Journal of Kufa for Mathematics and Computer Vol.2,no.1,may,2014,pp.76- 81 THE COMPACT OF THE COMPOSITION OPERATOR C  Ammar Ali Neamah † , Aqeel Mohammed Hussein ‡ Mathematical Department, Mathematical Department, Faculty of Mathematics and Computer Science College of Education Kufa University Qadisyia University adjoint of the operator C  induced by the map  and also discuss the compactness of the operator C  . Abstract Let U denote the unit ball in the complex plane, the Hardy space H 2 is the set of functions f (z)    f ^ (n ) z n n  0 holomorphic on U such that   2 f ^ (n )   1. Section One We are going study to the map  and properties of  , and also discuss  is an inner map. n  0 ^ with f (n) denotes the Taylor coefficient of f. Let  be a holomorphic self-map of U, the composition operator C  induced by  is defined on H 2 by the equation C f  f   (f  H 2 ) We have studied the composition operator induced by the bijective map  and discussed the adjoint of the composition operator .We have look also at some known properties of composition operator and tried to get the analogue properties in order to show how the results are changed by changing the map  in U. In order to make the work accessible to the reader , we have included some known results with the details of the proofs for some cases and proved some results. Definition (1.1) : [4] The set U = {z  C : z  1} is called unit ball in complex C and U = {z  C : z  1} is called boundary of U. Definition (1.2) : For   U, define (z)  z  1 z  z  U . Since the denominator equal zero only at z   , the function  is holomorphic on the ball { z   } . Since   U . Then this ball contain U .Hence  take U into U and holomorphic on U . Proposition (1.3) : For   U, z  2  1  1  z  1    2 2 z  2 Proof : Introduction : z  This search consists of two sections. In section one ,we are going study to the map  and properties of  , and also discuss  as inner map. In section two, we are going study to the composition operator C  induced by 2 z  1  z   z  76 2 2  z  1z  1  z  z   z  2  z  z  z  1  z  z  z   2  2 z  1 z  1 1  1  1  2 z  z  2 2 2 2 z  2 2 Ammar Ali Neamah and Aqeel Mohammed Hussein the map  and properties of C  , discuss the 1  z  1     2 We can define an inner product of the Hardy space functions as follows: 2 z  Remark (2.2) :[1] 2 Let f(z) =   f n  z   n n 0 Proposition (1.4) : If   U, then  take U into U .  f , g   f  n  g  z  . n 0 z  1 , hence Let z  U , then z  1 . By (1.3) z   1  0 , therefore 2 2 2 n 0 the inner product of f and g is defined by : Proof : z  and gz    g  n  z n , then  1, z   1 , hence z   U , hence  take Definition (2.3) :[9] U into U .   U , define K  z   Let hence 1 1  z z  U  . Since   U then  > 1 hence the geometric   series Definition (1.5) : [7] Let  : U  U be holomorphic map on U,  is called an inner map if z   1 almost everywhere on U .  2n is convergent and thus n 0   K   H 2 and K  z     n zn . n 0 Definition (2.4) : [4] Proposition (1.6) : Let  : U  U be holomorphic map on U, the composition operator C  induced by  is  is an inner map . Proof : From (1.4)  take U into z   1 . By (1.5)  is an inner. U and We are going study to composition operator C  induced by map  , and it’s properties, also discuss adjoint of the operator C  , and compactness of C  . the the the the f z     f n  z  Theorem (2.6) : [10] 1  0  for every f  H . 