Toeplitz and Translation Operators on the q-fock Spaces *

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1 Advces i Pure Mthemtics doi:436/pm659 Published Olie November ( Toeplit d Trsltio Opertors o the -Foc Spces * Abstrct Fethi Solti Higher College o Techology d Iormtics Tuis Tuisi E-mil: ethisolti@yhoocom Received Jue 5 ; revised July ; ccepted July 8 I this wor we itroduce clss o Hilbert spces < < with reproducig erel give by the -expoetil uctio o etire uctios o the dis D o ; d we prove some properties cocerig Toeplit opertors o this spce The deiitio d properties o the spce turlly those o the well-ow clssicl Foc spce Next we study the multiplictio opertor e Q exted d the -Derivtive opertor D o the Foc spce ; d we prove tht these opertors re djoit-opertors d cotiuous rom this spce ito itsel Lstly we study geerlied trsltio opertors d Weyl commuttio reltios o Keywords: -Foc Spces -Expoetil Fuctio -Derivtive Opertor -Trsltio Opertors -Toeplit Opertors -Weyl Commuttio Reltios by Itroductio I 96 Brgm [] itroduced Hilbert spce o etire uctios = o such tht = = := < O this spce the uthor study the dieretil opertor D =d d d the multiplictio opertor by d proves tht these opertors re desely deied closed d djoit-opertors o (see []) Next the Hilbert spce is clled Segl-Brgm spce or Foc spce d it ws the im o my wors [3] I this pper we cosider the -expoetil uctio: where = e := ; i ; := = i= We discuss some properties o clss o Foc spces * Author prtilly supported by DGRST project 4/UR/5- d CMCU progrm G 53 ssocited to the -expoetil uctio d we give some pplictios I the irst prt o this wor buildig o the ides o Brgm [] we deie the -Foc spce s the spce o etire uctios = = o the dis D o o ceter o d rdius d such tht ; := < = Let d g be i such tht = = d g = b = the ier product is give by g = The -Foc spce give by b ; = hs lso reproducig erel w = e w; w D o Copyright SciRes

2 36 F SOLTANI The i we hve w = w w D o Usig this property we prove tht the spce is Hilbert spce d we give Hilbert bsis Next we deie d study the Toeplit opertors o the -Foc spce I the secod prt o this wor we cosider the multiplictio opertor Q by d the -Derivtive opertor D o the Foc s pce d we prove tht these opertors re cotiuous rom ito itsel d stisy: D Q The we prove tht these opertors re djoit-opertors o : Q g = D g ; g Next we deie d study o the Foc spce -trsltio opertors: w:= e D w; w Do d the geerlied multiplictio opertors: M w:= eq w; w D o the Usig the previous results we deduce tht the ope- rtors d M or D o re cotiu- ous rom ito itsel d stisy: e M e Lstly we estblish Weyl commuttio reltios betwee the trsltio opertors d the multipli- ctio opertors M b where b D o These reltios re relied o the Foc spce The -Foc Spces d the Toeplit Opertors Prelimiries Let d be rel umbers such tht < <; the -shited ctoril re deied by i ; := ; := = i= Jcso [4] deied the -logue o the Gmm uctio s ; := x x x x ; It stisies the uctiol eutio where x x x = = x x := ; d teds to x whe teds to I prticulr or = we hve ; = = The -combitoril coeiciets re deied or d = by := The -derivtive D o suitble uctio (see [5]) is give by x := x x D x x d D = provided exists I is dieretible the D x teds to x s There re two importt -logues o the expo- etil uctio [5]: / E := e = := = Note tht the irst series coverges or < d the secod series coverges or < Copyright SciRes

