Products of Composition, Multiplication and Differentiation between Hardy Spaces and Weighted Growth Spaces of the Upper-Half Plane

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1 Global Journal of Pure and Alied Mathematics. ISSN Volume 3, Number 9 (207), Research India Publications htt:// Products of Comosition, Multilication and Differentiation between Hardy Saces and Weighted Growth Saces of the Uer-Half Plane Zaheer Abbas and Pawan Kumar 2 Deartment of Mathematical Sciences, Baba Ghulam Shah Badshah University Rajouri, Jammu, India. 2 Deartment of Mathematics, Govt. Degree College Kathua, Jammu, India. Abstract Let be a holomorhic function of the uer-half lane Λ + and a holomorhic self-ma of Λ +. Let C, M and D denote, resectively, the comosition, multilication and differentiation oerators. In this aer, we chacterize boundedness of the oerators induced by roducts of these oerators acting between Hardy and growth saces of the uer-half lane. Key words and hrases: Comosition oerator, Differentiation oerator, Multilication oerator, Growth sace, Hardy sace, Uer-half lane Mathematics Subject Classification: Primary 47B33, 46E0; Secondary 30D55... INTRODUCTION Let G be a non-emty set, X a toological vector sace, F(G, X) the toological vector sace of functions from G to X with oint-wise vector sace oerations and φ G G be a function such that foφ F(G, X) for all f F(G, X). Then the

2 6304 Zaheer Abbas and Pawan Kumar linear transformation C φ F(G, X) F(G, X), defined as C φ (f) = foφ for all f F(G, X), is known as the comosition transformation induced by φ on the sace F(G, X). If C φ is continuous, then it is called the comosition oerator or substitution oerator induced by φ on the sacef(g, X). Let Λ + = {x + iy: x, y R, y > 0 } be the uer half-lane and <. Then the Hardy sace H (Λ + ) is the collection of all analytic functions f: Λ + C such that + su y > 0 f(x + iy) dx <. It is well known that H (Λ + ) is a Banach sace under the norm + f H (Λ + ) = [ su y > 0 f(x + iy) dx] and H 2 (Λ + ) is a Hilbert sace under the inner roduct:, f, g = + 2π f (x) g dx (x), f, g H 2 (Λ + ), where which exists almost everywhere on f (x) = lim y 0 f(x + iy),. These Hardy saces fall under the category of the functional Banach saces which consist of bonafide functions with continuous evaluation functionals. For any ositive real number, the growth sace A (Λ + ) consists of analytic functions f Λ + C such that f A (Λ + ) = su{(imz) f(z) } <. With the norm. A (Λ + ), A (Λ + ) is a Banach sace. Note that A (Λ + ) is the usual growth sace. For H(Λ + ) the multilication oerator M is defined by M f = f. The roduct of comosition and multilication oerators, denoted by W, and defined as W, = M o C, is known as weighted comosition oerator and has been studied intensively in recent times. The differentiation oerator denoted by D is defined by Df = f'. As a consequence of the Little - wood Subordination rincile, it is known that every analytic self-ma φof the oen unit disk D induces a bounded comosition oerator on Hardy and weighted Bergman saces of the oen unit disk D

