On pairs of unbounded self-adjoint operators satisfying an algebraic relation
Unbounded pairs of self-adjoint operators A andB satisfying the algebraic relation F 1(A)B=BF 2(A) are studied. For these relations, various definitions of “integrable” pairs of operators are presented and the class of “tame” relations is indicated; for the “tame” relations, the irreducible pairs ar...
Saved in:
| Date: | 1993 |
|---|---|
| Main Authors: | , , , |
| Format: | Article |
| Language: | Russian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
1993
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/5927 |
| 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_ | 1860512148188299264 |
|---|---|
| author | Ostrovskii, V. L. Samoilenko, Yu. S. Островский, В. Л. Самойленко, Ю. С. Островский, В. Л. Самойленко, Ю. С. |
| author_facet | Ostrovskii, V. L. Samoilenko, Yu. S. Островский, В. Л. Самойленко, Ю. С. Островский, В. Л. Самойленко, Ю. С. |
| author_sort | Ostrovskii, V. L. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-19T09:21:13Z |
| description | Unbounded pairs of self-adjoint operators A andB satisfying the algebraic relation F 1(A)B=BF 2(A) are studied. For these relations, various definitions of “integrable” pairs of operators are presented and the class of “tame” relations is indicated; for the “tame” relations, the irreducible pairs are described and a structure theorem is presented. |
| first_indexed | 2026-03-24T03:24:10Z |
| format | Article |
| fulltext |
YJ].K517.986
B. JI. OttpOBCKHll, KaH.Q. q>H3.-MaT. Hayi<.,
IO. C. Car.t01lJ1eHKO, i.-p _q>H3.-MaT. HayK (HH-T MaTCMaTHKH AH YKpaHHbl, KHeB)
0 IIAPAX HEOrPAHlfqEHHbIX
CAMOCOilP1DKEHHbIX OilEPATOPOB,
CBR3AHHbIX AJifEJiPAlfqECKHM COOTHOIIlEHHEM
Unbounded pairs of self-adjoint operators A, B, which satisfy the algebraic relation F1 (A)B = BF2(A)
are studied. For such relations, the various definitions of "integrable" pairs of operators are given and
the class of "tame" relations is indicated; for these relations, the irreducible pairs are described and the
structure theorem is presented.
BHB'llllOTbC.11 napH HCOOMeJKeHHX caMocnp.llJKCHHX onepaTOpiB A, B, nos' Jl3aHHX anre6pai'!HHM cniB
Bi.QHOWCHHJIM BHrJJ.11.QY F1(A)B = BF2(.A). J].n.11 TaKHX cnissi.QHOWCHb HaBe.QeHi pi3HOMaHiTHi sapiaHTH
BH3Ha'!eHH.II .,iHTerpoBHHX" nap oneparopis, BH.QineHo Knac .. py'IHHX" cniBBi.QHOWeHb, .QR.II JIKHX
onHCaHO HC3Bi.QHi napH Ta HaBe,QeHO CTpYKTypHY TeopeMy.
B pa6orax [1, 2) H3Y'IMHCb napbl caMoconp10KeHHblX, soo6me rosop11, HeorpaHH
'leHHblX oneparopos A, B B rHJib6epTOBOM npocrpaHCTse H, CB113aHHbIX COOTHOwe-
AB = BF(A), (1)
r ,ne F: 1R l ➔ 1R I - H3MepHMa11 cpyHKQH11. nn11 TaKHX COOTHOllleHHA 6b1JIH OnHCa
Hhl HenpHBOAHMhle napbl H npuse,neHa crpyKTypHM reopeMa, ,na10ma11 pa3JIO)KeHHe
Ilp<)H3B0JlbHOA napbl A, B, YAOBJieTBOp1110meA (1), Ha HenpHBOAHMhle.
B Hacro11meA pa6ore paccMorpeHbl napbl, soo6me rosop11, HeorpaHH'leHHblX
CaMOCOilp11)KeHHblX oneparopos A, B, y,nosneTBOp1110IUHX 6onee o6meMY COOTHO
weHHIO
(2)
r .ne F 1, F 2: 1R 1 ➔ 1R 1 - H3MepHMbie <PYHKQHH. B n. 1 H3Y'la10TC11 sapuaHThl pa3-
JIH'IHblX onpe,neneHHA COOTHoweHHH (2) .nn11 HeorpaHH'leHHblX oneparopos. 3a,naqa
OilHCaHHH HenpHBOAHMblX nap BHAa (2) s o6meM cnyqae He no.n.naeTCH peweHHIO
(,,AHKa11"). B n. 2 npuse,neHbl Heo6XOAHMble H AOCTaTO'IHble ycJIOBHH Toro, 'ITO napa
HeorpaHH'leHHblX caMoconp11)KeHHblX oneparopos A, B, YAOBJieTBOp11IOIUHX (2) -
,,py'IHa11", AJIH ,,py'IHbIX" nap onHCaHbl HenpHBOAHMble npe,ncrasneHH11 H npuse,neHa
CTPYKTypHa11 TeOpeMa, a B n. 3 - pa3JlH'IHbie npHMepbl COOTHOllleHHA BH,na (2). B n .