3 F SOLTANI 37 Thereore the uctio settio [6]: x E hs the -itegrl repre- x = r r d r x > () where the -itegrl (itroduced by Jcso [4]) is deied by x d x = = Lemm The uctio e D o is the uiue lytic solutio o the -problem: Dy= y y = () Proo Serchig solutio o () i the orm y= The = Replcig i () we obti Thus Dy = We deduce tht We get Thereore = = x = = = = = = = y = = e which completes the proo o the lemm The -Foc Spces We deote by H D o the spce o etire uctios o D o m the mesure deied o D o i dm := E r drd = re π L D o m uctios o D o by the spce o mesurble D o L D o m stisyig := dm < Deiitio We deie the prehilberti spce to be the spce o uctios i H D o L D o m euipped with the ier product d the orm D o m g = g d / = dm D o Remr I the spce grees with the Segl-Brgm s spce (see []) Propositio ) For ll such tht = we hve = ) For ll = d g = b = we hve = (3) g such tht = = 3) For g we hve Proo Give = = g = g = b (4) g = D g g = g b = = d ) By domit ed covergece theorem s we hve m m D o m = = dm Copyright SciRes

4 38 F SOLTANI We put = re i the we deduce = = r E r d r But rom () we hve Thus re d = r r = = = ) We obti the result rom () by polritio 3) Sice the d d Thus D = D = (5) Dg g = = b Dg b = = Usig (4) d (6) we get Thus Dg = (6) g = D g = D g = = g = D g whi ch gives the desired result The ollowig theorem prove tht ducig erel spce Theorem The uctio give or w D o b y = e w w is reproducig erel or the -Foc spce is repro- tht is: ) or ll w Do the uctio w belogs to ) For ll w D o d we hve Proo ) Sice w = w = w = the rom (3) we deduce tht w ; w D o w e w w = = = < which proves ) ) I = rom (4) d 7) we = ( deduce (7) w = w = w w D o = This completes the proo o the theorem Remr From Theorem () or w D o we hve / d w w = e w (8) Propositio The spce euipped with the ier product is Hilbert spce; d th e set give by = D o orms Hilbert bsis or the spce Proo Let be Cuchy s euece i We put From (8) we hve = lim i / p w w e w p This ieulity shows tht the seuece is poitwise coverget to Sice the uctio / w e w is cotiuous o D o he t Copyright SciRes

5 F SOLTANI 39 o D o coverge s to uiormly o ll compct set Coseuetly is etire uctio o D o the belogs to the spce O the oth er hd rom the reltio (4) we get = m m where m is the Kroecer symbol This shows tht the mily is orthoorml set i Let = be eleme = t o such tht = From the reltio (4) we deduce tht = This completes the proo 3 Toepli Opertors o I this prgrph we study the Toeplit opertors o These opertors geerlie the clssicl Toeplit oper- tors [] First we deie the orthogol projectio opertor P rom L D o m ito by P w := K w w D o L D o m where K is the reproducig erel give by (7) Deiitio Let be mesurble uctio o D o The Toeplit opertor T is the opertor give by or every T := P DT := : L D o m Remr 3 Let L D o ) The opertor T is bouded d T ) By derivtio uder the itegrl sig d usig () we hve T = D Theorem I L D o hs compct support the T is compct opertor Proo For L D o we hve Sice T L D o m T D o w w m w = d T w K w m D o d = Applyig Fubii's theorem d Theorem we obti Thus T L D o m = L D o m T L D o m = L D o m = = Sice L D o with compct support there re positive costts d K so tht K e d = or ll > The or we get Thus L Do m = dm Copyright SciRes

6 33 F SOLTANI L D o m K K K But rom () we hve Hece ( )/ dm r E ( r)d r E r d r E r d r = = K Thus we obti L D o m T K e < L D o m = The T is Hilbert-Schmidt opertor [7] d coseuetly it is compct 3 The Multiplictio d Trsltio Opertors o 3 The Derivtive d Multiplictio Opertors o O we cosider the multiplictio opertor give by Q := By strightorwrd clcultio we obti Lemm D Q = DQ QD = where is the -shit opertor give by := This lemm is the -logous commuttio rule o [] Whe the D Q teds to the idetity opertor I We ow study the cotiuous property o the ope - rtors D d Q o Theorem 3 I the D d Q belog to d we hve ) Q ) D 3) Q Proo Let = = ) We hve d rom (3) we obti ) We hve = = = = = = = D = = (9) = = The rom (9) we get Sice we obti d coseuetly = = D = = = D = () D = () Usig the ct tht = D we obti / = 3) O the other hd sice the Q = By () we deduce = = = Q = = = () Q = (3) Usig the ct tht we obti Copyright SciRes