3 Products of Comosition, Multilication and Differentiation between Hardy 6305 (see [3] and [7]). However, if we move to Hardy and weighted Bergman saces of the uer half-lane Λ +, the situation is entirely different. In fact, there exist analytic self-mas of the uer half-lane which do not induce comosition oerators on the Hardy saces and weighted Bergman saces of the uer half-lane. Interesting work on comosition oerators on the saces of uer Half lane have been done by many authors, to cite a few, Singh [0], Singh and Sharma [, 2], Sharma [8], Matache [7, 8], Sharma, Sharma and Shabir [9, 20], Stevic and Sharma [22, 23, 24, 26], Sharma, Sharma and Abbas [ 6]. Recently, some attention have been aid to the study concrete oerators and their roducts between saces of holomorhic functions, for examle, Sharma and Abbas [4], Sharma, Sharma and Abbas [5], Sharma and Abbas [3], Bhat, Abbas and Sharma[2], Kumar and Abbas [6], Abbas and Kumar[], Kumar and Abbas [5] and [4, 9, 2, 23, 25, 27] and the related references therein. We can define the roducts of comosition, multilication and differentiation oerators in the following six ways. (M C Df )( z ) = ( z ) f ( ( z ) ), (M DC f ) ( z ) = ( z ) ( z ) f ( ( z ) ), (C M Df ) ( z ) = ( ( z ) ) f ( ( z ) ), (DM C f ) ( z ) = ( z ) f ( ( z ) ) + ( z ) ( z ) f ( ( z ) ), (C DM f ) ( z ) = ( ( z ) ) f ( ( z ) ) + ( ( z ) ) f ( ( z ) ), (DC M f ) ( z ) = ( ( z ) ) ( z ) f ( ( z ) ) + ( ( z ) ) ( z ) f ( ( z ) ), for z Λ + and f H(Λ + ). Note that the oerator M C D induces many known oerators. If =, then M C D = C D, while when ( z ) = ( z ), then we get the oerator DC. If we ut (z) = z, then M C D = M D, that is, the roduct of differentiation oerator and multilication oerator. Also note that M DC = M C D and C M D = M o C D. Thus the corresonding characterizations of boundedness and comactness of M DC and C M D can be obtained by relacing, resectively by ' and o in the results stated for M C D.

4 6306 Zaheer Abbas and Pawan Kumar In order to treat these oerators in a unified manner, we introduce the following oerator Tg, h, f(z) = g(z) f ( ( z ) ) + h(z) f ( ( z ) ) where g, h H(Λ + ) and a holomorhic self-ma of Λ +. It is clear that comosition, multilication, differentiation oerators and all the roducts of the comosition, multilication and differentiation oerators defined above can be obtained from the oerator Tg, h, by fixing g and h. More secifically, we have C =T, 0,, M =T, 0, z, D =T0,, z, M C = T, 0,, C M = T o, 0,, C D = T0,,, D C =T0, ',, M D = T0,, z, D M = T ',, z, M C D= T0,,, M D C = T0, ',, C M D =T0, o,, DM C = T ', ',, C DM = T 'o, o,, DC M = T( 'o ) ', ( o ) ',. In this aer we characterize the boundedness of the oerator Tg, h, acting between Hardy saces and growth saces of the uer-half lane. Throughout this aer, constants are denoted by C, they are ositive and not necessarily the same at each occurrence. 2. BOUNDEDNESS OF T g,h, H (Λ + ) A (Λ + ) In this section, we characterize boundedness of T g,h, H (Λ + ) A (Λ + ). Theorem 2.. Let < and be a holomorhic self-ma of Λ +. Then T g,h, H (Λ + ) A (Λ + ) is bounded if and only if (i) M = (ii) N = su su g(z) <, h(z) <. + Moreover if T g,h, H (Λ + ) A (Λ + ) is bounded, then T g,h, H (Λ + ) A (Λ + ) ~ M + N. Proof: Firstly, suose that (i)and (ii)hold, then T g,h, f = su{im(z) (T A (Λ + ) g,h, f)(z) }.

5 Products of Comosition, Multilication and Differentiation between Hardy 6307 Now Im(z) (T g,h, f)(z) = (Imz) f(φ(z))g(z) + h(z) f (φ(z)) (Imz) ( f(φ(z)) g(z) + h(z) f (φ(z)) ) C f H (Λ + ) ( g(z) + h(z) ) + C(M + N) f H (Λ + ). Thus, (T g,h, f)f A (Λ + ) C(M + N) f H (Λ + ) and so T g,h, H (Λ + ) A (Λ + ) is bounded and T g,h, H (Λ + ) A (Λ + ) C(M + N). (2.) Conversely, suose that T g,h, H (Λ + ) A (Λ + ) is bounded. Let z 0 Λ + be fixed and let ω = φ(z 0 ). Consider the function f ω (z) = (Im(ω))2 2i (Im(ω)) 3. π(z ω ) 2 π(z ω ) 3 Writing z = x + iy and ω = u + iv and using the elementary inequality (x + y) a 2 a (x a + y a ) which holds for all x, y 0and a > 0, we have f H (Λ + ) 2 [ su y > 0 v 2 dx π (x + iy) (u iv) 2 Again using the inequalities + su y > 0 2v 3 π (x + iy) (u iv) 3 dx (x + iy) (u + iv) 2 (v + y) 2 2 ((x u) 2 + (y + v) 2 ) ].