4 BKpaTQe paCCMOTpeHbl o6o6meHHH npuse,neHHblX noCTpoeHHA Ha cnyqa0 ceMe8CTB
caMOCOnp11)KeHHblX onepaTopos, CB113aHHbIX COOTtIOllleHH11MH BH,C{a (2).
I. TiocKOJibKY MbI He npe.nnonaraeM orpaHH'leHHOCTH oneparopos A, B , ,nn11 ycr
pattemu pa3HO'ITe1mA npH nOHHMaHHH COOTHOllleHHA (2) Heo6XOAHMO Bbl,C{eJIHTb
Knacc "HHTerpupyeMhlx" nap caMoconp11)KeHHbIX onepaTopos, y,nosnersop1110mux
(2). TipH 3TOM ecreCTBeHHO npe,nnOJIO)KHTb, 'ITO orpaHH'leHHbie napbl HBJI1IIOTC11
"HHTerpupyeMblMH".
TeopeMa 1. JJ/1.R ozpaH11'1eHHblX ca1't0conp.1VKe1111blX onepamopoe A. B. F 1(A),
F 2(A) C11eoy10u~ue yc/1081111 3K8U8G/lef111lHbl:
1) F 1(A)B =BF2(A);
2) EA(F1- 1(~))sintB=sintBEA(F21(M) Vte ]RI, \;/~e .l3(1RI);
3) f(F1(A))g(B) = g(B)f(F2(A)) OIi.Ji 11106blX ozpaH11'1efl/1blX U3MepuMblX f. g.
npu'leM g 11peiJ11011azaemc11 He'lenmoi1.
(3.nech H JJ,a.nee EA(- )- pa3JlO)KeHHe e,nHHHQbI onepaTopa A.)
aoKa.1ame✓lbC11l60. 1) => 3). lfa ycnOBH.ll 1 no COnp11)KeHHIO HMeeM TaK.>Ke
F2(A)B = BF 1(A), OTKyJJ.a
© B. !]_. OCl"pOBCKHH, 10. C. ~ AMOfiJlEHKO, 1993
1253 ISSN 004/ -6053. Y ,;p, MO/II . )l(Jplt., / 993 , Ill . ./5 , N • 9
1254 B. JI. OCTPOBCKHtl, 10. C. CAM0tlJ1EHKO
H .l.VIJf JIIo6oro Helfenmro U0JIHH0Ma Q(· )
F 1(A)Q(B) = Q(B)F2(A).
AnaJI0rHlfHO f.\JUI JIIOOOr0 U0JIHH0Ma P(- ) HMeeM P(F I (A))Q(B) = Q(B)P(F 2(A)).
ilJUI U0Jiy<leHml yCJI0BIUI 3 Tenepb M0)Kfl0 annpoKCHMHp0BaTb f H g U0JIHH0MaMH
H B0Cil0Jib30BaTbCJf cpyHKL(HOHaJlbHbIM HC\.fHCJieHHeM AJIH orpaHHlfeHHb[X caMOC0-
npJf)KetrnhlX oneparopos A, B.
3) ⇒ 2), 3) ⇒ 1) 01..feBHAH0, nOCK0JibKY ycJI0BHJf 2, 1 - \.faCHible C.IJ)'lfaH yCJJO
BHH 3.
2) ⇒ 1). Tipe)KAe scern, nocK0JibKY EF,ui(tl) = EA(F;- 1(!:i.)), H3 cneKrpanbnoro
pa3JI0)KeHHJf AJIJf F 1 (A), F i(A) HMeeM F 1 (A) sin tB = sin tBF 2(A) V t e JR 1.
llaJibHet!:wee LJ.0Ka3aTeJibCTBO CB0AHTCH K CHJibH0t!: annpoKCHMaIJ.HH orpaHHlfeH
noro caM0C0npJf)KeHH0f"O onepaT0pa B TpHI'0H0MeTpH'-leCKHMH Il0JIHH0MaMH OT
oneparopa B.
IlOCKOJlbKY a <POPMYJ!HpoBKe n. 2 Te0peMbl 1 pe'-lb HLJ.eT T0JibK0 o6 orpaHHlfeH
HblX o neparopax, npeACTaBJIJfeTCJf ecTeCTBeHHblM CJieAyIOw;ee 0npeAeJieHHe.
