The Products of Regularly Solvable Operators with Their Spectra in Direct Sum Spaces

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1 Advces Pure Mhemcs h://dxdoorg/436/m3346 Pulshed Ole July 3 (h://wwwscrorg/ourl/m) he Producs of Regulrly Solvle Oerors wh her Secr Drec Sum Sces Sohy El-Syed Irhm Derme of Mhemcs Fculy of Scece Beh Uversy Beh Egy Eml: sohyelsyed_55@homlcom Receved Jury 3; revsed Mrch 4 3; cceed Arl 9 3 Coyrgh 3 Sohy El-Syed Irhm hs s oe ccess rcle dsrued uder he Creve Commos Aruo Lcese whch erms uresrced use dsruo d reroduco y medum rovded he orgl wor s roerly ced ABSRAC I hs er we cosder he geerl qus-dfferel exressos ech of order wh comlex coef- fces d her forml dos o he ervl L I of I s show drec sum sces w fucos defed o ech of he sere ervls wh he cses of oe d wo sgulr ed-os d whe ll solu os of he equo wu d s do w v re L w (he lm crcle cse) h ll well-osed exesos of he mml oeror hve resolves whch re Hler- Schmd egrl oerors d cosequely hve wholly dscree secrum hs mles h ll he regulrly solvle oerors hve ll he sdrd essel secr o e emy hese resuls exed hose of formlly symmerc exresso suded [-] d hose of geerl qus-dfferel exressos [-9] Keywords: Produc of Qus-Dfferel Exressos; Regulr d Sgulr Edos; Regulrly Solvle Oerors; Essel Secr; Hler-Schmd Iegrl Oerors Iroduco he oerors whch fulfll he role h he self-do d mxml symmerc oerors ly he cse of formlly symmerc exresso re hose whch re regulrly solvle wh resec o he mml oerors d geered y geerl ordry qus-dfferel exresso d s forml do resecvely he mml oerors d form do r of closed desely-defed oerors he uderlyg -sce h s Such oeror S ssfes S d for some he oe- ror S I s Fredholm oeror of zero dex hs mes h S hs he desrle Fredholm roery h he equo S Iu f hs soluo f d oly f f s orhogol o he soluo sce of S Iu d furhermore he soluo sce of S Iu d S Iv hve he sme f- e dmeso hs oo ws orglly due o Vs [] Ahezer d Glzm [] d mr [] re showed h he self-do exeso S of he mml oeror geered y formlly symmerc dfferel exresso wh mxml defcecy dces hve resolves whch re Hler-Schmd egrl oerors d cosequely hve wholly dscree secrum I [5689] Irhm exed her resuls for geerl ordry qus-dfferel exresso of -h order wh comlex coeffces he sgulr cse I [38] Ever d Zel cosdered he rolem of egrle squre soluos of roducs of dfferel exressos d vesge he relosh ewee he defcecy dces of geerl symmerc dfferel exressos d hose of he rod uc exresso d [7] Irhm cosdered he rolem of he o secr d regulry felds for roducs of geerl qus-dfferel oerors Our oecve hs er s geerlzo of he resuls [675689] for he roduc qus-dffer el oerors d her sec- L I of fuc- r drec sum sces w Coyrgh 3 ScRes

