Spectral Problems of Two-Parameter System of Operators

Transcription

1 Pue ad Appled Matheatc Joual 5; 4(4-: Publhed ole Augut, 5 ( do: 648/pa5447 ISSN: (Pt; ISSN: (Ole Spectal Poble of Two-Paaete Syte of Opeato Rahhada Dhabaadeh Depatet of fuctoal aaly, Ittute of Matheatc ad Mechac of NAS of Aebaa, Bau, Aebaa Eal adde: ahhadadhabaade@ableu To cte th atcle: Rahhada Dhabaadeh Spectal Poble of Two-Paaete Syte of Opeato Pue ad Appled Matheatc Joual Specal Iue: Spectal Theoy of Multpaaete Opeato Pecl ad It Applcato Vol 4, No 4-, 5, pp do: 648/pa5447 Abtact: The autho ha poved the extece of the ultple ba of the ege ad aocated vecto of the two paaete yte of opeato Hlbet pace The poof eetally ue the theoe of the extece of ultple ba of opeato budle ad the oto of the abtact aalog of eultat of two opeato pecl, actg, geeally peag, dffeet Hlbet pace Codeable o-elfadot two paaete yte deped o both paaete a coplcated ae Keywod: Multpaaete, Spectu, Opeato, Space, Egevecto Itoducto Spectal theoy of opeato oe of the potat decto of fuctoal aaly The ethod of epaato of vaable ay cae tued out to be the oly acceptable, ce t educe fdg a oluto of a coplex equato wth ay vaable to fdg a oluto of a yte of oday dffeetal equato, whch ae uch eae to tudy FV Ato [] tuded the fagetay eult fo ultpaaete yetc dffeetal yte, bult ultpaaete pectal theoy of elfadot yte fte-deoal Eucldea pace Futhe, by tag the lt, Ato geealed the eult obtaed fo the ultpaaete yte wth elf-adot opeato fte deoal pace o the cae of the ultpaaete yte wth copact elfadot opeato Hlbet pace Late Bowe, Sleea, Roch ad othe atheatca bult the pectal theoy of elfadot ultpaaete yte fte-deoal Hlbet pace [],[3] I th wo the extece of ultple ba o ege ad aocated vecto of two paaete o-elfadot yte of opeato Hlbet pace poved Defto of the aocated vecto, ultple copletee of ege ad aocated vecto of two-paaete o-elfadot yte, ae toduced [4],[6] Pelay Defto ad Rea A( λ, µ A λa λ A µ A µ A λ µ A Let <,, < B( λ, µ B λb λ B µ B µ B λ µ B ( <,, < be the o-lea ultpaaete yte two paaete Opeato A (coepodgly, B act the Hlbet pace H (coepodgly, H, H H a teo poduct of paceh adh Fo olea algebac yte wth two vaable uffcet codto of extece of oluto ae gve The poof of thee tateet ae eceved a a coollay of oe coo evewg codeed th pape Defto ( λ, µ a egevalue of the yte ( depedg o two pectal paaete f thee ae uch o-eo pa vecto x H, y H that the equato (

2 34 Rahhada Dhabaadeh: Spectal Poble of Two-Paaete Syte of Opeato A( λ, µ x ( A λa λ A µ A µ A λ µ A x <,, < B( λ, µ y ( B λb λ B µ B µ B λ µ B y, ae atfed Decopoable eleet x y called a ege vecto of ultpaaete yte ( Defto [] Opeato A (coepodgly, B duced to the paceh H H by the opeato A (coepodgly B, actg the pace H (coepodgly, H, f the followg codto atfy: o decopoable teo x y, A ( x y ( Ax y ad, B ( x y x ( B y, o othe eleet of the pace H H H -o leaty ad cotuty Defto 3 ([4], [5] A teo, aed (, th the aocated vecto to a egevecto, x y, f the followg codto (3 ae atfed ( A ( λ, ε,!! λ, B ( λ, ε,!! λ, ;, ;, (3 (, - aageet fo et of the whole oegatve ube o wth poble ecug ad eo Ude caocal yte of eavecto (th defto geealato of the defto of caocal yte, toduced [] the cae polyoal pecl oe paaete fo ( λ, µ we udetad the yte ( { },,,, poeg the followg popete: eleet of a ege ubpace M λ ( ; thee (4 fo ba (, (, egevecto whch ultplcty eache a poble axa p ; thee egevecto whch ot expeg lealy though (, (, (,,, whch u of ultplcte eache a poble axa p Let' degate though M (( λ, µ a ubpace paed by ege ad aocated vecto of the yte (, coepodg to a egevalue λ Lealy-depedet eleet fo a cha of a et H ege ad aocated (ea vecto of ( The (, ultplcty of egevalue degate the geatet ube of aocated vecto to,, a plu The u p p p a ultplcty of a ege value ( λ, µ Eleet (4 fo a cha of ege ad aocated vecto fo evey fxed value,,,, Defto 4 Syte of eleet { } x, (,,, of Hlbet pace fo ultple bae th pace f ay eleet f, f,, f of pace ca be pead out ee, (,, wth coeffcet, ot depedet o f c x a dex of vecto,,, f f f If yte { } x cocde, wth the yte of ege ad aocated vecto of a x, opeato, ad yte { } fo equece o { } ae cotucted, poceedg x, accodg to oe ule the pea about -ultple ba o yte of ege ad aocated vecto of a opeato Defto5( [6],[7] Let be two opeato pecl depedg o the ae paaete ad actg, geeally peag, vaou Hlbet pace A A λa λ A λ A, B B λb λ B λ B Opeato Re ( A, B( λ peeted by the atx A E A E A E A E A E A E (5 E B E B E B E B E B E B whch act the ( H H - dect u of cope of the pace H H I a atx (5 ube of ow wth opeato A equal to leadg degee of the paaete λ pecl B ad the ube of ow wth B equal to the leadg degee of paaete λ A( λ Noto of abtact aalog of eultat of two opeato pecl codeed the [8] fo the cae of the ae leadg degee of the paaete both pecl ad the [7] fo, geeally peag, dffeet leadg degee of the paaete the opeato pecl Theoe Let all the opeato be bouded coepodg Hlbet pace, oe of opeatoa o B ha bouded vee The opeato pecl A( λ ad B( λ have a coo pot of pecta f ad oly f

3 Pue ad Appled Matheatc Joual 5; 4(4-: { } Ke Re ( A, B ϑ (6 Rea If the Hlbet pace H ad H ae the fte deoal pace the a coo pot of pecta of opeato pecl A ad B ae the coo egevalue(ee [6], [7] 3 Multple Ba of Ege ad Aocated Vecto of Multpaaete Syte wth Two Paaete Code the yte ( Opeato A, B act the Hlbet paceh ad H,coepodgly Fo tudy of the pectal popete of the yte ( we ue the oto of abtact aalog of eultat of A ad B( λ Fx the oe of the paaete ( Let t the paaete λ ad λ λ The we have two opeato pecl oe paaete µ A( λ, µ x ( A λ A λ A µ A µ A λ µ A x <,, < B( λ, µ y ( B λ B λ B µ B µ B λ µ B y (7 <,, < Aage the pecl o ceag of the degee of the paaete µ ad deote the opeato coeffcet of the paaete µ the degee the opeato pecl A( λ, µ though A λ A, Aɶ ad the pecl B( λ, µ < opeato coeffcet of the paaete µ degee we deote though B λ B, Bɶ Opeato, duced < to the pace H H H, we deote A ɶ ad Bɶ ( λ, coepodgly Itoduce the otato A ɶ ( λ λ λ, B ɶ ( λ B λb λ B A A A Cotuct the eultat of opeato pecl A( λ, µ ad B( λ, µ (the paaete λ fxed Aɶ ( λ Aɶ ( λ Aɶ Aɶ ( λ Aɶ ( λ Aɶ Aɶ ( λ Aɶ ( λ Aɶ ( λ Aɶ Re ( A( λ, µ, B( λ, µ B ɶ ( λ Bɶ ( λ Bɶ Bɶ ( λ Bɶ ( λ Bɶ Bɶ ( λ Bɶ ( λ Bɶ ( λ Bɶ (8 The ube of ow wth the opeato Aɶ equal to the leadg degee of the paaete µ the opeato pecl B( λ, µ, that ; ube of ow wth the opeato Bɶ equal to the leadg degee of the paaete µ tthe pecl A( λ, µ,that Let ax (, whee, eale at Let the geatet degee of λ the opeato coeffcet of µ (,, the opeato pecl A( λ, µ be By aalogy