OnpeiJe✓1e1tue 1. Eyoeu woopumb, ttmo napa HeoipaHutteHHbl.X caMoconpJLJKeu
Hbl.X onepamopoo A, B yooo✓1emoopJ1em coom1w 1uemt10 (2), ec11u V t e JR 1, V t:i. e
e .B(IR 1)
EA(F1- 1(t:i.l)sin tB = sin tBEA<F2- 1(t:i.)).
TeopeMa 2. /JAJI Heozpanu 1,e1-tHbl.X ca,.,wconpJVKeHHbl.X onepamopoe A, B c ✓1eoy-
10u1ue yCIIOl3Ull 3K8U8aAelmZHbt:
1) EA(Fl-l(t:i.))sin tB = sin tBEA(F2- 1(t:i.));
2) 01111 AI06bl.X OlpO!tll'le/lHblX ll3MepUMbl.X f, g
lOe geven U godd - ttemHall U Hettem/-lall 'IOCnlb g;
3) ,w n11om11oa e H 0611acmu <I>, cocnw111((eil u3 tfeAbl.X eeKmopoe 01111 F 1(A),
Fi(A), B u u,wapuawl!-toil om11ocume11b110 3mu.x onepanwpoe, eww,me110 coom110-
1ue1-1ue F 1(A)Bq>=BF2(A)q>, q> e <l>.
aoKil3atnl!✓lbCm60. 1) => 2). IlepeXOAJf B yCJI0BHH 1 K cneK-rpaJibH0MY HHTerpa
JIY, s cHJry orpanH'-lem1ocru oneparopa sin tB AJIH nlo6ott orpanuqeuuoa H3Mepu
Mott <t>YHKL(HH f HMeeM f(F1(A))sin tB = sin tBf(Fi(A)) Vt e JR 1. TiocKOJibKY
f (Fi(A))sin2 tB = sin tB f(Fi(A))sin tB = sin2 tB f (F1(A)),
IlpOB0l(Jf C00TBeTCTBYIOLUYIO annpoKCHMaIJ.HIO, AJI.ll JII06bIX '-leTH0A geven<· ) H
HelJ.eTH0t!: good<- ) 01·paH1-1lfeHHblX H3MepHMbIX q>yHKLJ,Ht!: noJiy\.faeM
O6paTHaJf HMUJIHKaQHJf 0\.feBHAHa.
1) ⇒ 3). Tipe)KAe BCero, noKa)KeM, \.fT0 Bbin0JIHeHHe yCJI0BHJf l BJielJ.eT cymecT
BOBaHHe nnoTnoa B H uusapna1ITH0t!: ornocHTeJibH0 B, F 1(A), Fi(A) o6nacru <I>,
ISSN 004/-(i()53. YKp. Mam. )l()lpH., 1993, m. 45, N' 9
0 TTAPAX HEOrPAHWIEHHblX CAMOCOTTP.SDKEHHblX OIIEPA TOPOB, ... 1255
COCTOjlll.(e0 H3 QeJlblX BeKTOpoB AJUI yJ(a3aHHblX onepaTopoB • .O:eRCTBHTeJibHO, Bbl6e
peM B KattecTBe TaKOR o6JiaCTH
rAe EF1(Af ), EF-1..Af ), E!J2(· ) - pa3JIO)Keffffjf eAHHHw,l COOTBeTCTBYIOU(HX CaMO
conpgJKeHHJ>IX oneparopos, a OObeAHHeuue 6epercg no BCeM KOMilaKTHblM MHO)Ke
CTBaM A1, A2, A3 e R 1. JlerKo nposepHTb, 'ITO rnKoe <I>0 YAOBJieTBopgeT BCeM ue
o6xOAHMbIM YCJIOBH.)IM.
IlyCTb Tenepb <l> - npoH3BOJibHOe llJIOTHoe B H HHBapHaHTHoe MHO)KeCTBO,
COCTOjlll(ee H3 QeJiblX BeKTOpoB. Jl,JUl JIIOOblX <p, "' e <I> H3 ycnOBffjf 1 HMeeM.
(EA(F1-1(A))<p, sin tB 'I') = (sin tB <p, EA(Fz-1(A))'\jf).
nocneAHee COOTHOwem1e MO)KHO npOAHcpcpepeHQHpOBaTb no t npH t = 0. B pe-
3YJibTaTe nony'IHM
TTocKOJibKY <I> BXOAHT B o6nacTu onpeAeneuu.)I onepaTopos F 1(A), F 2(A), HMeeM
TaK)Ke
IlOJib3Y .)(Cb HHBapHaHTHOCTblO H nJIOTHOCTblO <l>, H3 nocneA)1ero COOTHOWeffHjf
nony'laeM F1(A)B'\jf =BFz(A)'\jf.
