Analyticity of Semigroups Generated by Singular Differential Matrix Operators

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1 pple Mahemacs,,, o:.436/am..436 Publshe Ole Ocober (hp:// alycy of Semgroups Geerae by Sgular Dffereal Mar Operaors Oul hme Mahmou S hme, el Sa Deparme of Mahemacs, College of Scece, ljouf Uversy, ljouf, Sau raba Deparme of Mahemacs, College of Eucao for Grls Sara Ebeah, Kg Khal Uversy, bha, Sau raba E-mal: sahme@ju.eu.sa, ael.sa@fsg.ru. Receve March 9, ; ugus 3, ; ugus 6, bsrac I hs paper we prove he aalycy of he semgroups geerae by some sgular ffereal mar op- eraors of he form B( ) =, he Baach space C ([, ], M ( )), C wh suable bouary coos. To llusrae he work a eample s scusse. Keywors: Dsspave Operaors, Posve Operaors, Specrum, alyc Semgroups, Evoluo Equaos. Irouco s we wll see he sequel he problem of characerzg operaor marces geerag srogly couous semgroups or aalyc semgroups s que ffcul. The ma problem cosss fg approprae assumpos o he mar eres allowg geeral resuls bu sll clug he cocree eamples we have m. The evoluo of a physcal sysem me s usually escrbe a Baach space by a al value problem of he form U () U ()=, (.) U() = U. Problem of ype (.) s well pose a Baach space f a oly f he operaor (, D( )) geeraes a C -semgroup ( T) > o. Here he soluo U () s gve by U ()= TU for he al aa U D( ). For operaor semgroups we refer o [-5] a o [6] for he heory of operaor marces. The harmoc aalyss for a class of ffereal operaors wh mar coeffces was reae [7,8]. I hs work we are erese a geeralzao of he aalycy a he posvy of he semgroup geerae by a mar sgular ffereal operaor (, D( )). smlar suy was realze [9] for a class of ffereal operaors wh mar coeffces a erface. For he scalar case we refer o []. Ths paper s orgaze as follows. I he seco seco we rouce some oaos a gve prelmares resuls. I he hr seco we vesgae some properes of he operaor (, D( )) parcular we prove ha s close, esely efe, sspave a sasfes he posve mmum prcple. Some specral properes of he operaor (, D( )) are obae hs seco. I seco fourh we esablshe he aalycy of he semgroup geerae by a operaor he form (, D( )). I hs case he problem (.) has a uque soluo for all U. I seco fve we prese a cocree eample of applcao of he resuls obae.. Noaos a Prelmary Resuls Le M ( C ) be he space of marces wh comple coeffces a I he ey mar. The orm of a mar = a j M ( C ), j wll be efe by = ma a (.), j The reaso of hs specal choce wll be jusfe Lemma. a Lemma.. = a M ( C ) s calle oega- mar j, j ve (resp posve) f all eres a j are real umbers a oegave (resp posve). I hs case we wre, (. > ) = a resp a we call j, j j Copyrgh ScRes.

2 84 O.. M. S. HMED ET L. he absolue value of. = a M ( C ) M For j, ( C ), we wll wre j a = aj, j B f B. a > B f B >. mar-value fucos s sa o be couous, ffereable or egrable f all s elemes are couous, ffereable or egrable fucos. If he mar ( ) s egrable over [ ab,, ] he b a b a ( ) ( ). For wo mar fucos ( )= aj ( ), ( )= j ( ), B b j j a, we shall wre he eglgbly ( )= o( B( )), f, aj ( )= obj ( ) for all, j. Smlarly we efe he oos of omao ( )= O( B( )) a he equvalece ( )~ B( ) (see for more eals []). Le be he space C([, ], M of couous, mar-value fucos o [, ]. O he space we efe he orm., by f = sup f( ), for all f. Noe ha he orme space (,. ) s a Baach space. We eoe by C k (], ], M, k =,, he space of all k mes couously ffereable marvalue fucos U efe o ], [ such ha ( p) lm U ( ) ess a fe for all p k. Coser a sgular seco orer ffereal operaor (, D( )) wh mar coeffces efe o by B( ) U = U U, where U a B are mar-value fucos, wh he oma D( ) = { U C([, ], M (], ], M, lmu( ) = }. We assume ha B s a real value couous a boue mar-value fuco o [, [ a f B() = I we a he assumpo D R B( ) B() (.) a eghborhoo of, for oegave cosas D agoal marces R a D. Here s he agoal mar wh agoal eres =, =,,, where, =,,, are he agoal eres of D. We wll ow recall some resuls eee he sequel. Lemma. Le M ( C). The followg properes hol ) f a oly f B for all B. ) B B, hece B B f. Lemma. Le, B M ( C). The followg properes hol ) B mples B. ) =. 3) E, where E = e, e =,, j. j, j j Proposo. Le be a oegave mar wh specral raus r( ). ) The resolve R(, ) s posve wheever > r ( ). ) If > r ( ), he R(, ) R(, ). Proof. ) We use he Neuma seres represeao for he resolve R(, )= ( ) = ( I ) = for > r( ). If he, for all, hece for > r( ), we have (, )= k R lm k k= sce he fe sums are posve a covergece hols every ery. ) For > r( ), we have R(, ) = lm k = k k lm = R(, ). = k k Theorem. ([], Perros Theorem) If s a oegave mar, he r( ) s a egevalue of wh posve egevecor. Defo. ) operaor ( SDS, ( )) o a Baach Lace (, ) s calle posve f Su for all u D( S) ={ v D( S), v }. ) semgroup ( T ) > o s posve f a oly f T s posve for all >. Remark. operaor ( SDS, ( )) efe o a ( K) space (K compac) sasfes he posve mmum prcple f for every u D( S), u posve a K, u ( )=, he ( Su)( ). Theorem. ([]) k Copyrgh ScRes.