6 6308 Zaheer Abbas and Pawan Kumar and we get (x + iy) (u + iv) 3 (v + y) 3 2 ((x u) 2 + (y + v) 2 ), f H (Λ + ) su 2 2 [v y > 0 (y + v) 2 π y + v (x u) 2 + (y + v) 2 dx Also, Moreover, and f ω (φ(z 0 )) = 0. su 2 3 +v y > 0 (y + v) 3 π y + v (x u) 2 + (y + v) 2 dx ] = 2 su 2 [v y > 0 (y + v) f ω (z) = 2(Im(ω))2 π(z ω ) 3 2 +v3 su f ω (φ(z 0 )) = ( 4i + 3i 8 ) π 2 y > 0 (y + v) 3 ] i (Im(ω)) 3 π(z ω ) 4 Since T g,h, H (Λ + ) A (Λ + ) is bounded, we have (Im(ω)) + T g,h, f ω T A (Λ + ) g,h, H (Λ + ) A (Λ + ) f ω H (Λ + ) 2 +2 T g,h, H (Λ + ) A (Λ + ). This imlies for each, we have 2 +2 T g,h, H (Λ + ) A (Λ + ) (T g,h, f) (z) = (Imz) f(φ(z))g(z) + h(z)f (φ(z)).

7 Products of Comosition, Multilication and Differentiation between Hardy 6309 In articular, take z = z 0, we get (Im(z 0 )) h(z) + 3i 2 +2 T g,h, H (Λ + ) A (Λ + ) 4 Since z 0 Λ + is arbitrary, we have (Imφ(z 0 )) + 8 π N = su φ (z) 2 +2 T g,h, H + (Λ + ) A (Λ + ). (2.2) Again consider the function f ω (z) = 3i(Im(ω))2 π(z ω ) (Im(ω)) 3, ω = φ(z π(z 0 ). ω ) 3 Once again it is easy to rove that f H (Λ + ) 2 7. Also, Thus f ω (φ(z 0 )) = 0 and f ω (z) = 6i(Im(ω))2 π(z ω ) 3 2 (Im(ω)) 3 π(z ω ) 4 f ω (φ(z 0 )) = ( i π 4 ) (Im(ω)) Since T g,h, H (Λ + ) A (Λ + ) is bounded, there exists a ositive constant C such that T g,h, H (Λ + ) A (Λ + ) T g,h, f ω A (Λ + ) (Im(z 0 )) f (φ(z 0 ))g(z 0 ) + h(z 0 )f (φ(z 0 )) 3 8 (Im(z 0 )) g(z 0 ). (Im(φ(z 0 )))

8 630 Zaheer Abbas and Pawan Kumar Since z 0 Λ + is arbitrary, we have M = su g(z) C T g,h, H (Λ + ) A (Λ + ). (2.3) From (2.2) and (2.3), we have M + N C T g,h, H (Λ + ) A (Λ + ). (2.4) From (2.) and (2.4), we have Corollary 2.2. Let < and be a holomorhic self-ma of the uer halflane Λ +. Then T g,h, H (Λ + ) A (Λ + ) ~ M + N. C φ H (Λ + ) A (Λ + ) is bounded if and only if su <. Corollary 2.3.Let < and ψ H(Λ + ) and. Then M ψ H (Λ + ) A (Λ + ) is bounded if and only if X, where ψ A (Λ + ) if >. ψ X = { A if >. H if = Corollary 2.4.Let < and be a holomorhic self-ma of the uer halflane Λ +. Then Corollary 2.5.Let < and be a holomorhic self-ma of the uer halflane Λ +. Then C φ D H (Λ + ) A (Λ + ) is bounded if and only if su <. + DC φ H (Λ + ) A (Λ + ) is bounded if and only if

