IOSR Joul of Mthemtcs (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X Volume 11, Issue 1 Ve 1 (J - Feb 215), PP 13-18 wwwosjoulsog Spectl Cotuty: (p, ) - Α P Ad (p, k) - Q D Sethl Kum 1 d P Mhesw Nk 2 1 Post Gdute d Resch Deptmet of Mthemtcs, Govemt Ats College (Autoomous), Combtoe - 641 18, mlndu, Id 2 Deptmet of Mthemtcs, S Rmksh Egeeg College, Vttmlplym, Combtoe - 641 22, mlndu, Id Abstct: A opeto B s sd to be bsolute - (p, ) - poml opeto f p p fo ll H d fo postve l umbe p > d >, whee = U s the pol decomposto of I ths ppe, we pove tht cotuty of the set theoetc fuctos spectum, Weyl spectum, Bowde spectum d essetl sujectvty spectum o the clsses cosstg of (p, k) - qushypooml opetos d bsolute - (p, ) - poml opetos Keywods: bsolute - (p, ) - poml opeto, Weyl's theoem, Sgle vlued eteso popety, Cotuty of spectum, Fedholm, B Fedholm I Itoducto d Pelmes Let H be fte dmesol comple Hlbet spce d B deote the lgeb of ll bouded l opetos ctg o H Evey opeto c be decomposed to = U wth ptl somety U, whee = I ths ppe, = U deotes the pol decomposto stsfyg the keel codto N(U) = N( ) Ymzk d Ygd [23] toduced bsolute - (p, ) - poml opeto It s futhe geelzto of the clsses of both bsolute - k - poml opetos d p - poml opetos s pllel cocept of clss A(p, ) A opeto B s sd to be bsolute - (p, ) - poml opeto, deoted by (p, ) - Α P, f p p fo evey ut vecto o equvletly p p fo ll H d fo postve l umbes p > d > It s lso poved tht = U s bsolute - (p, ) - poml opeto fo p > d > f d oly f U 2 p U - (p + ) p 2 + p p + I fo ll l Evdetly, (k, 1) - Α P opeto s bsolute - k - poml; (p, p) - Α P opeto s p - poml; (1, 1) - Α P opeto s poml [23] A opeto B some < p 1 d tege k 1 f k ( 2 p - 2 p ) k Evdetly, s sd to be (p, k) - qushypooml opeto, deoted by (p, k) - Q, fo (1, k) - Q opeto s k - qushypooml; (1, 1) - Q opeto s qushypooml; (p, 1) - Q opeto s k - qushypooml o qus - p - hypooml ([8], [1]), (p, ) - Q opeto s p - hypooml f < p < 1 d hypooml f p = 1 If B, we wte N() d R() fo ull spce d ge of, espectvely Let ( ) = dm N() = dm ( -1 ()), ( ) = dm N( ) = dm (H / ), ( ) deote the spectum d ( ) deote the ppomte pot spectum he ( ) s compct subset of the set C of comple umbes he fucto vewed s fucto fom B to the set of ll compct subsets of C, wth ts husdoff metc, s kow to be uppe sem - cotuous fucto [14, Poblem 13], but t fls to be cotuous [14, Poblem 12] DOI: 1979/5728-11111318 wwwosjoulsog 13 Pge
Spectl cotuty: (p, ) - Α P d (p, k) - Q Also we kow tht s cotuous o the set of oml opetos B eteded to hypooml opetos [14, Poblem 15] he cotuty of o the set of qushypooml opetos ( B) hs bee poved by Eeveko d Djodjevc [1], the cotuty of o the set of p - hypooml hs bee poved by Duggl d Djodjevc [9], d the cotuty of o the set of G 1 - opetos hs bee poved by Luecke [17] A opeto B s clled Fedholm f t hs closed ge, fte dmesol ull spce d ts ge hs fte co - dmeso he de of Fedholm opeto s gve by () = ( ) - ( ) he scet of, sc (), s the lst o - egtve tege such tht - () = - ( + 1) () d the descet of, dsc (), s the lst o - egtve tege such tht = ( + 1) We sy tht s of fte scet (esp, fte descet) f sc ( - I) < (esp, dsc ( - I) < ) fo ll comple umbes A opeto s sd to be left sem - Fedholm (esp, ght sem - Fedholm), (esp, ) f H s closed d the defcecy de ( ) = dm ( -1 ()) s fte (esp, the defcecy de ( ) = dm (H \ H) s fte); s sem - Fedholm f t s ethe left sem - Fedholm o ght