the geatet degee of λ the opeato coeffcet at µ,,, the opeato peclb( λ, µ be So the paaete λ fxed abtaly, futhe, the yte we the dex of the paaete λ Let K deote the te of the expao of the eultat (7 fee fo paaete λ ad let K be a elfadot opeato wth the ek { ϑ} It ow that the elf-adot copletely cotuou opeato ha a dcete pectu, e the pectu of the opeato cota oly egevalue of fte ultplcty If the ee of odule of egevalue of copletely cotuou opeato oe potve degee covege the th opeato belog to cla σ p Aage the budle poduced by the decopoto of the eultat (7, to ceae the powe of the paaete λ ad deote the opeato coeffcet of λ though T Let' ed, that ude a ege value of polyoal budle L A λab λ A B λ B

4 36 Rahhada Dhabaadeh: Spectal Poble of Two-Paaete Syte of Opeato udetood uch o-eo vecto x,that equalty L( λ x x atfed -th aocated vecto x to a egevecto x the vecto, atfyg to codto d d x L( λ x L( x L( x ;,, dλ λ! dλ λ Let { x,} be yte of ege ad aocated vecto of a budle L( λ O yte of ege ad aocated vecto of polyoal budle L( λ devatve yte of vecto ae cotucted by ule d λt t t x, e ( x, x, x, dt!! t,, 4 Spectal Popete of Two Paaete Syte Fo the tudy of the pectal poble of two paaete yte ( we ue the followg eult I a epaable Hlbet pace we code a opeato pecl L A λab λ A B λ A B λ B whe the opeato pace H, λ C Let B ( { } KeB ϑ A, B ae copletely cotuou the be the oal opeato, chaactetc value of whch le o a fte ube of ay, eaatg fo the og Deote a ceag equece of odule chaactetc ube of the opeatob, tag to accout the ultplcty Theoe [5] Suppoe that oe of the followg two codto: p а < p <, l µ, opeato p AB (,,,, ae bouded p b < p <, l µ <, p AB (,,,, copletely cotuou opeato ae atfed The dffeet egevalue { λ} of L ca be aaged a equece uch that fo oe ceag equece { } ( of potve tege, ad fo all eleet f, f,, f of the pace H, atfyg the codto f R( B, the expao f p ( ( a Z whee p the ube of dffeet egevecto of ultplcty coepodg to egevalue λ wth egevecto L( λ ( d λt ( ( t t Z e Z Z,, Z dt!! ( { } t Z -caocal yte, coepodg ege value λ ae atfed Aage the budle poduced by the decopoto of the eultat (8, to ceae the powe of the paaete λ ad deote the opeato coeffcet of λ though T ad the odule of chaactetc ube of opeato S, S, S though µ S, µ, µ, coepodgly S, S, S ae defed below Theoe3 Let be, > S K A B K σ, p elfadot copletely cotuou opetato ad oe of followg codto ae fulflled p, p a < p <, l ( µ opeato ( T S,, KeS { ϑ} ae bouded, b, ( p < < l < p p µ opeato ( T S ae the copletely cotuou µ of Re ( A( µ ( λ, B( µ The chaactetc value { } ca be aaged a equece uch that fo oe ceag equece { } ( of potve tege, ad fo all eleet f, f,, f of the pace H, atfyg the codto f R( S, we have the expao whee f p ( ( a Z p the ube of dffeet egevecto of ultplcty coepodg to egevalue λ µ Poof of the Theoe3 If ay of the codto a, b atfed the codto of Theoe ae fulflled, theefoe, ultple ba of ege ad aocated vecto of the opeato budle Re ( A( λ, µ, B( λ, µ ext Stude coducted the pape [4],[5] ad [6] how that the egevecto of the budle (8(decopoto of eultat ae ethe egevecto o aocated vecto of the yte ( the defto of whch thee o dffeetato o λ The aocated vecto of the budle (8 (decopoto of the eultat (7 ae alo egevecto of the yte ( The eult of the theoe ea f the codto a, b of the theoe hold opeato pecl ( have the coo pot of pecta ( the fte deoal Hlbet pace H H the coo pot of pecta the coo egevalue Really, each codto a, b of the theoe ea the e Re ( A( λ, µ, B( λ, µ Code the decopoto of the eultat (8 whch the opeato