9 Products of Comosition, Multilication and Differentiation between Hardy 63 su ( z ) <. + Corollary 2.6.Let <, ψ H(Λ + )and uer half-lane Λ +. Then su be a holomorhic self-ma of the M ψ C φ H (Λ + ) A (Λ + ) is bounded if and only if Corollary 2.7.Let <, ψ H(Λ + ) and uer half-lane Λ +. Then su ψ(z) <. be a holomorhic self-ma of the C φ M ψ H (Λ + ) A (Λ + ) is bounded if and only if Corollary 2.8.Let <, ψ H(Λ + ) and uer half-lane Λ +. Then if su ψ(φ(z)) <. be a holomorhic self-ma of the M ψ C φ D H (Λ + ) A (Λ + ) is bounded if and only + Corollary 2.9.Let <, ψ H(Λ + ) and uer half-lane Λ +. Then if su ψ(z) <. be a holomorhic self-ma of the M ψ DC φ H (Λ + ) A (Λ + ) is bounded if and only + Corollary 2.0.Let <, ψ H(Λ + ) and uer half-lane Λ +. Then if su ψ(z) ( z ) <. be a holomorhic self-ma of the C φ M ψ D H (Λ + ) A (Λ + ) is bounded if and only + ψ( (z)) <.

10 632 Zaheer Abbas and Pawan Kumar Corollary 2..Let <, ψ H(Λ + ) and uer half-lane Λ +. Then su su be a holomorhic self-ma of the DM ψ C φ : H (Λ + ) A (Λ + ) is bounded if and only if + Corollary 2.2.Let <, ψ H(Λ + ) and uer half-lane Λ +. Then if su su ψ (z) < ψ(z) ( z ) <. be a holomorhic self-ma of the C φ DM ψ H (Λ + ) A (Λ + ) is bounded if and only + Corollary 2.3.Let <, ψ H(Λ + ) and uer half-lane Λ +. Then if Examle 2.4. Let su su ψ (φ(z)) < ψ(φ(z)) <. be a holomorhic self-ma of the DC φ M ψ H (Λ + ) A (Λ + ) is bounded if and only + ψ (φ(z)) ( z ) < ψ(φ(z)) ( z ) <. φ(z) = az + b cz + d, a, b, c, d R, ad bc > 0. Then DC φ H (Λ + ) A (Λ + ) is bounded if and only if c = 0 and = + /.

11 Products of Comosition, Multilication and Differentiation between Hardy 633 Proof: First suose that c = 0 and = + /. then su ( z ) = su y 2+ + ( a d y)+ a d, z = x + iy = ( a d )+ <. Thus C φ D H (Λ + ) A (Λ + ) is bounded. Again suose that c 0 or + /. Then Therefore, Im(φ(z)) = (ad bc)y ad (cx + d) 2 + c 2 and ( z ) = y2 su ( z ) + bc (cx + d) 2 + c 2 y 2. = = su y ((cx + d) 2 + c 2 y 2 ) + ((ad bc)y) + su y (+ ) ((cx + d) 2 + c 2 y 2 ) (ad bc) + ad bc (cx + d) 2 + c 2 y 2 =, and sodc φ H (Λ + ) A (Λ + ) is unbounded. Hence we are done Examle 2.5. Let φ(z) = az + b cz + d, a, b, c, d R, ad bc > 0. Then C φ D H (Λ + ) A (Λ + ) is bounded if and only if c = 0 and = + /. Proof: First suose that c = 0 and = + /. then for z = x + iy, we have