7 F SOLTANI 33 Q We deduce lso the ollowig orm eulity Theorem 4 ) I the Q = D ) The opertor Q : is ijective o Proo Let = = ) By (3) d usig the ct tht = we obti = Q = = D ) From () we hve Q Thereore Q = implies tht = The Q : is ijective cotiuous op ertor o Propositio 3 The opertors Q d D re djoit-opertors o ; d or ll g we hve Q g = D g Proo Cosider = = = b = i g d From (9) d () Dg = b = = Q = Thus rom (4) we get Q g = b = b which gives the result = = Dg = 3 The Trsltio Opertors o I this sectio we study geerlied trsltio opertors o We begi by the ollowig deiitio Deiitio 3 For d w D o we deie the -trsltio opertors o by := = w = w e D w D (4) For w D o the uctio e stisies the ollowig product ormul: e w= e e w Propositio 4 Let = = w D o The d w= w = = Proo Let = = hve From (4) we D w w= ; w D o = But rom (5) we hve Thus we obti = D w = w = = w = w = w = = Deiitio 4 For d w D o we deie: The geerlied multiplictio opertors o by = M w := e Q w = Q w The geerlied shit opertors o by := = = S w e w w Accordig to Theorem 3 we study the cotiuous property o the opertors M d S o Theorem 5 I d D o the M d S belog to d we hve ) e Copyright SciRes

8 33 F SOLTANI ) M e 3) S e Proo From (4) d Theorem 3 () we deduc e D / = = Thereore e which gives the irst ieulity d s i the sme wy we prove the secod d the third ieulities o this theorem From Propositio 3 we deduce the ollowig results Propositio 5 For ll g we hve We deote by by M g = g g = M g R := M M R the ollowig oper tor deied o = e D e Q e Q e D The we prove the ollowig theorem Theorem 6 For ll we hve = M R Proo From Propositio 5 we get = = M M M R = R 3 3 The Weyl Commuttio Reltios o Let b D o I this prgrph we estblish Weyl commuttio reltios betwee the trsltio opertors d the multiplictio opertors M b These reltios re relied o the Foc spce Lemm 3 For b D o we hve ) D Q = Q = ) D M b = bm b Proo ) From Lemm or = we deduce tht = = = = D Q Q D Q Q Q Q Sice we get Q = Q D Q = Q Which proves the irst eulity ) We hve Usig () we obti b D Mb = D Q = b D M Q b = = = b = bq = bmb [ ] Theorem 7 For b D o we hve M = M S b b b Proo From Lemm 3 () we hve The or Multiplyig by DM = M D b b b we deduce DM = M D b b b d summig we get Sice D = D rom [5] we get e D b = e D e b = S M = M e D b b b b which completes the proo o the theorem Remr 4 I we obti the clssicl commuttio reltios [8]: D bq b bq D DQ = I e e = e e e ; b 4 Reereces [] V Brgm O Hilbert Spce o Alytic Fuctios Copyright SciRes

9 F SOLTANI 333 d Associted Itegrl Trsorm Prt I Commuo Pure d Applied Mthemtics Vol 4 No ictios 3 96 pp 87-4 doi:/cp36433 [] C A Berger d L A Cobur Toeplit Opertors o the Segl-Brgm Spce Trsctios o the Americ Mthemticl Society Vol pp doi:9/s [3] F M Cholewisi Geerlied Foc Spces d Associted Opertors Society or Idustril d Applied Mthemtics Jourl o Mthemticl Alysis Vol 5 No 984 p p 77- doi:37/555 [4] G H Jcso O -Deiite Itegrls Qurterly Jourl o Pure d Applied Mthemtics Vol 4 9 pp 93-3 [5] T H Koorwider Specil Fuctios d -Commutig Vribles Fields Istitute Commuictios Vol pp 3-66 [6] G Adrews R Asey d R Roy Specil Fuctios Cmbridge Uiversity Press Cmbridge 999 [7] M Nimr Normed Rigs Noordho Groige 959 [8] T Hid Browi Motio Spriger-Verlg Berli 98 Copyright SciRes