3) => 1). JJ:ng npoH3BOJibHOro <p e <I> BblllOJIHeHbl COOTHOWeHH.)I
F{'(A)B<p = Fi"- I (A)BFz(A)<p = BF z"(A)
H B CHJIY QeJIOCTH BeKTOpoB <p, B<p HMeeM paBeHCTBO
eitF1(A)Bcp = BeitF2 (A)cp 'r/ t e (CI.
JJ:ng npoH3BOJlbHblX <p, '\jf e <1> HMeeM paBeHCTBO
J ei1Ad(EA(F1- 1(>..)B<p, 'I') = (eirF1(A)B<p, '\jf) =
= (eirF2<A>cp,B'\jf) = J eit¼(EA(F2-1('A,))<p,B'\jf).
B CHJIY emmcTBeHHOCTH npeo6pa30BaHH.)I <l>ypbe HMeeM COOTHOWenue
(EA(F,- 1 (A))B<p, 'I') = (EA(F 2-l (A))<p. B'\jf),
H3 Koroporo cne.nyeT ycnosue 1.
B pa6oTe [3) H3Y'laJIHCb napbl orpaHH'leHHblX caMocoap.)l)KeHHblX oneparopos A.
B, CB.)13aHHblX no11yJIHt1eRnbIM COOTtIOwe1meM o6mero BHJ).a. B 'laCTHOCTH, ,[lJl.)I co:
OTHOWeHHR BHAa (2) CJieAyeT TaKOR pe3yJihTaT. .
YTeep.lKtlCHHC I. llycmb r = {(t, s) e JR 2: F 1(t) = F2(s), F 1(s) = F 2(t)}. Ec.11u
otpam,'leHHble onepamopbl A, B C8113aHbl coomHmuem1eJ.t (2). mo V A, A' e .B (JR I),
~ x ~, n r=0
IlpHBe.UeM ne3aBHCHMoe /\OKa3aTeJibCTBO 3TOro cpaKTa, KOTOpoe llO3B0Jllff BKJIIO
'IHTb B paccMoTpenue ueo1·pa11H1-1euub1e oneparoph1.
JJ:eRCTBHTeJlbHO,
ISSN 0041-6053 . YKp . MO/II. )l(yp11., 1993 , Ill . 45, N" 9
1256 B. JI. OCTPOBCKHtt, 10 C. CAMOttJIEHKO
EA(F1- 1(F1(M)) BEA(Fz- I (Fi(!!!.'))) = BEA(Fz- I (F1(/J.))) EA(Fz- 1(Fz(!J.'))) =
= BEA(Fz-1(Fl(/J.) n Fz(/J.'))).
Atta.norH'IHO
EA(Fi I (Fz(/J.)))BEA(Fl-l(F1(/J.'))) = BEA(F1-1(Fz(!J.) n Fl(!J.'))).
ITpu :3TOM, no KpaAHeA Mepe. Ol\HO H3 npuaeaeHHblX Bbipa)l(eHHA paBHO nymo. noc
KOJJbKY /J. x /J.' n r = 0. TaK KaK /J. c F/ 1 (Fi (/J.)). OTCIOL\a CJieayeT
Ei!J.)BEill.') = 0.
0Ka3bIBaeTC.!I, BepHO H o6paTHoe YTBep)l(l\eHHe.
YTeeplKAenue 2. llycmb A, B - oipaHL1'4eHHble ca;,.wconp.JLJKeHHble onepamopbl
maiwe, '4mo
011.R 11106bLX /J., /J.' maKUX, '4mo t,. X /J.' n r = 0. Toioa 011.R onepamopo8A, B 8bl
no,1HeHO coom1-1omeHue (2).
l(ol<a.3Qm~n,cmao. ,lleACTBHTeJJMIO, V /J. e .B(1R I)
EA(F1- 1(/J.))B = EiFI- 1(/J.))BEA(1R 1) = EA(FI- 1(/J.))BEi!!),
r,ne L\ = n /J.a. H nepeCe'leHHe 6epeTC.!I nO BCeM /J.a. TaKHM, 'ITO (/J. X /J.J_ n r = 0.
HeTPYAHO 3aMeTHTb, 'ITO /J. = {s: 3 t e /J., F z(s) = t}, noaTOMY E A(F1-\ll.))B =
= EA(F1- I(!J.))BEA(Fz- l(ll.)).
AHa.TIOrH'IHO BEiF2-I (!-,.)) = EA(F1-1(/J.)) BEiF2-I (/J.)) H OKOH'laTenbHO
EiF1- 1(ll.))B = BEiF2- 1(ll.)).