sem - Fedholm, d s Fedholm f t s both left d ght sem - Fedholm he sem - Fedholm de of, d (), s the umbe d () = ( ) - ( ) A opeto s clled Weyl f t s Fedholm of de zeo d Bowde f t s Fedholm of fte scet d descet Let C deote the set of comple umbes he Weyl spectum ( w ) d the Bowde spectum ( b ) of e the sets ( w ) = { C : - s ot Weyl} d ( b ) = { C : - s ot Bowde} Let ( ) deote the set of Resz pots of (e, the set of C such tht - s Fedholm of fte scet d descet [7]) d let ) d so ( ) deotes the set of ege vlues of of fte geometc multplcty d solted pots of the spectum he opeto B s sd to stsfy Bowde's theoem f ( ) \ w( ) = ( ) d s sd to stsfy Weyl's theoem f ( ) \ w( ) = ) I [15], Weyl's theoem fo mples Bowde's theoem fo, d Bowde's theoem fo s equvlet to Bowde's theoem fo Bek [5] hs clled opetot B(X ) s B - Fedholm f thee ests tul umbe fo whch the duced opeto : (X) (X) s Fedholm We sy tht the B - Fedholm opeto hs stble de f d ( - ) d ( - ) fo evey, the B - Fedholm ego of () - ( ) he essetl spectum e( ) of B s the set B = { C : - s ot Fedholm} Let cc ( ) deote the set of ll ccumulto pots of ( ), the e() w( ) b( ) e( ) cc ( ) Let ( ) be the set of C such tht s solted pot of ( ) d < ( ) <, whee ( ) deotes the ppomte pot spectum of the opeto he ( ) ( ) ( ) We sy tht - Weyl's theoem holds fo f ( w () w = ( ) \ ( ), whee ) deotes the essetl ppomte pot spectum of (e, () w = ( K) : K K( H) wth K deotg the dl of compct opetos o H) Let = { B : ( ) < d s closed} d = { B : ( ) < } deote the semgoup of uppe sem Fedholm d lowe sem Fedholm opetos B d let = { : d () } he ) s the complemet C of ll those fo whch ( - ) w ( [19] he cocept of - Weyl's theoem ws toduced by Rkocvc [2] he cocept sttes tht - Weyl's theoem fo Weyl's theoem fo, but the covese s geelly flse Let ( b ) deote the Bowde essetl ppomte pot spectum of () b = { ( K) : K K d K K } = { C : - o sc ( - ) = } DOI: 1979/5728-11111318 wwwosjoulsog 14 Pge
Spectl cotuty: (p, ) - Α P d (p, k) - Q the ( w ) ( b ) We sy tht stsfes - Bowde's theoem f ( b ) = ( w ) [19] A opeto B hs the sgle vlued eteso popety t C, f fo evey ope dsc D ceteed t the oly lytc fucto f : D H whch stsfes ( - ) f( ) = fo ll D s the fucto f vlly, evey opeto hs SVEP t pots of the esolvet ( ) = C / ( ) ; lso hs SVEP t so ( ) We sy tht hs SVEP f t hs SVEP t evey C I ths ppe, we pove tht f { } s sequece of opetos the clss (p, k) - Q o (p, ) - Α P whch coveges the opeto om topology to opeto the sme clss, the the fuctos spectum, Weyl spectum, Bowde spectum d essetl sujectvty spectum e cotuous t Note tht f opeto hs fte scet, the t hs SVEP d ( ) ( ) fo ll [1, heoem 38 d 34] Fo subset S of the set of comple umbes, let S { : S} cojugte II M Results Lemm 21 () If (p, k) - Q, the sc ( - ) k fo ll () If (p, ) - Α P, the hs SVEP Poof: () Refe [13, Pge 146] o [22] () Refe [21, heoem 28] whee deotes the comple umbe d deotes the Lemm 22 If (p, k) - Q (p, ) - Α P d so ( ), the s pole of the esolvet of Poof: Refe [22, heoem 6] d [21, Poposto 21] Lemm 23 If (p, k) - Q (p, ) - Α P, the stsfes - Weyl's theoem Poof: If (p, k) - Q, the hs SVEP, whch mples tht ( ) = ( ) by [1, Coolly 245] he stsfes Weyl's theoem e, ( ) \ w( ) = ( ) = ) by [13, Coolly 37] Sce ) DOI: 1979/5728-11111318 wwwosjoulsog 15 Pge = ( ) = ( ), ( ) = ( ) = ( ) d w( ) = ( w ) = ( ) by [3, heoem 36()], ( ) \ ( ) = ( ) Hece f (p, k) - Q, the stsfes - Weyl's theoem If (p, ) - Α P, the by [21, heoem 218], stsfes - Weyl's theoem Coolly 24 If (p, k) - Q (p, ) - Α P d ( ) \ ( ) so ( ) Lemm 25 If (p, k) - Q (p, ) - Α P, the sc ( - ) < fo ll Poof: Sce - s lowe sem - Fedholm, t hs SVEP We kow tht fom [1, heoem 316] tht SVEP mples fte scet Hece the poof Lemm 26 [6, Poposto 31] If s cotuous t B, the s cotuous t Lemm 27 [12, heoem 22] If opeto B hs SVEP t pots w ( ), the s cotuous t w s cotuous t b s cotuous t Lemm 27 If { } s sequece (p, k) - Q o (p, ) - Α P whch coveges om to, the s pot of cotuty of Poof: We hve to pove tht the fucto s both uppe sem - cotuous d lowe sem - cotuous t But by [11, heoem 21], we hve tht the fucto s uppe sem - cotuous t So we hve to
pove tht s lowe sem - cotuous t e, tht Spectl cotuty: (p, ) - Α P d (p, k) - Q ( ) lm f ( ) Assume the cotdcto s ot lowe sem - cotuous t he thee ests >, tege, ( ) ( ) = fo ll Sce (H fo ll, the followg mplctos holds: - eghbouhood ( ) of such tht ll mples - ) d( - ), ( - ) < d ( - ) H s closed d( - ), ( - ) < d( - ) =, ( - ) < ( - ) < (Sce (p, k) - Q (p, ) - Α P d( - ) by Lemm 21 d Lemm 25 ) ( ) d ( ) fo fo ll he cotuty of the de mples tht d ( - ) = lm d ( - ) =, d hece tht( - ) s Fedholm wth d ( - ) = But the - s Fedholm wth d ( - ) = - s lowe sem - cotuous t Hece the poof, whch s cotdcto heefoe heoem 29 If { } s sequece (p, k) - Q o (p, ) - Α P whch coveges om to, the s cotuous t Poof: Sce hs SVEP by Lemm 21, ( ) ( ) Evdetly, t s eough f we pove tht ( ) lm f ( ) fo evey sequece { } of opetos (p, k) - Q o (p, ) - Α P such tht coveges om to Let ( ) he ethe ( ) o If ( ), the poof follows, sce ( ) lm f ( ) lm f ( ) ( ) \ ( ) If ( ) \ ( ), the so ( ) by Coolly 24 Cosequetly, lm f ( ) e, lm f ( ) fo ll by [16, heoem IV 316], d thee ests sequece ( ), such tht by pplyg Lemm 26, we obt the esult coveges to Evdetly lm f ( ) Hece lm f, ( ) Now Coolly 21 If { } s sequece (p, k) - Q o (p, ) - Α P whch coveges om to, the, w d b e cotuous t Poof: Combg heoem 29 wth Lemm 27 d Lemm 28, we obt the esults Let ( ) = { : - s ot sujectve} deote the sujectvty spectum of d let s = { : -, d ( - ) } he the essetl sujectvty spectum of s the set ( ) { : - } = Coolly 211 If { } s sequece (p, k) - Q o (p, ) - Α P whch coveges om to, the es () s cotuous t Poof: Sce hs SVEP by Lemm 21, ( ) = ( ) es by [1, heoem 365 ()] he by pplyg Lemm 28, we obt the esult Let K B deote the dl of compct opetos, B / K the Clk lgeb d let : B B / K deote the quotet mp he B / K beg C - lgeb, thee ests Hlbet spce H 1 d es DOI: 1979/5728-11111318 wwwosjoulsog 16 Pge
Spectl cotuty: (p, ) - Α P d (p, k) - Q sometc - somophsm : B / K B(H 1 ) such tht the essetl spectum ( e ) = ( ( )) B s the spectum of ( ) ( B(H 1 )) I geel, ( e ) s ot cotuous fucto of Coolly 212 If of s sequece (p, k) - Q o (p, ) - Α P whch coveges om to ( ), the ( e ) s cotuous t Poof: If B s essetlly (p, k) - Q o (p, ) - Α P, e, f (p, k) - Q o (p, ) - Α P, d the sequece { } coveges om to, the ( ) ( B(H 1 )) s pot of cotuty of by heoem 29 Hece e s cotuous t, sce ( e ) = ( ( )) () Let H( ()) deote the set of fuctos f tht e o - costt d lytc o eghbouhood of Lemm 213 Let B(X ) be vetble (p, ) - Α P d let f H( ()) he ( f ( )) f ( ( )), d f the B - Fedholm opeto hs stble de, the ( f ( )) f ( ( )) Poof: Let B(X ) be vetble (p, ) - Α P, let f H( ()), d let g() be vetble fucto such tht f ( ) ( 