2 46 S E-S IBRAHIM os defed o ech of he sere ervls wh he cses of oe d wo sgulr ed-os d whe ll soluos of he roduc equos wu d w v re L wi for some (d hece ll C he ed-os of I ssumed o e regulr or my e sgulr We del hroughou hs er wh qus-dfferel exresso of rrry order defed y Sh- Zel mrces [4] d he mml oeror ge- ered y w I where w s osve wegh fuco o he uderlyg ervl I he edos d of I my e regulr or sgulr edos oo d Prelmres We eg wh ref survey of do rs of oerors d her ssoced regulrly solvle oerors; full reme my e foud [7 Cher III] [ 568] he dom d rge of ler oeror cg Hler sce H wll e deoed y D d R resecvely d wll deoe s ull sce he ully of wre ul s he dmeso of d he defcecy of wre def s he co-dmeso of R H ; hus f s desely defed d R s closed he def ull he Fredholm dom of s ( he oo of [3]) he oe suse of 3 cossg of hose vlues of whch re such h I s Fredholm oeror where I s he dey oeror H hus 3 f d oly f I hs closed rge d fe ully d defcecy he dex of I s he umer d I ul I def I hs eg defed for 3 wo closed desely defed oerors A d B cg Hler sce H re sd o form do r f A B d cosequely B A ; equvlely Ax y ( xby ) for ll x DA d y DB where deoes he er-roduc o H Defo : he feld of regulry Π A of A s he se of ll for whch here exss osve cos K such h A I x K x for ll x D A () or equvlely o usg he Closed Grh heorem ul A I d R A I s closed he o feld of regulry Π A B of A d B s he se of whch re such h Π A ΠB d oh def A I d def B I re fe A do r A d B s sd o e comle f Π AB Defo : A closed oeror S H s sd o e regulrly solvle wh resec o he comle do r of A d B f A S B d Π AB 4 S where 4S : 3S ds I Defo 3: he resolve se S of closed oeror S H cosss of he comlex umers for whch S I exss s defed o H d s ouded he comleme of S s clled he secrum of S d wre S he o secrum S couous secrum c S d resdul secrum r S re he followg suses of S (see [5] d [6]) S S : S I s o ecve e he se of egevlues of S ; r : RS I RS I H c S S S I S s ecve s ec e S : S I v R S I H For closed oeror S we hve S S S S () c r A mor suse of he secrum of closed desely defed oeror S H s he so-clled essel secrum he vrous essel secr of S re defed s [ Cher 9] o e he ses: S S 345 (3) where d S 3 4 S hve ee defed erler Defo 4: For wo closed desely defed oerors A d B cg H f A S B d he resolve se S of S s oemy (see []) S s sd o e well-osed wh resec o A d B oe h f A S B d S he Π A d S ΠB so h f def A I d def B I re fe he A d B re comle hs cse S s regulrly solvle wh resec o A d B he ermology regulrly solvle comes from Vs s er [] whle he oo of well osed ws roduced y Zhhr hs wor o J -self do oerors [] Gve wo oerors A d B oh cg Hler sce H we wsh o cosder he roduc oeror AB hs s defed s follows ABx ABx xdab D AB x D B Bx D A d for ll I my he geerl h ; (4) D AB cos oly he ull eleme of H However he cse of my dfferel oerors he doms of he roduc wll e Coyrgh 3 ScRes

3 S E-S IBRAHIM 47 dese H he ex resul gves codos uder whch he defcecy of roduc s he sum of he defceces of he fcors I s geerlzo of h [3 heorem A] d [8] Lemm 5 (cf [7 Lemm 3]) Le A d B e closed oerors wh dese doms Hler sce H Suose h Π A B he AB s closed oeror wh dese dom d def AB I def A I def B I (5) Evdely Lemm 5 exeds o he roduc of y fe umer of oerors A A A 3 Qus-Dfferel Exressos Drec Sum Sces he qus-dfferel exressos re defed erms of Sh-Zel mrx F o ervl I he se ZI of Sh-Zel mrces o I cosss of -mrces F frs whose eres re comlex-vlued fucos o I whch ssfy he followg codos: f L I r s rs loc frr e o I r frs e o I r s (3) For F ZI he qus-dervves ssoced wh F re defed y: y : y r r r s y : frr y frsy r s y : y frs y s (3) where he rme ' deoes dffereo he qus-dfferel exresso ssoced wh F s gve y: : y (33) hs eg defed o he se: r loc V : y: y AC I r where ACloc I deoes he se of fucos whch re soluely couous o every comc suervl of I he forml do of s defed y he mrx F gve y: y for ll y V : (34) r loc V : y: y AC I r r where y he qus-dervves ssoced wh he mrx Z I F r F s rs s r F f f for ech r d s (35) oe h: F d so We refer o [39] d [7-] for full ccou of he ove d suseque resuls o qus-dfferel exressos For u V v V d I we hve Gree s formul where v u u v dx uv uv rs r r uv x u x v x r v u u u J x; v (36) (37) see [49] d [4-8] Le he ervl I hve edos d le w : I w L I e o-egve wegh fuco wh loc d w (for lmos ll x I ) he H I deoes he Hler fuco sce of equvlece clsses of Leesgue mesurle fucos such h w I f ; he er-roduc s defed y: he equo f g : w f x g x dx I w f g L I (38) u w u o I (39) s sd o e regulr he lef ed-o X ll w f L X rs rs ; f for Oherwse (39) s sd o e sgulr If (39) s regulr oh ed-os he s sd o e regulr; hs cse we hve w f L rs rs ; Coyrgh 3 ScRes