ITycTb Tenepb A, B - HeorpaHH'leHHbie caMoconp.!l)l(eHHbie onepaTOpbl, CB.!13aH
Hble COOTHOWeHHeM (2) B CMbJCJJe onpeaeneHH.11 1. KaK noKa3aHO npu ,lJ.OKa3aTeJJb
CTBe TeOpeMbl 2, BbJIIOJJHeHHe COOTHOWeHHA (2) :3KBHBa.TieHTHO HX Bb!IlOJJHeHHIO Ha
MHO)l(eCTBe COBMeCTHblX orpaHH'leHHblX BeKTOpoB KOMMyTHpylOIL(HX onepaTOpOB
F 1(!J.), Fz(!J.), B2. 06o3Ha'IHM :no MHO)l(ecrno qepe3 <I>.
TeopeMa 3. a11.R 1-1eozpa1-1u'4eHHbLX ca;,.wconp.JLJKeHHbLX onepamopo8 A, B 8bmo11-
HeHbl COOmHOUteHlt.R (2) mozi)a lt mOllbKO mozoa, KOlOa \;/ <p e <l>
011.R 8CeX MHOJKeCm8 /J., /J.' e JR I maKUX, '4mO /J. X /J.' n r =0.
,lloKa3aTeJJbCTBO nOBTOpReT apryMeHTbl YTBep)l(aenuA 1, 2 c y'leTOM Toro, 'ITO
see COOTHOWeHHR CnpaBel\nHBbl Ha BeKTOpax <p e <I>.
2. PaccMoTpHM Tenepb 3aaaqy on0ca1m;i: scex nap A, B, noo61.1..1e rosop.ll, ueorpa
HH'leHHbIX caMOCOnp.11:>KeHHblX onepaTOpOB, CB.!13aHHblX COOTHOWeHHeM (2) (scex ca
MOCOnp.11:>KeHHblX peweHHtt onepaTOpHoro ypaeHeHHR (2) ). Ey aeM OTO:>Kl\eCTBJJ.!ITb
yHHTaptto aKBHBa.TieHTHble peweHHll (KaK OObI'IHO, napa A, B oneparopoe B H na-
3bIBaeTCR yHHTapno aKBueai1eHT11ott nape A, B onepaTopoe e H , ecJm cymeCTeyeT
yHHTapnhlA oneparnp U:-H ➔ H TaKOA, 'ITO A= U* AU, B = u• BU).
Ilpe)Kae ecero onuweM npoCTeAwue (nenpuooanMble) napb1 A.B (napaA,B na-
3hlBaeTCR ttenpHBOAHMOA, ecnu anR C eIJ..H) H3 [A, C]=[B. C] =0 cneayeT C = ')J).
Cneay_g [4], cocrnBHM xapaKTepnCTH'leCKHe <PYHKUHH <1> 1(-,··) H <1>2(-, ·) cooTHO-
ISSN 0041-6053 . YKp. Mam. ;,cyp1t., 1993, m. 45, N• 9
0 ITAPAX HEOrPAHHt.J:EHHblX CAMOCOITP5DKEHHhlX OITEPATOPOB, . ..
we1uu1 (2):
<1>1(t,s) = F 1(t) -Fi(s), <1>2(1,s) = F 2(t)-F1(s), t,se JR!.
XapaKTepHCTH'leCKoe 6 HttapHoe OTHOwemre
r = {(t , s): <1> 1(1, s) = <1>2(t, s) = O}
1257
MO)KHO pacci-1arp1maTb KaK MHO.lKeCTBO Ayr HeOpHeHTHpoBaHHOro rpacpa (AJUI KpaT
KOCTH o6o3Ha'IHM ero TaK)Ke qepe3 r H tta30BeM rpacpoM COOTHOWeHHH (2)). B "py
'IHOl:f' curyau;mi CBji:3Hble KOMnoneHTbl rpacpa 1 HMelOT BHA ., Q HJIH ...... , a CHC-
{
fi(t) = F2(s),
F2(t) = fj(s)
(3)
AJIH mo6oro t e JR I He HMeeT pewe1rnn HJIH HMeeT eAHHCTBeHHoe peweHHe. 3ati.a-
F
Ba.H AHHaMH'leCK YIO CHCTeMy JR :) Cf 3 t ~ S E Cf C JR AJIR Tex f, AJUI KOTO-
pbIX 3TO peweHHe cymecTByeT, 3alIHWeM COOTHOllleHHe (2) B mme (1). Bee uenpHBO
AHMbie IlpellCTaBJieHHH COOTHOWeHHH (2) TOrAa OLIHO- HJIH L1ByMep11b1, H cnpaBeAJIH
Ba CTPYKTypua.H TeopeMa [ 1].