3 O.. M. S. HMED ET L. 85 Le ( SDS, ( )) be he geeraor of a semgroup ( T ) > o CK ( ), he he semgroup s posve f a oly f ( SDS, ( )) sasfes he posve mmum prcple. 3. The Dagoal Case I hs seco we assume ha he mar fuco B s agoal. 3.. Characerzao of he Operaor (, D( )) The proofs of Proposo 3. a Proposo 3. ca be euce from he scalar case gve Proposo. a Proposo.3 from []. Proposo 3. Le I > B() a U C (], ], M. The U D( ) f a oly f U ([, ], M for some > a U() = U () =. Proposo 3. Le B() = I a U C (], ], M. The U D( ) f a oly f U C ([, ]), M for some >, U () = a lm ( U ( ) U ( )) =. Moreover, f U D( ) he U ( )= o(log E) a U ( )= ologe as, where E s he cosa mar rouce Lemma.. Proposo 3.3 The operaor (, D( )) s a esely efe, close, sspave a sasfes he posve mmum prcple. b ( ) b ( ) Proof. Pu B ( )= b ( ) a for =,,, le wh D( )= b ( ) u = u u { uc([, ], C) (], ], C), lm u( ) = }. The j, U = u D( ) f a oly f j uj D( ) for all, j =,,. Hece from ([]. Lemma.4), he operaor (, D( )) s a esely efe a close. Le us show ha (, D( )) s sspave: U U U, for > a U D( ) Le U = u ( ) j D a >. ccorg, j o [], for all, j we have u u u j j j he U U U, a hece he sspavy hols. I orer o prove ha (, D( )) sasfes he posve U = u D( ) mmum prcple, assume ha j, j a posve such ha U( )= for some [, [. If >he uj a uj ( )=, for all, j =,,,. Hece uj'( ) = a u '( j ) for all, j =,,,. Tha meas U( ). If = he U( )=. 3.. Specral alyss of he Operaor (, D( )) The purpose of e heorem s o euce uer reasoable hypohess o he coeffces of B a precse escrpo of he specrum of he operaor (, D( )). Theorem 3. If I B(), he he specrum of (, D( )) s coae ],]. CU, = u D ( ) Proof. Le j, j a V =( vj ), j. We have ( ) U = V ( ) u = v for all, j =,,, j j u v for all j a j =( ) j, =,,, = ( ). = Ths s suffce o oe ha he specrum of = (, D( )) verfes ( ) ( ) a he he = resul hol by ([], Lemma.6). Theorem 3. The operaor (, D( )) wh I B(), geeraes a aalyc semgroup of agle. Moreover, he semgroup s posve a coracve. Proof. For < <, pu * ={ zc ; arg( z) }. I s clear from heorem 3. ha ( ) a he for a =,,,, he operaor ( ) s verble C a verfes ( ) for some posve cos- a C (see []). So ( ) s verble a ver- Copyrgh ScRes.

4 86 O.. M. S. HMED ET L. ma C fes ( ). Hece he operaor (, D( )) wh I B(), geeraes a aalyc semgroup of agle. For he posvy usg he rela- o bewee he operaors, =,,, a a from he fac ha each operaor (, D( )), =,,,, geeraes a aalyc posve a coracve semgroup, we euce he resul. 4. The No Dagoal Case I hs seco we coser he operaor ( ) = '' B U U U ' ssume ha B ( )= PDP ( ) ( ) ( ), where D s agoal mar fuco a P s osgular mar fuco. If P s cosa, pu V = P U, he ' ' '' '' U = PV, U = PV a U = PV so we oba '' D ( ) ' P PV = V V Hece, from Theorem 3. we ca easly verfy he followg Proposo 4. The operaor (, D( )) wh I D(), geeraes a aalyc semgroup of agle. Remark ha he coo I D() s equvale o he fac ha he specrum ( B()) of he cosa mar B () verfes ( B()) ], ]. We ur ow o he geeral case whch we procee wh a perurbao argume. For hs we recall he followg efo. Defo 4. ([].Defo.). Le : D ( ) be a lear operaor o he Baach space. y operaor B: D( B) s calle -boue f D) D( B) a f here es cosas ab, such ha BU a U b U for all U D( ). (5.) The -bou of B s a =f{ a : here ess b such ha (5.) hols} Proposo 4. ([].Theorem.). Le he operaor (, D ( )) geeraes a aalyc semgroup o a Baach space. The here ess a cosa > such ha ( BD, ( )) geeraes a aalyc semgroup for every -boue operaor B havg -bou a <. Irouce ow he operaors (, D( )) a (, D( )) gve by B() B( ) B() = a = wh, D( )= D( )={ U C([, ], M C (], ], M, lm U( )= }. The we have = a f we choose D( ) = D( ), we oba he prcpal resul of he paper. Theorem 4. ssume ha B () s agoalzable a ( B()) ], ] or B() = I a (.) hols. The he operaor (, D( )), geeraes a aaly- c semgroup of agle. Proof. Le U D( ) a observe ha B( ) U = U U B() B( ) B() = U U U = U U, Le >, here ess > such ha for all < < we have B ( ) B() <. The formula U( ) = U ( ) mples ha RU ( ) U, < <. O he oher ha from he Taylor epaso o orer a > a for all U D( ) here ess a cosa C > such ha RU ( ) U C U B() U U C U. Sce B () s agoalzable a ( B()) ], [ or B sasfes he coo (.) for B() = I, he map B() U U, from D( ) o C([, ], M s couous (see []. Remark.5), so we euce ha he operaor s -boue wh -bou s equal o zero. Hece, he esre resul follows by applyg Theorem 3. a Proposo 4.. Copyrgh ScRes.