4 48 S E-S IBRAHIM We shll e cocered wh he cse whe s regulr ed-o of (39) he ed-o eg llowed o e eher regulr or sgulr oe h vew of (35) ed-o of I s regulr for (39) f d oly f s regulr for he equo v w v o I (3) oe h regulr ed-o sy r r u v r s defed for ll u V v V Se: w D : u: u V u d w u L w D : v: v V v d w v L (3) he susces D d D of re doms of he so-clled mxml oerors d resecvely defed y: u: w u u D d v: w v v D For he regulr rolem he mml oerors d re he resrcos of w u d w v o he susces: D : r r u: ud u u D : r r v: vd v v resecvely he susces D d dese d d (3) D re re closed oerors (see [59 Seco 3] [36]) I he sgulr rolem we frs roduce he oer- ors d ; eg he resrco of w o he susce: D : u : u D su u d wh desely-defed d closle w defe he mml oerors d (33) defed smlrly hese oerors re L ; d we o e her resecve closures (see [369]) We deoe he doms of d y D d D resecvely I c e show h: r ud u r ; r vd v r ; (34) ecuse we re ssumg h s regulr ed-o Moreover oh regulr d sgulr rolems we hve ; (35) see [8 Seco 5] he cse whe d comre wh reme [ Seco III3] d [6] geerl cse I he cse of wo sgulr ed-os he rolem o s effecvely reduced o he rolems wh oe sgulr ed-o o he ervls c d c c ; d where We deoe y ; he mxml oerors wh doms D ; d D ; d deoe ; ; he closures of he oerors ; ; defed y: D ; d d : u: ud ; su u (36) o he ervls c d c resecvely see e ([593] d [6]) Le he orhogol sum s: ; ; L L c L c w w w s desely-defed d closle L d s closure s gve y: w ; ; Also ull I ull ; I ull ; I def I ; ; def I def I Coyrgh 3 ScRes

5 S E-S IBRAHIM 49 d R ; I s closed f d oly f R I R ; I closed hese resuls mly rculr h d Π Π ; Π ; re oh We refer o [ Seco 34] [6] d [8] for more dels Remr 3: If S s regulrly solvle exeso d S s regulrly solvle exe- of ; so of ; he S S S s re- gulrly solvle exeso of We refer o [ Seco 34] [6] d [8] for more dels ex we se he followg resuls; he roof s smlr o h [] [ Seco 34] [6] d [8] heorem 3: ; ; d dm D D I he If Π 3 d I d rculr f Π def I def ; I def ; I def ; I def ; I Remr 33: I c e show h D : r u: ud u c D r : v: vd v c ; (37) see [ Seco 34] Le H e he drec sum H H L w he elemes of H wll e deoed y f f f f wh f H f H f H Remr 34: Whe I I ; he drec L c e urlly defed sum sce w wh he sce I where w w o I hs remr s of sgfcce whe I my e e s sgle ervl see [5] d [7] We ow eslsh y [835] d [8] d some furher oos D D D D (38) D D D D f f f f ; f D f D f D g g g g ; g D g D g D Also f f f f; f D f D f D gd gd g D g g g g ; We summrze few ddol roeres of he form of Lemm Lemm 35: We hve ) I rculr D D D D ull I ull I ) ull I ull I re gve y: 3) he defcecy dces of def I def I for ll Π for ll Π def I def I Proof: Pr () follows mmedely from he defo of d from he geerl defo of do oeror he oher rs re eher drec cose- Coyrgh 3 ScRes

