Specu of The Diec Su of Opeaos by E.OTKUN ÇEVİK ad Z.I.ISMILOV Kaadeiz Techical Uivesiy, Faculy of Scieces, Depae of Maheaics 6080 Tabzo, TURKEY e-ail adess : zaeddi@yahoo.co bsac: I his wok, a coecio bewee soe specal popeies of diec su of opeaos i he diec su of Hilbe spaces ad is coodiae opeaos has bee ivesigaed. Keywods: Diec su of Hilbe spaces ad opeaos; specu ad esolve ses; coiuous ad copac opeaos; disceeess of specu; asypoics of eigevalues; 000 MS Subjec Classificaio: 470.. Ioducio I is kow ha ifiie diec su of Hilbe spaces H, ad ifiie diec su of opeaos i H, ae defied as ad H H u u : u H,, u u H H, D u u H : u D,, u u H, : D H H (see []). The geeal heoy of liea closed opeaos i Hilbe spaces ad is applicaios o physical pobles has bee ivesigaed by ay aheaicias (fo exaple, see []). Howeve, ay physical pobles of oday aisig i he odellig of pocesses of ulipaicle uau echaics, uau field heoy ad i he physics of igid bodies suppo o sudy a heoy of liea diec su of opeaos i he diec su of Hilbe spaces (see []-[6] ad efeeces i i). I his pape, a coecio bewee specu, esolve ses, disceeess of he specu (sec. ) ad asyoical behaviou of he eigevalues (sec. 3) of diec su of opeaos defied i he diec su of Hilbe spaces ad suiable popeies of coodiae opeaos has bee esablished. The obaied esuls has bee suppoed wih applicaios.
These ad elaed pobles i he case coiuous diec su of he Hilbe space opeaos have bee ivesigaed i woks Bu i hese woks has o bee cosideed a coecio bewee pas of he specu of diec su opeao ad suiable pas of he specu hei coodiae opeaos.i his pape give shap foulaes i he his sese.. O he specu of diec su of opeaos I his secio, he elaioship bewee he specu ad esolve ses of he diec su of opeaos ad is coodiae opeaos will be ivesigaed. Fis of all i will be ivesigaed he coiuiy ad copacess popeies of he opeao i H H i case whe LH fo each. I is easy o see ha he followig poposiios ae ue i geeal. Theoe.. Le, H H ad fo ay he ecessay ad sufficie codiio is sup. LH. I ode fo LH I addiio, i his case whe LH i is ue sup (see Theoe.. Le C H li 0. fo each Fuheoe, he followig ai esul ca be poved. C H if ad oly if. I his case Theoe.3. Fo he pas of specu ad esolve ses of he opeao H H he followig clais ae ue i Hilbe space
Poof. The validiy of fis clai of give elaios is clea. Moeove, i is easy o pove he fouh eualiy usig he heoe.. Now we pove he secod elaio o he coiuous specu. Le c. I his case by he defiiio of coiuous specu E is a oe-ooe opeao, R E H ad R E is dese i H. Coseuely, fo ay a opeao E is a oe-o-oe opeao i H, hee exiss such ha R E H ad fo ay bu liea aifold R E This eas ha is dese i H o O he coay, ow suppose ha fo he poi he above elaio is saisfied. Coseuely, eihe fo ay c, o ad hee exis such ha c. Tha is, fo ay d fo his i iplies ha he opeao. Hece R E H is a oe-o-oe opeao, R E H ad. c R E H. is a oe-o-oe opeao, R E H ad 3
O he ohe had he siple calculaios show ha c c c c p c. By he siilaly idea ca be poved he validiy of he hid eualiy of he heoe. Exaple.4. Coside he followig ulipoi diffeeial opeao fo fis ode u u, H L, a, b, a b a... : D H H, D u W : u a u b,, ad H H. Fo ay opeao ki ad ae oal, p : k b a eigevecos accodig o he eigevalue k,, k ae i he fo ad u c exp a,, c \ 0 []. k k k k I his case uk ck k a d ck b a exp. H The coefficies c k ay be choose such ha he las seies o be covege. This eas ha. Fo his ad Theoe. i is obaied ha k p p p. Defiiio.5[]. Le T be a liea closed ad desely defied opeao i ay Hilbe space. If T ad fo T he esolve opeao R T C T : DT is called a opeao wih discee specu. Noe ha he followig esuls ae ue., he opeao I is clea ha if he opeao fo evey he opeao is a opeao wih discee specu i is also i H. The followig poposiio is poved by usig he heoe.. H H, he 4
