Multiplication and Translation Operators on the Fock Spaces for the q-modified Bessel Function *

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1 Advces i Pure Mthemtics 0-7 doi:0436/pm04039 Pulished Olie July 0 ( Multiplictio d Trsltio Opertors o the Fock Spces or the -Modiied Bessel Fuctio * Astrct Fethi Solti Higher College o Techology d Iormtics Street o the Employers Tuis Tuisi E-mil: ethisolti0@yhoocom Received Ferury 8 0; revised April 8 0; ccepted April 0 0 We study the multiplictio opertor M y d the -Bessel opertor o Hilert spces o etire uctios o the disk Do 0< <; d we prove tht these opertors re djoit-opertors d cotiuous rom ito itsel Net we study geerlied trsltio opertors o Keywords: Geerlied -Fock Spces - I Modiied Bessel Fuctio -Bessel Opertor Multiplictio Opertor -Trsltio Opertors Itroductio I 96 Brgm [] itroduced Hilert spce etire uctios o such tht 0 :! O this spce the uthor studied the dieretil opertor D d d d the multiplictio opertor y d proved tht these opertors re desely deied closed d djoit-opertors o (see []) Net the Hilert spce is clled Segl-Brgm spce or Fock spce d it ws the im o my works [] I 984 Cholewiski [3] itroduced Hilert spce o eve etire uctios o the ier product is weighted y the modiied Mcdold uctio O the Bessel opertor d d : > d d d the multiplictio y closed d djoit-opertors o re desely deied * Author prtilly supported y DGRST project 04/UR/5-0 d CMCU progrm 0G 503 I this pper we cosider the uctio: I ; : - I modiied Bessel ; re give lter i Sectio We deie the -Fock spce s the Hilert spce o eve etire uctios o the disk Do o ceter o d rdius d such tht Let d g e i : ; 0 d g c such tht the ier product is give y g c ; 0 Net we cosider the multiplictio opertor M y d the -Bessel opertor o the Fock spce d we prove tht these opertors re cotiuous rom ito itsel d stisy: Copyright 0 SciRes

2 F SOLTANI The we prove tht these opertors re djoitopertors o : g g ; g Lstly we deie d study o the Fock spce the -trsltio opertors: / T w: I w; w Do d the geerlied multiplictio opertors: M w: I M w; w Do Usig the previous results we deduce tht the opertors T d M or Do re cotiuous rom ito itsel d stisy: T M I I The -Fock Spces α Let d e rel umers such tht 0< <; the -shited ctoril re deied y ; : ; : i 0 i0 Jckso [5] deied the -logue o the Gmm uctio s ; : 0 It stisies the uctiol eutio d teds to whe or we hve teds to I prticulr ; The -comitoril coeiciets re deied or k k 0 y ; : k k k k k The -derivtive D o suitle uctio (see [6]) is give y : D 0 d D 0 0 provided 0 eists I is dieretile the D teds to s Tkig ccout o the pper [4] d the sme wy we deie the - I modiied Bessel uctio y : I we put U I ; : ; ( ) U U ; the Thus the - I modiied Bessel uctio is deied o D o d teds to the I modiied Bessel uctio s I [4] the uthors study i gret detil the opertor deoted y [ ] : D D [ ] : () () -Bessel The -Bessel opertor teds to the Bessel opertor s Lemm : ) The uctio I Do is the uiue lytic solutio o the -prolem: y y y 0 d Dy 0 0 (3) Copyright 0 SciRes

3 F SOLTANI 3 ) For we hve ; ( ) ( ) ; 3) The costts ; stisy the ollowig reltio: ; Let The -Fock spce is the Hilert spce o eve etire uctios o Do such tht ; The ier product i : < (4) is give y () d g is give or c y c ; g Remrk : I the spce grees with the geerlied Fock spce ssocited to the Bessel opertor (see [3]) Theorem : The uctio give or w Do y w I w ; is reproducig kerel or the -Fock spce tht is: ) For ll w Do the uctio w elogs to ) For ll w Do d we hve w w Remrk : From Theorem ) or w Do we hve (5) d w w I w ; / 3 Multiplictio d -Bessel Opertors o α O d N we cosider the multiplictio opertors M give y : N : D We deote lso y deied or etire uctios o We write the -Bessel opertor Do M M M By strightorwrd clcultio we oti the ollowig result Lemm : M B W d : B W : D Remrk 3: The Lemm is the logous commu- ttio rule o Cholewiski [3] Whe d the M teds to 4 I 4 I d is the idetity opertor Lemm 3: I the B N d W elog to d ) B ) N 3) W Proo Let B the (6) (7) N (8) d rom (4) we oti Copyright 0 SciRes