6 4 S E-S IBRAHIM queces of r () or follow mmedely from he defos Π Lemm 36: For def I def I s cos d def I def I I he rolem wh oe sgulr ed-o def I def I for ll Π I he regulr rolem def I def I for ll Π Proof: he roof s smlr o h [ Lemm 4] [7] d [9] d herefore omed Lemm 37: Le e closed desely-defed oeror o H he Π Proof: he roof follows from Lemm 35 d sce R I s closed f d oly f R I re closed Remr 38: If S s regulrly s regulrly solvle solvle exeso of ; exeso of S he ; exeso of S S S 6] d [9] for more dels s regulrly solvle We refer o [ 4 he Produc Oerors he roof of geerl heorems wll e sed o he resuls hs seco We sr y lsg some roeres d resuls of qus-dfferel exressos For roofs he reder s referred o [387] d [9] d for comlex umer (4) A cosequece of Proeres (4) s h f he P P for P y olyoml wh comlex coeffces Also we oe h he ledg coeffces of roduc re he roduc of he ledg coeffces Hece he roduc of regulr dfferel exressos s regulr Lemm 4: (cf [9 heorem ]) Suose s regulr dfferel exresso o he ervl d Π he we hve ) he roduc oeror sely-defed d s closed de def I def I def I def I ) d oe r () h he come my e roer e he oerors d re o equl geerl Lemm 4: Le e regulr dfferel exressos o d suose h Π he (4) f d oly f he followg rl sero codos re ssfed: s f f ACloc where s s he order of roduc exresso d f L ogeher mly h: w f L w Furhermore f d oly f (43) d def I def I def I def I We wll sy h he roduc s rlly sered exressos wheever Proery (43) holds Lemm 43: For we hve Π Π Π Proof: Le Π (44) he from defo of he feld of regulry we hve Π d Π e ech of he oerors d hs closed rge d desely-defed o H wh fe defcecy dces Cosequely y Lemm 4 ech of he oerors I d Coyrgh 3 ScRes

7 S E-S IBRAHIM 4 Coyrgh 3 ScRes I hs closed rge d her defc- ecy dces re fe e Π he res of he roof follows from defo d Lemm 4 Corollry 44: Le s e regulr dfferel exresso o for If ll soluos of he dfferel equo Iu d Iv o re for d ; he ll soluos of I u d Iv o re for ll Proof: Le = order of = order of for he y Lemm 5 we hve def I def I for ll Π Hece y Lemm 4 we hve def I def I r order of orde of hus def I order of d cosequely ll soluos of he equos I u d Iv re Reeg hs rgume wh relced y we coclude h ll soluos of re w I v L he secl cse of Corollry 44 whe for d s symmerc ws eslshed [9] I hs cse s esy o see h he coverse lso holds If ll soluos of Iu re he ll soluos of Iu mus e I geerl f ll soluos of Iu re he ll soluos of Iu re sce hese re lso soluos of Iu If ll soluos of he do equo I v re lso he follows smlrly h ll soluos of Iv re Le e he soluos of he homogeeous equo ssfyg Iu (45) for ll r r for fxed he r s couous for d for fxed s ere Le deoe he soluos of he do homogeeous equo I v (46) r r ssfyg r r for ll Suose c By [8] soluo of he roduc equo ssfyg I u wf f (47) r u c r s gve y f s w s d s where d for ech sds for he comlex couge of s cos whch s deede of (u does deed geerl o ) he ex lemm s form of he vro of rmeers formul for geerl qus-dfferel equo s gve y he followg Lemm Lemm 45: Suose f fuco d loclly egrle s he soluo of he Equo (47) ssfyg: r r for r s gve y d f s w s s (48) for some coss where d re soluos of he Equos (45) d (46) resecvely s cos whch s deede of Proof: he roof s smlr o h [9357] Lemm 45 cos he followg lemm s secl cse Lemm 46: Suose f loclly egrle fuco d s he soluo of Equo (47) ssfyg: r r for r he r r r f s w s ds (49)

