IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. 2 Issue 12, December

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1 IJISET - Iteratial Jural f Ivative Sciece, Egieerig & Techlgy, Vl Issue, December 5 wwwijisetcm ISSN Psirmal ad * Pararmal mpsiti Operatrs the Fc Space Abstract Dr N Sivamai Departmet f athematics, Tamiladu llege f Egieerig,imbatre Tamiladu, Idia sivamaitce@gmailcm I this paper, psirmal, * pararmal, quasi psirmal ad qusi * pararmal cmpsiti peratrs Fc space are charecterised athematics Subject lassificati 47, 478, D5 Keywrds: mpsiti peratr, Fc space, psirmal, quasipsirmal quasi * pararmal INTRODUTION mpsiti peratrs spaces f aalytic fuctis have bee studied i may settigs uch has bee writte abut the prperties f these peratrs the Hardy, ergma, ad lch spaces the uit dis i the cmplex plae r uit ball i (see, fr example [],[] ad [7] ) Already the paper [] discussed buded ad cmpact cmpsiti peratr Fc space I paper [6] We have discussed sme classes f cmpsiti peratrs the Fc space I this paper psirmal, * pararmal, quasi psirmal ad qusi * pararmal cmpsiti peratrs Fc space are charecterised with ier prduct The Fc space F is the Hilbert space f all hlmrphic fuctis ( ) π z f, g = f ( z) g( z) e dv( z) Where v detes Lebesgue measure (refer [] ad [7] ) Let e! ( z) = z fr a psitive iteger The the {e } frms a rthrmal basis fr F Sice each pit evaluati is a buded liear fuctial F, fr w there

2 IJISET - Iteratial Jural f Ivative Sciece, Egieerig & Techlgy, Vl Issue, December 5 wwwijisetcm ISSN exists a uique fucti w F such that f, w = f ( w) which hlds fr all f F The reprducig erel fuctis fr the Fc space are give by ( z) = e w z, w / where z, w = z w j j Nte that the substituti f = w it the reprducig frmula f, w = f ( w) which hlds fr all f F ad ( w / 4) w leads t the idetity w = exp Thrughut this paper we use f = w is the reprducig erel fucti fr the Fc space F ad = be the pit evaluati F (Refer [], [4], [5] ad [7]) Fr a give hlmrphic mappig :, the cmpsiti peratr : F F is give by ( f ) = f The paper [4] already prved that if the peratr is buded,the must be f the frm ( z ) = Az where A is a matrix ad is a vectr Furthermre it will fllw that A fr buded ad that will restricted by the cditi that Aς, = fr ay ς i with A ς = ς I paper [] Therem shws that if is cmpact, the A < with restricti PRILIINARIES Let F be a Fc space ad be a cmpsiti peratr F The is psirmal iff c, quasipsirmal iff * ( ), fr sme c >,* pararmal iff fr, quasi * pararmal if ad ly if fr ( ), R, It is w that the (,) class is f ( ), class if ( ) cicides with the class f quasi hyprmal peratrs ut the class f hyprmal peratrs des t cicide with the (, ) class fr ay Hwever, if we defie a class (, ) as : ( ) AIN RESULTS: Therem F is psirmal if ad ly if c Prf: is psirmal if ad ly if

3 IJISET - Iteratial Jural f Ivative Sciece, Egieerig & Techlgy, Vl Issue, December 5 wwwijisetcm ISSN c fr c > hece Therem c F is psirmal if ad ly if Prf: is psirmal iff c fr c > c We have rllary Frm therems ad ad if c = we get ad are psirmal peratrs if ad ly if is rmal Therem 4 F is quasipsirmal if ad ly if ( ) ( ) ( ) where ( ) ( =, ) = ο ο Prf: is quasipsirmal if * ( ) fr sme c > Nw ( ) = ( ) Ad ( ) = = = ( ο ) ο ο = = ο ο ο ( ) ο ( ) S is quasipsirmal iff ( ) ( ) ( ) ο ο

4 Therem 5 F is * pararmal if ad ly if ( ) ( ) = Prf: is * pararmal if ad ly if fr f ( ) ( ) y therem (4) ( ) ( ) ( ) which reduces t ( ) ( ) ( ) Hece we have ( ) ( ) = Therem 6 F is quasi * pararmal if ad ly if ( ) ( ) ( ) ( ) Prf : is quasi * pararmal if ad ly if ( ) R, Usig therem(4) we have ( ) ) ( ο ο = Nw ( ) = = ( ) ( ) ( ) = ( ) ( ) ( ) IJISET - Iteratial Jural f Ivative Sciece, Egieerig & Techlgy, Vl Issue, December 5 wwwijisetcm ISSN

5 IJISET - Iteratial Jural f Ivative Sciece, Egieerig & Techlgy, Vl Issue, December 5 wwwijisetcm ISSN , Sice ( ) R ( ) ( ) ( ) ( ) Hece ( ) ( ) ( ) ( ) Therem 7 F is class (,) if ad ly if is * pararmal Prf F is class (,) Nw = ( ) ( ) the ( ) ο ( ) = ( ) Hece ο ( ) = ο ( ) ( ) ( ) ο y therem (5) which reduces t * pararmal Refereces [] J arswell, D acluer ad A Schuster, mpsiti peratr the Fc Space, Acta Sci ath (Szeged),69,, pages [] we ad acluer, mpsiti Operatrs Spaces f Aalytic Fuctis, R Press,ca Rat, FL, 995 [] S Paayappa, D Sethilumar, R haraj, Quasihyprmal mpsiti Operatrs Weighted Hardy Spaces, It Jural f ath Aalysis, Vl 8, 4, 6-7 [4] S Paayappa ad D Sethilumar, lass A mpsiti peratrs, ullalathsc, 96()(4), 6 [5] S Paayappa ad NSivamai Sme lasses f mpsiti Operatrs the Fc Space ItJural f temp athscieces, Vl7,, N 6, [6] Sei ichir Uei, Hilbert- Schmidt Weighted mpsiti Operatr the Fc Space, It Jural f ath Aalysis, Vl, 7, 6, [7] J Sharpi, mpsiti Operatrs ad lassical Fucti, Spriger, New Yr, 99 4