4 4 F SOLTANI d Usig the ct tht B ; 4 ; ; N we deduce N ; O the other hd rom (6) we hve d W (9) W Usig the ct tht W ; 4 we deduce tht ; Thereore we coclude tht W which completes the proo o the Lemm Theorem : I the d elog to d we hve ) ) Proo Let ) From Lemm ) 0 ; ; ; ; ( ) (0) The rom (0) we get ; ; ; Usig Lemm 3) we oti d coseuetly Usig the ct tht oti / () we the ) O the other hd sice () ; ; By Lemm 3) we deduce ; (3) Usig the ct tht we oti We deduce lso the ollowig orm eulities Theorem 3: I the ) ) 3) 4) W N B N N B N B N B B W Copyright 0 SciRes

5 F SOLTANI 5 0 g c c g Proo Let ) Follows rom (7) (8) d (9) ) From () we get Usig the ct deduce N 3) By (3) d usig the ct tht we oti N B 4 B N N B we 4) Follows directly rom ) ) d 3) Remrk 4: Let Sice W 0 the 0 B Thereore 0 implies tht 0 The M : is ijective cotiuous opertor o Propositio : The opertors M d re djoit-opertors o ; d or ll g we hve Proo Cosider c g d g g d i From (0) d () g c ; Thus rom (5) we get 0 which gives the result 4 Geerlied Multiplictio d Trsltio Opertors o α I this sectio we study geerlied multiplictio d trsltio opertors o Deiitio : For d w Do we deie: -The -trsltio opertors o y w w: (4) ; 0 -The geerlied multiplictio opertors o y M w M w: (5) ; For w Do the uctio I ; stis- ies the ollowig product ormuls: ; ; ; ; ; ; I w I I w M I w I w I w Remrk 5: I we oti the geerlied trsltio opertor give i ([3] pge 8) Propositio : Let d w Do The ) w 0 k0 k w k k w k ) M k k w w k0 k ; Copyright 0 SciRes

6 6 F SOLTANI Proo ) Let 0 From (4) ) M I we hve w 0 ; ; Proo From (4) d Theorem ) we deduce w w Do T Sice rom Lemm ) ; we c write ; k k ( k) w w ( k ) ; ; k k ( k) w k w k ( ) ; k Thus we oti w ; ; ; w k0 k k O the other hd rom () d () we get ; ; ; k k k which gives the ) ) From (5) we hve k k ; ; ( k) k M w M w w Do But rom () we hve Thus we oti M w w k k k M w k0 k ; k k w Accordig to Theorem we study the cotiuous property o the opertors T d M o Theorem 4: I d Do the T d M elog to d we hve ) T I Thereore ; T I which gives the irst ieulity d s i the sme wy we prove the secod ieulity o this theorem From Propositio we deduce the ollowig results Propositio 3: For ll g we hve We deote y y M g T g T g Mg R the ollowig opertor deied o / / / / ; ; R : T M M T I I M I M I The we prove the ollowig theorem Theorem 5 For ll we hve M T R Proo From Propositio 3 we get M 5 Reereces T M M T R R T [] V Brgm O Hilert Spce o Alytic Fuctios d Associted Itegrl Trsorm Prt I Commuictios o Pure d Applied Mthemtics Vol 4 No 3 96 pp 87-4 doi:000/cp [] C A Berger d L A Cour Toeplit Opertors o the Segl-Brgm Spce Trsctios o the Americ Mthemticl Society Vol pp doi:0090/s Copyright 0 SciRes

7 F SOLTANI 7 [3] F M Cholewiski Geerlied Fock Spces d Associted Opertors SIAM Jourl o Mthemticl Alysis Vol 5 No 984 pp 77-0 doi:037/05505 [4] A Fitouhi M M Hm d F Boueour The -j α Bessel Fuctio Jourl o Approimtio Theory Vol 5 No 00 pp doi:0006/jth [5] G H Jckso O -Deiite Itegrls The Qurterly Jourl o Pure d Applied Mthemtics Vol 4 90 pp [6] T H Koorwider Specil Fuctios d -Commutig Vriles Fields Istitute commuictios Vol pp 3-66 Copyright 0 SciRes