8 4 S E-S IBRAHIM for r We refer o [] for more dels Lemm 47: Suose h for some ll soluos of he equos (4) I I re he ll soluos of he equos (4) re for every comlex umer Proof: he roof s smlr o h [7 heorem 53] Lemm 48: Suose h for some comlex umer ll soluos of he equos (4) re Suose f he ll soluos of he equo (47) re for ll Proof: he roof s smlr o h [9 Lemm 38] Remr 49: Lemm 48 lso holds f he fuco f s ouded o Lemm 4: Le f Suose for some h: ) All soluos of I re L w r re ouded o ) for some r r he for y soluo of he equo I wf for ll Lemm 4: Suose h for some comlex ll soluos of he equo Iv re L c where c f L w Suose w he f swsd s uous for d for ll s co- Proof: I follows from Lemm 48; see [8 Lemm 36] Lemm 4: he o secr d of he roduc oerors d re emy Proof: See [7 heorem 46] Lemm 43: If I wh he for y he oeror hs closed rge zero ully d defcecy Hece 3 45 (4) Proof: he roof s smlr o h [ Lemm IX9]; see [7 Lemm 49] 5 he Produc Oerors Drec Sum Sces ex we cosder our ervl s I d deoe y d he mml d mxml oerors We see from (35) d Lemm 4 h d hece d form do r of closed desely defed oerors From Lemms 35 d 4 we hve he followg: Lemm 5: For Π we hve: ) ) ull I ull I ull I ull r I ull I ull I 3) he defcecy dces of re gve y: d def I def I def I def I def I def I Lemm 5: For Π def I def I s cos d def I def I I he rolem wh oe sgulr ed-o Coyrgh 3 ScRes

9 S E-S IBRAHIM 43 def I def I for ll Π I he regulr rolem def I def I Π Proof: he roof s smlr o h [ Lemm 4] [7] d [9] d herefore omed For Π we de fe rs d m s follows: for ll d Also : : r r def I def I r s s def I def I s (5) (5) m rs r s m For Π (53) he oer- ors whch re regulrly solvle wh resec o d re chrcerzed y he followg heorem whch roved for geerl qus-dfferel oeror [ heorem 5] heorem 53: For Le Π rs d m e defed y (5) d (5) d le r r m e rrry fucos ssfyg: ) r D re lerly deede modulo D d re lerly - r m D deede modulo D ) r r m ; he he se u: u D u u r m Φ Φ (54) s he dom of oeror S whch s regulrly solvle wh resec o d d he se v: v D v v r (54) s he dom of he oeror S ; moreover 4 S Coversely f S s regulrly solvle wh resec d d o he Π S 4 wh r d s defed y (5) d (5) here exs fucos r r m whch ssfy () d () d re such h (54) d (55) re he doms of S d S resecvely S s self-do f d oly f r s d r r m; S s J self-do f J J (J s comlex couge) s r r r m Proof: he roof s erely smlr o h of [3 68] d [9] d herefore omed d 6 he Cse of Oe Sgulr Ed-Po We see from (35) d Lemm 4 h d hece d form d- o r of closed closed-desely oerors By Lemms 3 3 [ Lemm 4] d [7 Lemm 3] def I def I s cos o he o feld of regulry Π d we hve h def I def I Coyrgh 3 ScRes

