Some Classes of Composition Operators. on the Fock Space

Transcription

1 It. J. otemp. ath. Scieces, Vol. 7, 01, o. 14, Some lasses of ompositio Operators o the Foc Space S. Paayappa Departmet of athematics Govermet Arts ollege, oimbatore Tamiladu, Idia paayappa@gmail.com N. Sivamai Departmet of athematics Tamiladu ollege of Egieerig, oimbatore Tamiladu, Idia sivamaitce@gmail.com Abstract I this paper ormal, quasiormal, hypoormal, quasihypoormal, class A ad paraormal compositio operators o Foc space are characterized. athematics Subject lassificatio 4733, 4738, 30D15 Keyords: Foc space, compositio operator, ormal, quasiormal, hypoormal, quasi hypoormal, class A ad paraormal operators.

2 650 S. Paayappa ad N. Sivamai 1. Itroductio ompositio operators o spaces of aalytic fuctios have bee studied i may settigs. uch has bee ritte about the properties of these operators o the Hardy, ergma, ad loch spaces o the uit dis i the complex plae or o the uit ball i.( [],[5] ad [6]. Already i the paper [1] bouded ad compact compositio operator o Foc space is discussed. We ill determie hich compositio operators are ormal, quasiormal,hypoormal, quasi hypoormal, class A ad paraormal. The Foc space F is the Hilbert space of all holomorphic fuctios o ith ier product 1 1 z f, g = f ( z g( z e dv( z ( π here v deotes Lebesgue measure o. (refer [1] ad [4]. 1 Let e ( z = z for a positive iteger. The {e } forms a! orthoormal basis for F. Sice each poit evaluatio is a bouded liear fuctioal o F, for there exists a uique fuctio F such that f, = f ( hich holds for all f F. The reproducig erel fuctios for the Foc space are give by z, / ( z = e here z, = z j j. Note that the substitutio f = ito the reproducig 1 formula f, = f ( hich holds for all f F ad leads to the idetity exp( / 4 =. Through out this paper e use f = for the reproducig erel fuctio for the Foc space F ad let 0 = 1 be the poit evaluatio o F ( [3] ad [4]. For a give holomorphic mappig :, the compositio operator : F F is give by ( f = f o. I the paper [4] it is proved that if the operator is bouded,the must be of the form ( z = Az + here A is a matrix ad is a 1 vector. Further more it

3 ompositio operators o Foc space 651 ill follo that A 1 for bouded ad that ill be restricted by the coditio that Aς, = 0 for ay ς i ith A ς = ς. I paper [1] Theorem 1 shos that if is compact, the A < 1 ith o restrictio o. I sectio 3 e characterize ormal ad other classes of compositio operators here is bouded.. PRILIINARIES be a compositio operator o the Foc space F. The ormal iff = I paper [1] it is shos that = τ τ ( z = A z ad quasiormal iff hypoormal iff 1 is multiplicatio by the erel fuctio.we say ( ( ( = (, hypoormal iff is here is,quasi, lass A operator iff ( ad paraormal iff + + I > 0, > AIN RESULTS : Theorem 3.1 Let be a compositio operator o F.The is ormal if ad oly if = ο τ ο. is ormal if ad oly if = here ( z = Az + so that is bouded. The = τ here τ ( z = A z ad is multiplicatio by the erel fuctio. Sice = τ [1] it follos that τ = τ Sice = e have as desired. θ θ ο = ο τ ο

4 65 S. Paayappa ad N. Sivamai orollary 3. Let be a compositio operator o F.If is ormal the =. ο y the above theorem, is ormal if ad oly if = ο τ ο ( f ( f = ο τ ο for all f F. ( fοοτ = ο ( fοτο Let f = 0 = 1 be the poit evaluatio o F. We have ( 0ο οτ = ο ( 0οτο = ο orollary 3.3 Let be a compositio operator o F here The is ormal if ad oly if A A= AA ( z = Az, A is matrix. οτ ( z = τ ο ( z is equivalet to A Az = AA z. Sice = 1, the result follos. Theorem 3.4 Let be a compositio operator o F.The is hypoormal if ad oly if ο. is hypo ormal if ad oly if Sice = ο τ τ ο it follos that orollary 3.5 be a compositio operator o F ad let f = be the reproducig erel fuctio for the Foc space ad put f = = 0 1 be the poit evaluatio o F. If is hypoormal the. ο

5 ompositio operators o Foc space 653 Theorem 3.6 Let be a compositio operator o. The is quasi ormal if ad oly if (. ο ο = ο is quasi ormal if ad oly if ut ad ( = ( ( ( = ( = = ο = ο τ ο = ο ο so is quasi ormal if ad oly if (. ο ο = ο orollary 3.7 Let be a compositio operator o F here ( z = The is quasiormal if ad oly if A commutes ith y the above theorem, ο ο Az ad A is matrix. A A. is quasiormal if ad oly if = hich is equivalet to AA A= AAA Theorem 3.8 Let be a compositio operator o F.The is quasihypoormal if ad oly if ο ( ( ( here ( = ο. is quasihypoormal if ad oly if ( ( No ( = ( =

6 654 S. Paayappa ad N. Sivamai = = ο ο ο ( ( ad = = = ( ο = τ ο ( ( ad so the compositio operator o o F is quasihypoormal if ad oly if. ο ( ( ( ( orollary 3.9 Let o F be a compositio operator here quasihypoormal if ad oly if Observe that ( (οτ ( z = ( AA z + AA A A ad 0 ( z = Az +.The is A. οτ ( z = A A z + A + ad hece the result. Theorem 3.10 Let A operator if ad oly if No be a compositio operator o F. The is quasihypoormal. is of class A operator if ad oly if ( ( ( = ( ο ( ad ( = ad so ( ( hich reduces to ( ( ( ο ( belogs to class

7 ompositio operators o Foc space 655 hece it is quasihypoormal. Theorem 3.11, a compositio operator o F is paraormal if ad oly if is quasihypoormal. is paraormal if ad oly if + + 0, for all real hich is equivalet to Which shos that ( ( ( ( ( οτ hich reduces to ο ( ( ( ( Usig the previous theorem, the result follos. Refereces [1].J. arsell,.d. acluer ad A. Schuster, ompositio operator o the Foc Space, Acta Sci. ath. (Szeged,69 pages( , 003. []. oe ad. acluer, ompositio Operators o Spaces of Aalytic Fuctios, R Press,oca Rato, FL, [3] S. Paayappa, D. Sethilumar, R. oharaj, Quasihypoormal ompositio Operators o Weighted Hardy Spaces, It. Joural of ath. Aalysis, Vol.. 008, o 4, [4] S. Paayappa ad D. Sethilumar,lass A ompositio operators, ull.al.ath.soc.,96(1(004, [5] Sei ichiro Uei, Hilbert- Schmidt Weighted ompositio Operator o the Foc space, It. Joural of ath. Aalysis, Vol. 1, 007, o. 16,

8 656 S. Paayappa ad N. Sivamai [6] J. Sharpio, ompositio Operators ad lassical Fuctio, Spriger, Ne Yor, Received: October, 011