2 The goal of this theorem C : H 2  H 2 . n holomorphic on U, where z ∈ C such that  f  n  2   with f n  0 Taylor coefficient of f . equation Let T be a bounded operator on a Hilbert space H, then the norm of an operator T is defined by T  sup Tf : f  H, f  1. n  0  the then f    H 2 and f    1  0 f Let U denote the unit ball in the complex plane, the Hardy space H 2 is the functions f  H  . by 2 If  : U  U is holomorphic map on U, Definition (2.1) : [4] of C f  f   H2 on Definition (2.5) : [2] 2. Section Two set defined  n  denotes the Definition (2.7) : The composition operator C  induced by  is defined on H 2 as follows C f  f   . 77 Journal of Kufa for Mathematics and Computer Vol.2,no.1,may,2014,pp.76- 81 Definition (2.14) : [6] Let g  H  , the Toeplits operator Tg is Proposition (2.8) : For each f  H 2 we have f    H 2 and and f    1  0 1  0 1  0 1  0 f T f z  gz f z f  H f . , 2 g Proof : Since  : U  U is holomorphic map on U, then by (2.6) f    H 2 and f   the operator on H 2 given by : hence C :  , zU . Theorem (2.15) : [6] If  : U  U is holomorphic map on U, then C  Tg  Tg   C  (g  H  ) Remark ( 2.16) : [8] For each f  H 2 , it is well- know that Th f  Th f , such that h H  . H2  H2 Remark ( 2.9) : [4] 1) One can easily show that and hence C  C  C Proposition (2.17) : If   U , then C  = Tg C  Th* , where Cn  C C C h z   z   , gz    C  Cn 2) C  is the identity operator on H 2 if and only if  is identity map from U into U and holomorphic on U. 3) It is simple to prove that C  C if and only if    . 1 , z  1 .  z   z  z  Proof : By (2.16), Th f  Th f for each f  H 2 . Hence for all   U , Th f , K   Th f , K   f , Th K  On the other hand , Th f , K   f , Th K   f , h K  Definition (2.10) : [3] Let T be an operator on a Hilbert space H, The operator T  is the adjoint of T if Tx, y  x, T  y for each x, y  H .  (1) From (1) and (2) one  (2) can see  h Calculation give C k  z  = k    z  Theorem (2.11) : [5] K  U that T k   h  k  . Hence T k   h  k  .  h  2 forms a dense subset of H . Theorem (2.12) : [9] = If  : U  U is holomorphic map on U, then for all   U 1 1     z 1     z  z   C K   K    =     Definition (2.13) : [10] Let H  be the set of all bounded holomorphic maps on U . 1  1  1  z    1 z     z     z 1 1   z 1  1     z   = h  Tg k  z   Tg h k  z   Tg h  C  k  z  = Tg C  h  k  z  78 Ammar Ali Neamah and Aqeel Mohammed Hussein Tg = Tg C  Th k  z  , therefore C k  z   Tg C  T k  z     h z  U . But K  U  H 2 , then C  Tg C  Th    and Th are bounded operators then C  C  is compact by ( 2.18) Conversely, Suppose that C  C compact. C  C Note  C  Tg C  Th Let T be an operator on a Hilbert space H , T is called compact, if every sequence x n in H is weakly converges to W i.e. x n  x x n , u  x, u , u  H )) then Tx n in H (( if is S strongly converges to Tx (( i.e. x n   x if x n  x  0 )) Theorem (2.19) : [9] If  : U  U is holomorphic map on U, then C is not compact if and only if  take U into U . Proposition (2.20) : If  U , then C  is not compact composition operator. Proof : From (1.4),  take U into U . By (2.19) C  is not compact composition operator. Theorem (2.21) : If  : U  U is holomorphic map on U, then C C is compact if and only if C C  is compact , where C  Tg C  Th , z  1 .  z   z   h C C  T (2.15) ).  = C Th C Tg = Th   C C Tg Since C C is compact operator , Th   and Tg are bounded operators by (2.13) and (2.14) then C C  is compact by ( 2.18). Corollary (2.22) : If  : U  U is holomorphic map on U, then C C is not compact if and only if there exist points z1 , z 2  U such that    z1   z 2 for each z 2  U . Proof : By (2.21) C C is not compact if and only if C C   C  is not compact. Since  : U  U and  : U  U are holomorphics on U, then also    . Thus by (2.19) C  is not compact if and only if    take U into U . So, there exist points z1 , z 2  U such that    z1   z 2 for each z 2  U . Theorem (2.23) : If  : U  U is holomorphic map on U, then C C is compact if and only if C  C is   compact , where C  Tg C  Th ,  z   z  1 . z  