10 44 S E-S IBRAHIM for ll Π We shll use he oo uv lm uv x x (6) ud v D f s sgulr ed-o of d smlrly for uv f s sgulr oe h follows from (36) h hese lms exs for u D d v D sce he v u d u v re oh egrle y Cuchy-Schwrz equly We shll ow vesge he cse of oe sgulr ed-o h he resolve of ll well-osed exesos of he mml oeror d we show h he mxml cse e whe def I def I hese re- Π solve re egrl oerors fc hey re Hler-Schmd egrl oerors y cosderg h he fuco f e e s qudrclly egrle over he ervl heorem 6: Suose for oeror wh oe sgulr ed-o h for ll def I def I d le S e rrry closed oeror whch s well-osed exeso of he mml oeror d S he he resolves R d R of S d S resecvely re Hler-Schmd egrl oerors whose erels re couous fucos o d ssfy: for ll Π K s K s d K s w s w dsd (6) Remr A exmle of closed oeror whch s well-osed wh resec o comle do r s gve y he Vs exeso (see [67 heorem III33] [9] d [ heorem ]) oe h f S s well-osed he d re comle do r d S s regulrly solvle wh resec o d Proof: Le def I def I he we for ll Π choose fudmel sysem of soluos I I of he equos o so h (63) elog o L e hey re qudrclly egrle he ervl Le R S I e he resolve of y well-osed exeso of he mml oeror For f we u R f he sequely hs soluo I wf d co he form f s w s d s (64) (see f d L for some he for some coss Lemm 45) Sce w f L w for some d hece he egrl he rgh-hd of (64) wll e fe o deerme he coss le e ss for DS / D he ecuse DS S 4 S we hve from heorem 4 h o w (65) Coyrgh 3 ScRes

11 S E-S IBRAHIM 45 d hece from (64) (65) d o usg Lemm 46 we hve: d f s w s s By susug hese exressos o he codos (65) we ge: (66) d f s w s s hs mles h he sysem f swsd s (67) he vrle he deerm of hs sysem does o vsh (see [6 heorem 37] d [9]) If we solve he sysem (67) we o: where h d s s soluo of he sysem: h s f sws s (68) (69) h s Sce he deerm of he ove sysem (69) does o vsh d he fucos s re couous he ervl he he fucos h s re lso couous he ervl By susug formul (64) for he exres- sos we ge R f ( ) h s f swsds ow we u K s h s f swsds (6) h s for s h s for s (6) Formul (6) he es he form R f K s f s w s ds (6) for ll e R s egrl oeror wh he erel K s oerg o he fucos f soluos of he equo hs he form: I wg s s Smlrly he s s g w d where d (63) s re soluos of he equos (66) he rgume s efore leds o fr w Rg K s gwd o g L e R s egrl oeror wh he erel K s g where oerg o he fucos (64) Coyrgh 3 ScRes

12 46 S E-S IBRAHIM K s for s for s d h s h s h s soluo of he sysem h s From defos of R d R follows h (65) (66) R f g K s f s w s ds g w d K s gwd f swsds f R g (67) for y couous fucos f g H d y cosruco (see (6) d (65)) K s d K s re couous fucos o d (67) gves us for ll K s K s s Sce s for d for fxed como of whle for fxed K s s ler como of we hve K s w d (68) sk s s ler s K s w s d s s d (68) mles h he K s w s d s K ( s ) w s d s K s w d K s w d ow s cler from (69) h he fucos h s elog o sce h s s ler como of he fucos s whch le d hece h elog o Smlrly h elog o By he uer hlf of he formul (6) d (65) we hve: K s w s d s w d for he er egrl exss d s ler como of he roducs s d hese roducs re egrle ecuse ech of he fcors elogs o he y (68) d y he uer hlf of (65) K s w s ds w d Hece we lso hve: K s w s ds w d d d K s w w s s d he heorem s comleely roved for y well-osed exeso Remr 6: I follows mmedely from heorem 6 h f for oeror wh oe sgulr ed-o h def I def I d S s for ll Π well-osed wh resec o S d wh he R S I s Hler-Schmd egrl oeror hus s comleely couous oeror d cosequely s secrum s dscree d cosss of soled egevlues hvg fe lgerc (so geomerc) mullcy wh zero s he oly ossle o of ccumulo Hece he secr of ll well-osed oerors S re dscree e for S 345 (69) We refer o [67 heorem IX3] [5] [6] d [8] d for more dels 7 he Cse of wo Sgulr Ed-Pos For he cse of wo sgulr ed-os we cosder our ervl o e I d deoe y d he roduc of mml d mxml oerors We see from (35) d Lemm 4 h d hece d form do L r of closed desely-defed oerors w Coyrgh 3 ScRes

