 – y}, de y ∈[ ]1 3, ,
pry τ2 = ( – )y 2 2
. Takym çynom, operator
y
y
0
0 4 –
[ sumog ortoproektoriv
P3, P4 ta Q4, pryçomu P4 ta Q4 ortohonal\ni.
Pobudu[mo 5 poslidovnostej ortoproektoriv P i
1
( )
, P i
2
( )
, P i
3
( )
, P i
4
( )
ta Q i
4
( )
z umovog P i
4
( )
⊥ Q i
4
( )
, i = 1, 2, 3,1… , tak, wob na koΩnomu kroci otrymuvaty
P i
1
( )
+ P i
2
( ) =
x
x
i
i
0
0 2 –
,
P i
3
( )
+ P i
4
( )
+ 2 Q i
4
( ) =
y
y
i
i
0
0 4 –
, abo P i
3
( )
+ P i
4
( )
+ 2 Q i
4
( ) =
y
y
i
i
0
0 2 –
.
Ostanng rivnist\ otryma[mo, poklavßy Q i
4
( )
= 0. Pry c\omu vymahatymemo, wob
poslidovnosti z nevid’[mnyx dijsnyx çysel xi, yi zadovol\nqly spivvidno-
ßennq
y1 = γ , y i2 =
4 4 3
2 2 4
2 1 2 1
2 1 2 1
– , – ,
– , – – ,
– –
– –
y y
y y
i i
i i
γ
γ γ
≤ ≤
≤ ≤
y i2 1+ = γ – x i2 ,
(4)
x i2 1– = γ – y i2 , x i2 = 2 2 1– –x i .
Pry takij pobudovi oçevydno, wo xi –1 + yi = γ (x
0
= 0). Zalyßylos\ dovesty
korektnist\ takoho zadannq, tobto pokazaty, wo taki poslidovnosti moΩna po-
buduvaty z vykorystannqm konstrukci], navedeno] vywe. Dlq c\oho dostatn\o
dovesty, wo xi ∈[ ]0 2, , a yi ∈[ ]γ γ– ,2 = γ γ– , –2 4[ ) ∪ 4 – ,γ γ[ ], i = 1, 2,1… .
ZauvaΩymo, wo x i2 = 2 – x i2 1– ∈ 0 2,[ ] ekvivalentno x i2 1 0 2– ,∈[ ], i ≥ 1, a
ce vklgçennq vykonu[t\sq, z uraxuvannqm (4), todi i til\ky todi, koly y i2 ∈
∈ γ γ– ,2[ ].
PokaΩemo, wo z toho, wo y i2 1– ∈ γ γ– ,2[ ], vyplyva[, wo y i2 ∈ γ γ– ,2[ ]:
y i2 1 2– – ,∈[ ]γ γ ⇒
y
y
i
i
2 1
2 1
2 4
4
–
–
– , – ,
– , .
∈[ )
∈[ ]
γ γ
γ γ
Qkwo y i2 1– ∈ γ γ– , –2 4[ ), to y i2 = 2 – y i2 1– ∈ γ γ– , –2 4( ] � γ γ– ,2[ ]. Qk-
wo Ω y i2 1– ∈ 4 – ,γ γ[ ], to y i2 = 4 – y i2 1– ∈ 1, γ[ ] � γ γ– ,2[ ]. Oskil\ky y1 = γ,
to za indukci[g otrymu[mo, wo xi ∈[ ]0 2, , a yi ∈[ ]γ γ– ,2 , i = 1, 2,1… .
Teper vyznaçymo ortoproektory Pk , k = 1, 2, 3, 4 , ta Q4 u prostori l2 za
pravylom
Pk =
( ) ( )0
1
⊕
=
∞
�
i
k
iP , k = 1, 2,
Pk =
�
i
k
iP
=
∞
1
( ), k = 3, 4, Q4 =
�
i
iQ
=
∞
1
4
( ) .
Za pobudovog zi spivvidnoßen\ (4) ma[mo P1 + P2 + P3 + P4 + 2Q4 = γ I , wo i
potribno bulo pokazaty.
Krok indukci]. Pry zbil\ßenni kil\kosti verßyn hrafa Γ moΩlyvi dva vy-
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
KOLY SUMA SAMOSPRQÛENYX OPERATORIV IZ ZADANYMY … 477
padky:
1. Doda[t\sq hilka odynyçno] dovΩyny. Ce oznaça[, wo v rivnosti (1) doda-
[t\sq odyn ortoproektor.
2. ProdovΩu[t\sq odna z hilok, a ce oznaça[, wo odyn z operatoriv Ai zami-
ng[t\sq operatorom Âi takym, wo σ Âi( ) � {0, 1, 2, … , mi , mi+ 1}.
V obox vypadkax k zaming[t\sq na k + 1.
PokaΩemo, wo v prypuwenni indukci] Γ∑( � 2
2
, k
dlq hrafa Γ̂ , otry-
manoho z Γ odnym iz vkazanyx vywe sposobiv, Γ̂∑ � 2
1
2
,
k +
.
U perßomu vypadku, obyragçy „novyj” proektor rivnym nulg, abo odynyç-
nym operatorom, otrymu[mo Γ̂∑ � Γ∑ + 0 1,{ } � 2
2
, m
∪ 3
2
1, m +
�
� 2
1
2
,
m +
.
Rozhlqnemo vypadok, koly hraf Γ̂ otrymano z Γ podovΩennqm l-] hilky.