TeopeMa 4. llycmb A u B - caMoconpJVKem-tbte onepamopbl 8 H, C8Jl:JaHHble
coomHoLUenueM (2) maKuM, 'lmo pea11UJyemc.J1 "py'lna.JI" cumyaL{UJI. ToziJa oiJH03Ha-
'll-tO onpeiJe11eHbt pa31tOJKeHue H = H 0 EB H 1 EB (<C 2 ® H +) u opmowHaAbHble pa3-
11OJKe1-tu.J1 eiJuHUL{bt : l) £ 0(-) Ha JR 1 co 3Ha'leHUJIMU 8 npoeKmopax na noiJnpocm
paHcm8a H0; 2) £ 1(·,·) Ha M 1 ={(A,b): F(A)=A,bi'O} co311G'leHUJ1MU8npo
eKmopax Ha noiJnpocmpaHcnwa H 1; 3) £ 2(-, ·) na M 2 = {(A, b): F(F(A)) = A,
F(A) > A, b > O} co 3Ha'lem111Mu 8 npoeKnzopax na noiJnpocmpaHcmoa H + maKue,
A = J AdE00 .. ) + f AdEi('A,, b) + f ( ~ F?~ .. )) ® dE2()., b).
R M1 M2
B == f bdE1()-.., b) + (? 6) ® J bdE2(A, b).
M1 M2
EcJIH )Ke CBR3I-lbie rpacphl xapaKTepHCTH'-leCKOro 6HI-Iapnoro OTHOWeHml ycrpo
eHbl CJIO)KI-Iee H B Ka'leCTBe cpparMeHTOB COAep)KaT IlOArpacpbl BHAa c;l. HJIH
••• (CHCTeMa (3) npH HeKOTOpbIX t HMeeT ABa HJIH 6onee peweHttit), TO 3ati.a'la
OnHCaHH.H BCex HenpHBOAHMblX peweHHit (2) - "AHKaH" (T. e. cymeCTBYIOT pewe
HHH (2) TaKHe, 'ITO CJia6o 3aMKfIYTa.H anre6pa, IlOp0)KAeHHaH A H B, - cpaKTOp ne
rnna I (cM. [4])).
3. llpuMepbl. a). Coon,owettHe AB= BF(A), F: JR 1 ➔ JR 1. CttcreMa (3)
{
I = F(s),
s = F(l)
HMeeT eAHHCTBeHHoe peweHHe (I. F(I)), npH 3TOM AOJIJKHO 6b1Tb F(F(I)) = I.
6). CooTHOllleHHe [A2, BJ= 0- " nHKoe", TaK KaK 3aJ:1a'!a onHCaHH.SI nap A, B conepJKHT B ce6e
3ana'ly onttcantt.si nap onepaTopoe ettna A = P - p 1., B = Q1 + 2Q2 + 3Q3 (P. Qi, Q2, Q3 -
npoeKTOpbl, Q1 j_ Q2 j_ Q 3, Qi + Q2 + Q3 = /). B CBOIO O'!epenb, :na 3aAa'!a '.lKBHBaJieHTHa "AHKOR"
(CM. [S, 6]) 3ana'le OHHCaHH>I TpoeK npoeKTOpoe, nea H3 KOTOpblX oprorOH:1./IbHbl.
XapaKTepHCTH'l.eCKoe OTHOllleHHe 1 = {(t, s): t2 = s 2} HMeeT rpacp co CB.H3HbIMH
KOMUOHeHraMH BHAa c;>..Q.
4. AHaJIOrH'IHO [l] MO.lKHO paccMaTpHBaTb KOMMYTHpyiomee ceMeitCTBO A =
ISSN 0041-6053. YKp. Mam. JKJpH., /993, m. 45, N• 9
1258 B. JI. OCTPOBCKJ-ltt, IO. C. CAMOltJIEHKO
= (Ak)k•A CaMOConpH)KeHHbl.X onepaTOpoB B H' CBH3aHHhlX C onepaTOpoM B co
OTHOWeHHHMH
(4)
Jl_ng TaKHX ceMeRCTB onepaTOpoB IlOAOOHO n. l onpeAeJIHeTCH KJiacc "nHTerpupye
MbIX" ceMeRCTB, ecJIH BMeeTO pa3JIO)KeHH.H eAJfHHQbl £ /·) onepaTopa A HCilOJlb30-
BaTb COBMeeTHoe paJJIO)KeHHe eAHHHQbl EA O KOMMYTHpyIOLQero ceMeRCTBa A.
IlpH 3TOM ecJIH ua6op A KOHe'{eH HJIH eqeTeH, enpaBeAJIHBbl see YTBep)Kl).eHHH n . 1,
AJIH uecqeTHblX ua6opoB MOryT BO3HHKttyTb TPYAHOCTH npH noc-rpoeHHH o6naeTH <l>
( ocuameuna) .