1)( ) g( ) If f ( ( )), the f ( ) ( 1)( ) g( ) d ( ), = 1, 2,, Cosequetly, - s B - Fedholm opeto of zeo de fo ll = 1, 2,,, whch, by [5, heoem 32], mples tht f () s B - Fedholm opeto of zeo de Hece, ( f ( )) Suppose ow tht hs stble de, d tht ( f ( )) he, f ( ) ( 1)( ) g( ) s B - Fedholm opeto of zeo de Hece, by [4, Coolly 33], the opeto g() d -, = 1, 2,,, e B - Fedholm d = d (f() - ) = d ) + + d ) + d g() ( 1 ( ( Sce g() s vetble opeto, d (g()) = ; lso d ) hs the sme sg fo ll = 1, 2,, hus d ( ) =, whch mples tht ( ) f ( ( )) Lemm 214 Let B(X ) C such tht - s B - Fedholm fo ll = 1, 2,,, d hece be vetble (p, ) - Α P hs SVEP, the d ( - ) fo evey Poof: Sce hs SVEP by [21, heoem 28] he M hs SVEP fo evey vt subspces M X of Fom [4, heoem 27], we kow tht f - s B - Fedholm opeto, the thee est - vt closed subspces M d N of X such tht X = M N, ( - ) M s Fedholm opeto wth SVEP d (- ) N s Nlpotet opeto Sce d ( - ) M } by [18, Poposto 22], t follows tht d ( - ) Refeeces [1] Ae P, Fedholm d Locl Spectl heoy wth Applctos to Multples, Kluwe Acd Pub, 24 [2] Ae P d Moslve O, Opetos whch do ot hve the sgle vlued eteso popety, J Mth Al Appl, 25 (2), 435 -- 438 [3] Apostol C, Flkow L A, Heeo D A, Voculescu D, Appomto of Hlbet spce opetos, Vol II, Resch Notes Mthemtcs, 12, Ptm, Bosto (1984) [4] Bek M, O clss of qus - Fedholm opetos, Ite Equt opeto heoy, 34 (1999), 244 -- 249 [5] Bek M, Ide of B - Fedholm opetos d geelzto of the Weyl theoem, Poc Ame Mth Soc, 13 (22), 1717 -- 1723 [6] Buldo L, Nocotuty of the djot of opeto, Poc Ame Mth Soc, 128 (2), 479 -- 486 DOI: 1979/5728-11111318 wwwosjoulsog 17 Pge
Spectl cotuty: (p, ) - Α P d (p, k) - Q [7] Cdus S R, Pfffebege W E, Betm Y, Clk lgebs d lgebs of opetos o Bch spces, Mcel Dekke, New Yok, 1974 [8] Cmpbell S I, Gupt B C, O k - qushypooml opetos, Mth Jpoc, 23 (1978), 185 -- 189 [9] Djodjevc S V, O the cotuty of the essetl ppomte pot spectum, Fct Mth Ns, 1 (1995), 69 -- 73 [1] Djodjevc S V, Cotuty of the essetl spectum the clss of qushypooml opetos, Vesk Mth, 5 (1998), 71 -- 74 [11] Djodjevc S V, Duggl B P, Weyl's theoem d cotuty of spect the clss of p - hypooml opetos, Stud Mth, 143 (2), 23 -- 32 [12] Djodjevc S V, H Y M, Bowde's theoem d spectl cotuty, Glsgow Mth J, 42 (2) 479 -- 486 [13] Duggl B P, Resz pojectos fo clss of Hlbet spce opetos, L Alg Appl, 47 (25), 14 -- 148 [14] Hlmos P R, A Hlbet spce poblem book, Gdute ets Mthemtcs, Spge - Velg, New Yok, 1982 [15] Hte R E, Lee W Y, Aothe ote o Weyl's theoem, s Ame Mth Soc 349 (1997), 2115 -- 2124 [16] Kto, Petubto theoy fo L opetos, Spge - velg, Bel, 1966 [17] Luecke G R, A ote o spectl cotuty d spectl popetes of essetlly G 1 opetos, Pc J Mth, 69 (1977), 141 -- 149 [18] Oudgh, Weyl's theoem d Bowde's theoem fo opetos stsfyg the SVEP, Stud Mthemtc, 163 (24), 85 -- 11 [19] Rkocevc V, O the essetl ppomte pot spectum II, Mt Vesk, 36(1) (1984), 89 -- 97 [2] Rkocevc V, Opetos obeyg - Weyl's theoem, Rev Roume Mth Pues Appl, 34 (1989), 915 -- 919 [21] Sethlkum D, Mhesw Nk P, Absolute - (p, ) - poml opetos, Itetol J of Mth Sc & Egg Appls (IJMSEA), Vol 5, No III (My, 211), 311 -- 322 [22] hsh K, Uchym A, Cho M, Isolted pots of spectum of (p, k) - qushypooml opetos, L Alg Appl, 382 (24), 221 -- 229 [23] Ymzk d M Ygd, A futhe geelzto of poml opetos, Scete Mthemtce, Vol 3, No: 1 (2), 23 -- 31 DOI: 1979/5728-11111318 wwwosjoulsog 18 Pge