Nexaj γ ∈∑ Γ , tobto isnugt\ hil\bertiv prostir H ta n samosprqΩenyx
operatoriv Ai taki, wo σ( )Ai � {0, 1, 2, … , ki } i Aii
n
=∑ 1
= γ I . Porqd z Al
rozhlqnemo samosprqΩenyj operator B = Al + I zi spektrom σ( )B = {1, 2, …
… , kl + 1}. Todi
A Bi
i
i l
n
+
=
≠
∑
1
= ( )γ + 1 I ⇒ γ + ∈∑1 Γ̂ .
Zvidsy, qk i vywe, ma[mo Γ̂∑ � Γ∑ + 0 1,{ } � 2
1
2
,
m +
, wo i zaverßu[ dove-
dennq.
Avtor vyslovlg[ wyru podqku profesoru G. S. Samojlenku za postanovku
zadaçi ta korysni porady wodo zmistu statti.
1. Crawley-Boevey W., Holland M. P. Noncommutative deformations of Kleinian sigulatities // Duke
Math. J. – 1998. – 92, # 3. – P. 605 – 635.
2. Kruglyak S. A., Roiter A. V. Locally scalar graph representations in the category of Hilbert spaces
// Funct. Anal. and Appl. – 2005. – 39, # 2. – P. 91 – 105.
3. Kruglyak S. A., Popovich S. V., Samoilenko Yu. S. The spectral problem and ∗-representations of
algebras associated with Dynkin graphs // J. Algebra and Appl. – 2005. – 4, # 6. – P. 761 – 776.
4. Kruhlqk S. A., Rabanovyç V. Y., Samojlenko, G. S. O summax proektorov // Funkcyon. ana-
lyz y eho pryl. – 2002. – 36, # 3. – S. 20 – 25.
5. Mellit A. S., Rabanovich V. I., Samoilenko Yu. S. When is a sum of partial reflections equal to sca-
lar operator // Funct. Anal. and Appl. – 2004. – 38, # 2. – P. 157 – 160.
6. Kruglyak S. A., Popovich S. V., Samoilenko Yu. S. The spectral problem and ∗-representations of
algebras associated with Dynkin graphs. – Getteborg, 2007. – Preprint.
7. Ostrovs\kyj V. L., Samojlenko G. S. Pro spektral\ni teoremy dlq simej linijno pov’qza-
nyx samosprqΩenyx operatoriv iz zadanymy spektramy, wo asocijovani z rozßyrenymy hra-
famy Dynkina // Ukr. mat. Ωurn. – 2006. – 58, # 11. – S. 1556 – 1570.
8. Albeverio S., Ostrovsky V., Samoilenko Yu. On functions on graphs and representations of a certain
class of ∗-algebras // J. Algebra. –2006. – 308, # 2. – P. 567 – 582.
9. Yusenko K. A. On existence of ∗-representations of certain algebras related to extendend Dynkin
graphs // Meth. Funct. Anal. and Top. – 2006. – 12, # 2. – P. 197 – 204.
OderΩano 24.09.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 4
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| id | umjimathkievua-article-3169 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:37:32Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/e9/ce0fd713712e51c6063609e59066a8e9.pdf |
| spelling | umjimathkievua-article-31692020-03-18T19:47:27Z On conditions under which the sum of self-adjoint operators with given integer spectra is a scalar operator Коли сума самоспряжених операторів із заданими цілочисельними спектрами є скалярним оператором Hrushevoi, R. V. Грушевой, Р. В. We describe the set ∑M1 ,...,Mn и of parameters γ for which there exists a decomposition of the operator γIH into a sum of n self-adjoint operators with the spectra belonging to the sets M1 ,...,Mn . The description of this set is performed for Mi = {0,1, ...,ki } in the case of n ≥ 4 and in some cases for n = 3. Описано множество ∑M1 ,...,Mn и параметров γ, для которых существует разложение оператора γIH в сумму n самосопряженных операторов со спектрами из множеств M1 ,...,Mn , для Mi = {0,1, ...,ki } в случае n ≥ 4 и некоторые случаи при n = 3. Institute of Mathematics, NAS of Ukraine 2008-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3169 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 4 (2008); 470–477 Український математичний журнал; Том 60 № 4 (2008); 470–477 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3169/3085 https://umj.imath.kiev.ua/index.php/umj/article/view/3169/3086 Copyright (c) 2008 Hrushevoi R. V. |
| spellingShingle | Hrushevoi, R. V. Грушевой, Р. В. On conditions under which the sum of self-adjoint operators with given integer spectra is a scalar operator |
| title | On conditions under which the sum of self-adjoint operators with given integer spectra is a scalar operator |
| title_alt | Коли сума самоспряжених операторів із заданими цілочисельними спектрами є скалярним оператором |
| title_full | On conditions under which the sum of self-adjoint operators with given integer spectra is a scalar operator |
| title_fullStr | On conditions under which the sum of self-adjoint operators with given integer spectra is a scalar operator |
| title_full_unstemmed | On conditions under which the sum of self-adjoint operators with given integer spectra is a scalar operator |
| title_short | On conditions under which the sum of self-adjoint operators with given integer spectra is a scalar operator |
| title_sort | on conditions under which the sum of self-adjoint operators with given integer spectra is a scalar operator |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3169 |
| work_keys_str_mv | AT hrushevoirv onconditionsunderwhichthesumofselfadjointoperatorswithgivenintegerspectraisascalaroperator AT gruševojrv onconditionsunderwhichthesumofselfadjointoperatorswithgivenintegerspectraisascalaroperator AT hrushevoirv kolisumasamosprâženihoperatorívízzadanimicíločiselʹnimispektramiêskalârnimoperatorom AT gruševojrv kolisumasamosprâženihoperatorívízzadanimicíločiselʹnimispektramiêskalârnimoperatorom |