13 S E-S IBRAHIM 47 we defe rs d m s follows: d For Π : ; ; r r def I def I def I r r : ; ; s s def I def I def I s s mrs r r s s rsr s m m Also sce m (7) (7) (73) he y Lemm 5 we hve h m For oeror wh wo sgulr ed-os heorem 6 rems rue s erely h s ll well-osed exesos of he mml oeror he mxml cse e whe r r d s s (7) d (7) hve resolves whch re Hler-Schmd egrl oerors d cosequely hve wholly secrum d hece Remr 6 lso rems vld hs mles s Corollry 7 elow h ll he regulrly solvle oerors hve sdrd essel secr o e emy We refer o [675] d [6] for more dels ow we rove heorem 6 he cse of wo sgulr ed-os heorem 7: Suose for oeror wh wo sgulr ed-os h def I def I for ll Π d le S e rrry closed oeror whch s well-osed exeso of he mml oeror d S he he resolve R d R of S d S resecvely re Hler-Schmd egrl oerors whose erels re couous fucos o d ssfy (6) Proof: Le def I def I he we for ll Π choose fudmel sysem of soluos d s: o c o c o c o c of he equos (63) so h Φ d elog o (74) e hey re qudrclly egrle he ervl () Le R S I e he resolve of y well osed exeso S S S of he mml oeror For f L w c c we u R f he I wf d hece s (64) we hve R f f s w s d s for some coss o c o c (75) where By roceedg s heorem 6 we ge (68) (76) s h d s f s w s s where o o h c h h c (77) By susug (76) for he coss we ge for ll R f K s f s w s d s (78) Coyrgh 3 ScRes

14 48 S E-S IBRAHIM d K o o K s c K s K s c (79) s c e oed s (6) Smlrly ( ) d r w Rg K s gsws s fo g L K o K s c s K s c From (6) d (65) we hve h d (68) mles h K s w d o K ( s) w s d s s (7) K s w s d s K s w s d s K ( s ) w d K s w d he res of he roof s erely smlr o he corresodg r of he roof of heorem 6 We refer o [ 56756] d [8] for more dels Corollry 7: Le Π wh he def I def I for (7) S 3 (7) of ll regulrly solvle exesos S wh resec o he comle do r d Proof: Sce def I def I he we for ll Π hve from [5 heorem III35] h dm / D S D def I dm / D S D def I hus S s -dmesol exeso of d so y [ Corollry IX4] S 3 (73) From Lemms 4 d 43 we ge 3 (74) Hece y (74) we hve h S 3 Remr 73: If S s well-osed (sy he Vs exeso see []) we ge from (69) d (73) h 3 O lyg (73) g o y regulrly solvle exesos S uder cosdero hece (7) Corollry 74: If for some here re lerly deede soluos of he equos w u (75) w v d hece Π d Π 3 s he o es- where sel secr of s he o feld of regulry Π defed Proof: Sce ll soluos of he equos (75) for some he re def I def I From Le- for some Π mm 3 we hve h hs o egevl- ues d so I exss d s dom s closed susce of R I Coyrgh 3 ScRes