IlpH HJyqeHHH npeACTaBJieHHR COOTHOWeHHR (4) nonyqaeM 6uuapuoe OTHOUleHHe
r = {(t, s) e JRA x JRA: F(t) = G(s), F(s) = G(t)}
(3Aech FO = (Fi(-), Fi(·), .. . ), GO= (G 10 , G2(·), ... ): JR A ➔ JRA) H coorneTeT
sy10mHR rpacp r . "Pyquaa" CHTYaQIDI peaJIH3YeTc.H npu Tex )Ke ycnosuax ua r, '{TO
0 B cnyqae OAHOro onepaTopa A.
IlpHBeAeHHble paeeMOTpeHHH o6o6LQaIOTeB H Ha enyqaR KOHe•-rnoro ua6opa B =
= (Bj) j=l KOMMYTHpYIOLQHX caMoconpH)KeHHblX onepaTopoB, eBB3aHHbIX C onepaTO
pOM A (HJIH KOMMYTaTHBHblM ceMeRCTBOM A) COOTHOUleHHBMH BHAa
F/A)Bj = BP/A), j = 1, ... , n (5)
(COOTBeTCTBeHHO eOOTHOUleHHHMH BHAa
F1,jA)Bj = B/ikJA), j = 1, ... , n; k e A).
nna TaKHX Ha6opoB onpeAeJIHeTCJI IlOHJITHe "HHTerpupyeMOCTH" H sepHbl auanorH
YTBep)KAeHHR n. 1. l.JTo KaeaeTeB onueamuI uenpHBOAHMblX npeAeTaBJieHHR, TO
3Aeeb eHTYaQHB HeCKOJlbKO eJIO)t<Hee - ueo6XOAHMO paeeMaTpHBaTb n 6HHapHbIX
OTHOWeHHR rj, COOTBeTeTByIOLQHX Ka)KAOMY H3 onepaTOpoB Bj' IlyeTh see onepa
TOpbI Bj HeBbipo)KAeHbl (3TOrO MO)KHO AOOHTbCJI, onuenJljfJI HHBapHaHTHbie IlOA-
npocrpaHCTBa KerBj). Ilpu 3TOM, noeKOJihKY Bj H B1 KOMMyTupyIOT, 03 (t, s)e
e rj• (t, t') e r 1 eneAyeT, '{TO eymeernyeT s' TaKOe, '{TO (t', s') e rj• {f, s') e r I •
KpnTepuR Toro, '{TO 3al).aqa onueaHHJI HenpHBOAHMblX npeACTaBJieHRR eOOTHOllle
HHR (5) "pyquag", eOCTOHT B TOM, '{TOObl eBJl3Hble KOMilOHeHTbl Ka)KAOro H3 rpacpos
rj HMeJIH BHA, yKaJaHHblfi a n. 2. HenpHBOAHMble npell,CTaBJieHIUI a "pyquoR" CHrya
U.HH MOryT HMeTb pa3MepHOCTb 1, 2, ... , zn.
1. OcmpoecKuil B. JI., Ca,-wiiAeww JO. C. CeMdlCTsa HeorpaHH'leHHblX caMoconpJ1JKeHHblX onepa
TOpoB, CBJl3aHHblX HeJJHeBCKHMH COOTHOWeHHJIMH // <l>yHKUHOH. aHaJJH3 Hero npHJI. - 1989. -
23, Bbln. 2. - C. 67 - 68.
2. Ostrovskii V. L., Samoilenko Yu . S. Unbounded operators satisfying non-Lie commutation relations
// Reports Math. Phys. - 1989. - 28, N" I. - P. 91 - 104.
3. Samoilenko Yu. S. Spectral theory of collections of selfadjoint operators. - Dordrecht; Boston;
London: Kluwer Acad. Pub!. , 1990. - 310 p.
4. Eecna11oeJO. H., CaMoil/leHJ(o JO . C .. Wy11bMOk B. C. 0 Ha6opax onepaTopos, CBJl3aHHblX nony
JIHHellHblMH COOTHOWeHHJIMH // npHMeHCHHe MeTOJ:108 q>yHKUHOH. aHaJJH3a 8 MaT. q>H3HKe. -
KHeB: liH-T MaTeMaTHKH AH YKpaHHbl, 1991. -C. 28- 51.
5. KpyZ11J1K C. A ., Ca,-wf111eHJ(O JO . C. 06 YHHTapuoll 3KBHBaJICHTHOCTH Ha6opos caMoconpJ1JKeH
HblX onepaTOpoB // <l>yHKI.\HOH. aHaJJH3 H ero npHJI. - 1980. - 14, Bbln. 1. - C. 59 - 62.