Proof : Suppose that C  C is compact . Note C C  =   ( since C Tg  Tg   C by Tg C  Th C ( since  h C  Tg C  T by ( 2.17) ) = Tg C  Th C ( by (2.16) ) = (since Tg T C  C Note that C C = C Tg C  Th ( since by (2.17) ) = C  Tg C  Th    = that Proof : Suppose that C C  is compact . Tg  C C  = (since C Th  Th   C by (2.15) ). Definition (2.18) : [11] x  that is h   C  Th  T h   C  by(2.15)). Since C  C is compact operator , Tg and T h   79 Journal of Kufa for Mathematics and Computer Since C  C  is compact operator , are bounded operators then C C is compact by ( 2.18) Conversely, Suppose that C C is compact . Note that   C  C = C  C = Tg C Th  C (since C  Tg C Th ) = Th C Tg C Note that , by ( 2.11) it is enough to prove the compactness on the family K     U . Hence for each z  U we have C  C K  z  = Th C Tg C K  z  = Th C Tg K  z  = Th C g K  z  (since Tg K   g  K  ) = g  Th C K  z  Vol.2,no.1,may,2014,pp.76- 81 [2] Appell, M. J. , Bourdon , P.S. & Thrall, J. J. , “ Norms of Composition Operators on the Hardy Space ” , Experimented Math. , pp.111-117, (1996). [3] Berberian, S. K. , “ Introduction to Hilbert Space ” , Sec. Ed. , Chelesa publishing Com., New York , (1976). [4] Bourdon, P.S. & Shapiro, J.H., “ Cyclic Phenomena for Composition Operators ” , Math. Soc., (596), 125, (1999). [5] Cowen, C. C. “ Linear Fraction Composition Operator on H 2 ”, Integral Equations Operator Theory , Vol(11) , pp. 151 -160, (1988). = g  Th C C K  z  Since C C is compact , Th is bounded and g  H  , then C  C  is compact by ( 2.18 ). [6] Deddnes, J. A. “ Analytic Toeplits and Composition Operators ” , Con . J. Math. , Vol(5), pp. 859- 865, (1972). Corollary (2.24) : [7] Duren, P. L., “ Theory of H p Space ” , Academic press , New york , (1970). If  : U  U is holomorphic map on U, then C C  is not compact if and only if there exist points z1 , z 2  U such that    z1   z 2 for each z 2  U . Proof : By (2.23) C C is not compact if and only if C C  C is not compact . Since  : U  U and  : U  U are holomorphic on U, then also    . Thus by (2.19) C  is not compact if and only if    take U into U . So, there exist points z1 , z 2  U such that    z1   z 2 for each z 2  U . References [1] Ahlfors, L. V. , “ Complex Analysis ”, Sec , Ed., McGraw-Hill Kogakusha Ltd , (1966). [8] Halmos , P. R. , “ A Hilbert Space Problem Book ” , Springer- Verlag , New York , (1982). [9] Shapiro, J. H. , “ Composition operators and Classical Function Theory ”, Springer- Verlage, New York, (1993). [10] Shapiro, J. H. , “ Lectures on Composition operators and Analytic Function Theory ” . www.mth.mus.edu. shapiro  pubrit  downloads  computer  complutro . pdf . [11] Zorboska , N. , “ Closed Range Essentially Normal Composition Operators are Normal ”, Acta Sic .Math. (Szeged), Vol(65), pp.287-292, (1999). 80 ‫‪Aqeel Mohammed Hussein‬‬ ‫‪Ammar Ali Neamah and‬‬ ‫المستخلص‪:‬‬ ‫نُكه ‪U‬‬ ‫‪‬‬ ‫‪(n ) z n‬‬ ‫^‬ ‫‪f‬‬ ‫َسمز إنً كسة انىحدة فٍ انمستىي انعقدٌ‪ ،‬إن فضاء هازدٌ ‪ H 2‬هى مجمىعت كم اندوال‬ ‫‪ f (z) ‬انتحهُهُت عهً ‪ U‬بحُث أن‬ ‫‪2‬‬ ‫‪f ^ (n )  ‬‬ ‫‪‬‬ ‫‪‬‬ ‫و ) ‪َ f ^ (n‬سمز إنً تُهس نهدانت ‪.f‬‬ ‫‪n  0‬‬ ‫‪n  0‬‬ ‫نتكه ‪  : U  U‬دانت تحهُهُت عهً ‪ ، U‬انمؤثس انتسكُبٍ انمعسف بـ ‪َ ‬عسف عهً فضاء هازدٌ ‪ H 2‬بانشكم‬ ‫انتانٍ ‪(f  H 2 ) :‬‬ ‫‪C f  f  ‬‬ ‫دزسىا فٍ هرا انبحث انمؤثس انتسكُبٍ انمعسف مه اندانت انمتقابهت ‪ ‬حُث واقشىا انمؤثس انمسافق نهمؤثس انتسكُبٍ‬ ‫انمعسف باندانت ‪ . ‬باإلضافت إنً ذنك وظسوا إنً بعض انىتائج انمعسوفت وحاونىا انحصىل عهً وتائج مىاظسة نىتمكه مه‬ ‫مالحظت كُفُت تغُس انىتائج عىدما تتغُس اندانت انتحهُهُت ‪ . ‬ومه أجم جعم مهمت انقازئ أكثس سهىنت‪ ،‬عسضىا بعض انىتائج‬ ‫انمعسوفت عه انمؤثساث انتسكُبُت وعسضىا بساهُه مفصهت وكرنك بسهىا بعض انىتائج ‪.‬‬ ‫‪81‬‬