15 S E-S IBRAHIM 49 Hece sce s closed oeror he I s ouded d hece Π Smlrly Π herefore Π d hece def I def I From Corol- for ll Π lry 7 we hve for y regulrly solvle exeso S of h S we ge 3 d y (74) 3 Smlrly 3 Hece 3 Remr 75: If here re lerly deede soluos of he Equos (75) for some he he comlex le c e dvded o wo dso ses: Π 3 We refer o [67689] for more dels REFERECES [] I Ahezer d I M Glzm heory of Ler Oerors Hler Sce Frederch Uger Pulshg Co ew Yor 963 [] M mr Ler Dfferel Oerors ew Yor Ugr Pr I 967 Pr II 968 [3] W Ever d A Zel he umer of Iegrle Squre Soluos of Producs of Dfferel Exressos Proceedgs of he Royl Socey of Edurgh Edurgh Vol 76 A [4] W Ever d A Zel Geerlzed Symmerc Ordry Dfferel Exressos I he Geerl heory ew Archef Voor Wsude Vol o [5] A Krll d A Zel Sgulr Self-Ado Surm- Louvlle Prolems Jourl of Dfferel d Iegrl Equos Vol o [6] D Rce O he Loco of he Essel Secr d Regulry Felds of Comlex Surm-Louvlle Oerors Proceedgs of he Royl Socey of Edurgh Edurgh Vol 85A 98-4 do:7/s [7] D Rce O he Essel Secr of Ler d Order Dfferel Oerors wh Comlex Coeffces Proceedgs of he Royl Socey of Edurgh Edurgh Vol 9A do:7/s [8] A Zel Defcecy Idces of Polyomls Symmerc Dfferel Exressos II Proceedgs of he Royl Socey of Edurgh Edurgh Vol 73A o 974 (975) 3-36 [9] A Zel Formlly Self-Ado Qus-Dfferel Oerors Rocy Mou Jourl of Mhemcs Vol 5 o do:6/rmj [] S E Irhm O he Producs of Self-Ado Surm- Louvlle Dfferel Oerors Drec Sum Sces Jourl of Iformcs d Mhemcl Sceces Vol 4 o 93-9 [] D E Edmuds d W D Evs Secrl heory d Dfferel Oerors Oxford Uversy Press Oxford 987 [] W D Evs Regulrly Solvle Exesos of o- Self-Ado Ordry Dfferel Oerors Proceedgs of he Royl Socey of Edurgh Edurgh Vol 4 A [3] W D Evs d S E Irhm Boudry Codos for Geerl Ordry Dfferel Oerors Proceedgs of he Royl Socey of Edurgh Edurgh Vol 97A do:7/s [4] W Ever d D Rce Some Remrs o Ler Ordry Qus-Dfferel Exressos Jourl of Lodo Mhemcl Socey Vol 3 o [5] S E Irhm o-self-ado Qus-Dfferel Oerors wh Dscree Secr Rocy Mou Jourl of Mhemcs Vol 5 o do:6/rmm/874 [6] S E Irhm he Secr of Well-Posed Oerors Proceedgs of he Royl Socey of Edurgh Edurgh Vol 5 A [7] S E Irhm he Po Secr d Regulry Felds of Producs of Qus-Dfferel Oerors Id Jourl of Pure d Aled Mhemcs Vol 3 o [8] S E Irhm O he Essel Secr of Geerl Dfferel Oerors Il Jourl of Pure d Aled Mhemcs Y o [9] S E Irhm O he Essel Secr for Producs of he Geerl Qus-Dfferel Oerors d her Ados Ierol Jourl of Pure d Aled Mhemcs Vol 7 o [] M I Vs O Geerl Boudry Prolems for Ellc Dfferel Equos Amerc Mhemcl Socey rsl Vol o [] A Zhhr he heory of Exeso of J-Symmerc Oerors Ur M Z XI Vol Coyrgh 3 ScRes