Theoe.6. If, R is a opeao wih discee specu i H,, ad li 0, he is a opeao wih discee specu i H. Poof: I his case fo each we have R C H, :. Now we defie he opeao K R i H. I his case fo evey u u D have K Eu R E u R E u R E u u ad E K u E R u E R u E R u E R u u These elaios show ha R R Fuheoe, we defie he followig opeaos K : H H, i he fo K u : R u, R u,..., R u,0,0,..., u u H. we Now he covegece i opeao o of he opeaos ivesigaed. Fo he u u H we have K o he opeao K will be K u Ku R u R u R u sup H H H H sup R u Fo his i is obaied ha H K K sup R,. 5
This eas ha seuece of he opeaos K coveges i opeao o o he opeao K. The by he ipoa heoe of he heoy of copac opeaos K C H ay K C H. Exaple.7. Coside ha he followig faily of he opeaos i he fo d : S, S S 0, S C, d : D L L, a, b, sup b a, D u W, : u b W u a, W W, whee [], because fo L L,,, is ay Hilbe space ad W is a uiay opeao i, (fo his see []). Fo ay a opeao Fo he is oal wih discee specu ad ad sufficiely lage a siple calculaio shows ha S E a S E b a S E b s R f e E W e W e f s ds SEs e f sds, f L, a O he ohe had he followig esiaes ae ue. S E s S E s a L a e f s ds e f s ds d S E s S E s e ds d f s ds e ds d f L a a s S s s e e ds d f e ds d f L L a a b a a b e f L,. 4 6
E W e W W e e S E b a S E b a S E b a S E b s 0 0 b a S E b a e e,. b a L e f s ds e f,.3 Hece fo. ad.3 we have S E a S E b a S E b s e E W e W e f s ds L a S a S E b a S E b s e e d E W e W e f s ds b a b a L ba e e e f,.4 4 L whee, is he eal pa of ad is he fis eigevalue of he opeao S,. Theefoe, fo esiaes. ad.4 he followig esul is obaied., sup b a ad S as, he Poposiio.8. If 0 as. R Coseuely, he opeao is a opeao wih discee specu i L L. 3. sypoical behaviou of he eigevalues I his secio asypoical behaviou fo he eigevalues of he opeao will be ivesigaed i a special case. i H H 7
Theoe 3.. ssue ha eh opeao specu ad fo i, j, i j i H ad i H, ae opeaos wih discee, i j c, 0 c,,, c, is saisfied. If ad hee exiss such ha if 0, he, 0, as. Poof: Fis of all oe ha by he Theoe.3 Hee i is deoed by NT. p p, ha is, a ube of eigevalues of he soe T ; :, 0 liea closed opeao T i ay Hilbe space wih odules of hese eigevalues less ha o eual o, 0. This fucio akes values i he se of o-egaive iege ubes ad i case of ubouded opeao T i is odeceasig ad eds o Sice fo evey i, j, i j I his case i is clea ha, i j c c c as., he N ; N ; N ;, 0. The las seies is uifoly covege i, o. The li c c Theefoe ;, 0, as N c c c. The he followig asypoic behaviou of eigevalues of he opeao, 0, as is ue. i H Reak 3.. If i he above heoe he coefficies, saisfy he followig codiio if 0, he fo evey 0 if, N ; as Reak 3.3. If he evey fiiely ay ses of he faily, 8 i coplex plae iesec i he fiiely ay pois, he i ca be poved ha clai of he Theoe 3. is valid i his case oo. Exaple 3.4: Le H H, H,, : D H H, u : c u,,,,,,,,, 0, u u D c c c k k c k k k
as, k is covege ad hee exiss such ha if. I his case, fo ay Now we obai he esolve opeao of is a liea oal opeao ad c. Le p. The fo he elaio E u v,, v H, i.e c u u v,. v I is esablished ha u,, c i.e R v,. v c O he ohe had sice c k, as, he fo ay v H H c c c R v v v Coseuely, fo ay v v H we have R (3.) c Moeove, i is kow ha a esolve opeao R, 9 is copac if ad oly if.sice c,, ad codiios o c, he he las codiio is saisfied. Hece fo ay R C H O he ohe had sice he seies. k is covege, he fo he ieualiy (3.) fo he, i is easy o see ha li R 0, R Hece by he Theoe.6 fo he i is esablished ha R C H he Theoe.3 i is ue ha p p. Fuheoe, he validiy of he elaio i j Theefoe by he Theoe 3., 0, as, i, j, i j is clea.. The by
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