6. Eecna11oe JO . H ., CaMOillleHJ(O JO . C. Anrc6paH'leCKHe onepaTOpbl H napbl CaMOCOflpJIJKeHHblX
oneparopos, CBJl3aHHblX nOJIHH0MHaJJbHblMH COOTHOllleHHJIMH // TaM >Ke. - 1991. - 2S, Bbln. 4. -
C. 72-74.
noJiy'leHO )0. 06.92
ISSN 0041 -6053. Y1<p. Mam. ;,cyp'<._, 1993, m. 45, N• 9
0067
0068
0069
0070
0071
0072
|
| id | umjimathkievua-article-5927 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:24:10Z |
| publishDate | 1993 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/79/66e16356cb1f341a786194823886d779.pdf |
| spelling | umjimathkievua-article-59272020-03-19T09:21:13Z On pairs of unbounded self-adjoint operators satisfying an algebraic relation О парах неограниченных самосопряженных операторов, связанных алгебраическим соотношением Ostrovskii, V. L. Samoilenko, Yu. S. Островский, В. Л. Самойленко, Ю. С. Островский, В. Л. Самойленко, Ю. С. Unbounded pairs of self-adjoint operators A andB satisfying the algebraic relation F 1(A)B=BF 2(A) are studied. For these relations, various definitions of “integrable” pairs of operators are presented and the class of “tame” relations is indicated; for the “tame” relations, the irreducible pairs are described and a structure theorem is presented. Вивчаються пари необмежених самоспряжених операторів $А$, $В$, пов'язаних алгебраїчним співвідношенням вигляду $F_1(A)B = BF_2(A)$. Цля таких співвідношень наведені різноманітні варіанти визначення „інтегровних” пар операторів, виділено клас „ручних” співвідношень, для яких описано незвідні пари та наведено структурну теорему. Institute of Mathematics, NAS of Ukraine 1993-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/5927 Ukrains’kyi Matematychnyi Zhurnal; Vol. 45 No. 9 (1993); 1253–1258 Український математичний журнал; Том 45 № 9 (1993); 1253–1258 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/5927/8537 https://umj.imath.kiev.ua/index.php/umj/article/view/5927/8538 Copyright (c) 1993 Ostrovskii V. L.; Samoilenko Yu. S. |
| spellingShingle | Ostrovskii, V. L. Samoilenko, Yu. S. Островский, В. Л. Самойленко, Ю. С. Островский, В. Л. Самойленко, Ю. С. On pairs of unbounded self-adjoint operators satisfying an algebraic relation |
| title | On pairs of unbounded self-adjoint operators satisfying an algebraic relation |
| title_alt | О парах неограниченных самосопряженных операторов, связанных алгебраическим соотношением |
| title_full | On pairs of unbounded self-adjoint operators satisfying an algebraic relation |
| title_fullStr | On pairs of unbounded self-adjoint operators satisfying an algebraic relation |
| title_full_unstemmed | On pairs of unbounded self-adjoint operators satisfying an algebraic relation |
| title_short | On pairs of unbounded self-adjoint operators satisfying an algebraic relation |
| title_sort | on pairs of unbounded self-adjoint operators satisfying an algebraic relation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/5927 |
| work_keys_str_mv | AT ostrovskiivl onpairsofunboundedselfadjointoperatorssatisfyinganalgebraicrelation AT samoilenkoyus onpairsofunboundedselfadjointoperatorssatisfyinganalgebraicrelation AT ostrovskijvl onpairsofunboundedselfadjointoperatorssatisfyinganalgebraicrelation AT samojlenkoûs onpairsofunboundedselfadjointoperatorssatisfyinganalgebraicrelation AT ostrovskijvl onpairsofunboundedselfadjointoperatorssatisfyinganalgebraicrelation AT samojlenkoûs onpairsofunboundedselfadjointoperatorssatisfyinganalgebraicrelation AT ostrovskiivl oparahneograničennyhsamosoprâžennyhoperatorovsvâzannyhalgebraičeskimsootnošeniem AT samoilenkoyus oparahneograničennyhsamosoprâžennyhoperatorovsvâzannyhalgebraičeskimsootnošeniem AT ostrovskijvl oparahneograničennyhsamosoprâžennyhoperatorovsvâzannyhalgebraičeskimsootnošeniem AT samojlenkoûs oparahneograničennyhsamosoprâžennyhoperatorovsvâzannyhalgebraičeskimsootnošeniem AT ostrovskijvl oparahneograničennyhsamosoprâžennyhoperatorovsvâzannyhalgebraičeskimsootnošeniem AT samojlenkoûs oparahneograničennyhsamosoprâžennyhoperatorovsvâzannyhalgebraičeskimsootnošeniem |