5 Pue ad Appled Matheatc Joual 5; 4(4-: pecl the paaete λ Coequetly, ege ad aocated vecto of th opeato pecl fo the ultple ba the teo poduct pace H H ad the ultplcty of th ba cocde wth the geatet degee of the paaete λ the opeato pecl Re ( A( λ, µ, B( λ, µ (the paaete λ fxed Eale the [4],[5],[6] t poved that the yte of ege ad aocated vecto of obtaed pecl, depedg o paaete λ ad cocde wth the yte of ege ad aocated vecto of the yte (Theefoe, the ege ad aocated vecto of the two paaete yte ( fo the Max(, ultple ba H H Thu, the yte of ege ad aocated vecto of ( ad the eultg expao of the eultat Re ( A( λ, µ, B( λ, µ ultaeouly Max(, ultple ba wth bacet Theoe4 Let be >, opeato, p S K A B K σ ha vee, elfadot copletely cotuou ad oe of followg codto p < p <, l ( µ a p TS ɶ (,,, ae bouded opeato p b < p <, l ( µ < p TS ɶ (,,, ae copletely coplete opeato atfed The the aeto of theoe 3 fulflled Theoe 5 Let be, opeato (, (, p S K A B A B K σ ha vee, elfadot copletely cotuou ad oe of followg codto < p <, ( µ c p l p TS ɶ (,,, ae bouded < p <, ( µ c p l p TS ɶ (,,, copletely cotuou opeato atfed The the tateet of the theoe 3 fulflled The ext theoe the pecal cae of the theoe Theoe6 Let all opeato A (coepodgly, B act fte deoal Hlbet pace H (coepodgly, H, ad oe of followg codto: a, ( Ke A B ϑ, A A B B,, ; b b > { } >, opeato A,, B ha vee, ad elfadot,opeato c ( A B ha vee ad elfadot, A B,, atfed The the ege ad aocated vecto of the yte ( fo Max(, ultple ba the teo poduct of the pace H H Rea Ege ad aocated vecto ae the ft copoet of the eleet of the eel of the eultat of opeato pecl A( λ, µ ad B( λ, µ Rea If the H H fte deoal Hlbet pace the the yte of ege ad aocated vecto of the yte ( cocde wth the yte of ege ad aocated vecto of the opeato pecl obtag a eult of decopoto of the eultat of opeato Theoe 4,5,6 ae poved by aalogy wth the poof of the Theoe 3 5 Cocluo I th pape t gve the codto of ultple ba of ege ad aocated vecto of two paaete yte of opeato Hlbet paceh H Refeece [] Ato FV Multpaaete pectal theoy BullAeMathSoc968, 74, -7 [] Bowe PJ Multpaaete pectal theoy Idaa Uv Math J,4, 3, 974 [3] Sleea BD Multpaaete pectal theoy Hlbet pace Pta Pe, Lodo, 978, p8 [4] Dhabaadeh RM Spectal theoy of two paaete yte fte-deoal pace Taacto of NAS Aebaa, v , p-8 [5] Dhabaadeh RM About expao o ege ad aocated vecto of opeato pecl polyoally depedg o paaete Scetfc ote of Aebaa State Uvety 964, 3,с75-8 [6] Dhabaadeh RMSpectal theoy of ultpaaete yte of opeato Hlbet pace, Taacto of NAS of Aebaa, -, 999, 33-4 [7] Bal AI Geeato of oto of Beutat ad Reultat DAN of U SSR, eph-ath ad tech of cece,98, ( Rua [8] Khayq (Хайниг Г Abtact aalog of Reultat fo two polyoal budle Fuctoal aalye ad t applcato, 977,, o 3, p94-95 [9] Dhabaadeh RM O oluto of olea algebac yte wth two vaable Pue ad Appled Matheatc, Joual, vol, No, pp 3-37, 3 [] Keldh M V O copletee of ege fucto of oe clae of lea oelfadot opeato Succee of Matheatcal Scece (УМН, 97, v7, ue4, pp5-47 ( Rua