5 O.. M. S. HMED ET L pplcao a Eample ssume ow ha he operaor (, D( )) sasfes he assumpos of Theorem 4., geeraes so a aalyc semgroup, a coser he evoluo equao problem U () U ()= f (), (5.) U() = U Corollary 5. If f = he he problem (5.) has a uque soluo for all U. Ths soluo s of fely couously ffereable o ], [. For geeral case we have by Pazy [3] a R.aury [3]. Corollary 5. If f =( fj ), j, a for all, j, fj L(], T[, C) a for every ], T[ here s a, j a a couous fuco, j :[, [ [, [ such ha, j j, j ( ) () ( ) ( ) fj fj s s a <. The he problem (5.) has a classcal soluo. EPLE Le he Baach space = C([, ], M a. Pu ( ) = a efe he lear rasformao P o self by PU = Uo. I s clear ha P s verble a ( P ) = P. Coser ow he operaor (, D( )) efe o by wh U = U B( ) U D U C M C U, lm( op ) U( )= }, ( ) = { (], ], ( )), a B s a agoal mar real value fuco sasfyg B() < ( ) I. smple calculus gves = P ( P ) (5.3) where B ( ) U = U U. Pu D U C M C C M, lmu( )= }. ( ) = { ([, ], ( )) (], ], From Theorem 3. he operaor (, D( )) geeraes a aalyc semgroup of agle moreover, he sem- group s posve a coracve. Hece he relao (5.3) mples ha he operaor (, D( )) geeraes a aalyc semgroup of agle a he semgroup s coracve f. 6. ckowlegemes The uhors wsh o hak Professor. Rha for may helpful scussos a commes o he mauscrp. 7. Refereces [] R. Nagel, Oe-Parameer Semgroups of Posve Operaors, Lecure Noes Mah, Sprger-Verlag, 986. [] K. J. Egel a R. Nagel, Oe-Parameer Semgroups for Lear Evoluo Equaos, Sprger-Verlag,. [3]. Pazy, Semgroups of Lear Operaors a pplcaos o Paral Dffereal Equaos, pple Mah Sceces 44, Sprger, 983. [4] K. Io a F. Kappel, Evoluo Equaos a ppromaos, Seres o vaces Mahemacs for pple Sceces, Vol. 6,. [5] E. M. Ouhabaz, alyss of Hea Equaos o Domas, Prceo Uversy Press, New Jersey, 5. [6] K. J. Egel, Operaor Marces a Sysems of Evoluo Equaos. (Prepr). [7] N. H. Mahmou, Paral Dffereal Equaos wh Marcal Coeffces a Geeralse Traslao Operaors, Trasacos of he merca Mahemacal Socey, Vol. 35, No. 8,, pp [8] N. H. Mahmou, Hea Equaos ssocae wh Mar Sgular Dffereal Operaors a Specral Theory, Iegral Trasforms a Specal Fucos, Vol. 5, No. 3, 4, pp [9]. Sa a O.. M. S. hme, alycy of Semgroups Geerae by a Class of Dffereal Operaors wh Mar Coeffces a Ierface, Semgroup Forum, Vol. 7, No., 5, pp. -7. [] G. Meafue, alycy for Some Degeerae Oe-Dmeoal Evoluo Equaos, Sua Mahemaca, Vol. 7, No. 3, 998, pp [] Z. S. graovch a V.. Marcheko, The Iverse Problem of Scaerg Theory, Kharkov Sae Uversy, Goro a Breach, NewYork a Loo, 963. [] R. B. Bapa a T. E. S. Raghava, Noegave Marces a pplcaos, Cambrge Uversy Press, Cambrge, 997. [3] R. Dauray a J.-L. Los, alyse Mahémaque e Calcul Numérque pour les Sceces e les Techques, Tome 3, Sére Scefque, Masso, 985. Copyrgh ScRes.