12 634 Zaheer Abbas and Pawan Kumar su + = su y 2+ = ( a ( a d )+ <, d y)+ Thus C φ D H (Λ + ) A (Λ + ) is bounded. Again suose that c 0 or + /. Then Therefore, Im(φ(z)) = (ad bc)y (cx + d) 2 + c 2 y 2. su = + su y ((cx + d) 2 + c 2 y 2 ) + ((ad bc)y) + = su y (+ ) ((cx + d) 2 + c 2 y 2 ) + (ad bc) + =, and soc φ D H (Λ + ) A (Λ + ) is unbounded. Hence the roof. REFERENCES [] Z. Abbas, P. Kumar, Product of multilication, comosition and differentiation oerators from weighted Bergman-Nevanlinna saces to Zygmund saces, SS International Journal of Pure and Alied Mathematics, Vol., Issue 2, 205 [2] A. Bhat, Z. Abbas, A. K. Sharma, Comosition followed by differentiation between Weighted Bergman Nevanlinna Saces, Mathematica Aeterna, Vol. 2, no. 5, , 202 [3] C. C. Cowen and B. D. MacCluer, Comosition oerators on saces of analytic functions, CRC ress, Boca Raton, New York, 995. [4] R. A. Hibschweiler and N. Portnoy. Comosition followed by differentiation between Bergman and Hardy saces. Rocky Mountain J. Math. 35 (2005), [5] P. Kumar, Z. Abbas, Comosition oerators between weighted Hardy tye saces, International journal of ure and alied mathematics, Vol. 07, no. 3: , 205

13 Products of Comosition, Multilication and Differentiation between Hardy 635 [6] P. Kumar, Z. Abbas, Product of multilication and comosition oerators on weighted Hardy saces, SS International Journal of Pure and Alied Mathematics, Vol., Issue 2, 205 [7] V. Matache, Comact comosition oerators on H of a half-lane, Proc. An. Univ. Timisoara Ser. Stiint. Mat. 27 (989) [8] V. Matache, Comact comosition oerators on Hardy saces of a half-lane, Proc. Amer. Math. Soc. 27(999) [9] S. Ohno. Products of comosition and differentiation between Hardy saces, Bull. Austral. Math. Soc 73 (2006), [0] R. K. Singh, A relation between comosition oerators on H (D) and H (Λ + ), Mathematika Sciences, (975), -5. [] R. K. Singh and S. D. Sharma, Comosition oerators on a functional Hilbert sace, Bull. Austral. Math. Soc. 20(979), [2] R. K. Singh and S. D. Sharma, Non comact Comosition oerators, Bull. Austral. Math. Soc. 2(980), [3] A. K. Sharma Z. Abbas, Weighted Comosition Oerators between Weighted Bergman-Nevanlinna and Bloch-Tye Saces, Jour. Al. Math. Sci., 4(4): , 200 [4] A. K. Sharma, Z. Abbas, Z. (2009), Comosition Proceeded and Followed by Differentiation between Weighted Bergman Nevanlinna and Bloch saces, Journal of Advanced Research in Pure Mathematics, (2): 53-62, online ISSN: [5] A. K. Sharma, R. Sharma and Z. Abbas, Weighted Comosition Oerators between Weighted Bergman-Nevanlinna and Growth Saces, Int. Journal of Math. Analysis, 4(26): , 200 [6] S. D. Sharma, A. K. Sharma, Z. Abbas, Weighted Comosition Oerators on Weighted Vector valued Begman saces, Alied Mathematical Sciences, Vol. 4, 200, no. 42, [7] J. H. Shairo Comosition oerator and classical function theory, Sringer- Verlag, New York, 993. [8] S. D. Sharma, Comact and Hilbert Schmidt comosition oerators on Hardy saces of the uer-half lane, Acta. Sci. Math. (Szeged), 46(983), [9] S. D. Sharma, A. K. Sharma, S. Ahmed, Carleson measures in a vector-valued Bergman sace, J. Anal. and Al. 4(2006), [20] S. D. Sharma, A. K. Sharma, S. Ahmed, Comosition oerators between Hardy and Bloch-tye saces of the uer half-lane, Bull. Korean Math. Soc. 43(2007), [2] S. Stevic, A. K. Sharma, Comosition oerators from the sace of Cauchy transforms to Bloch and the little Bloch-tye saces on the unit disk, Al. Math. Comut.27 (20),

14 636 Zaheer Abbas and Pawan Kumar [22] S. Stevic, A. K. Sharma and S. D. Sharma, Weighted comosition oerators from weighted Bergman saces to weighted tye saces of the uer halflane, Abstr. Al. Anal.(20), Article ID , 0 ages. [23] S. Stevic, A. K. Sharmaand A. Bhat, Products of comosition multilication and differentiation between weighted Bergman saces, Al. Math. Comut.27 (20), [24] S. Stevic, A. K. Sharma, Weighted comosition oerators between Hardy and growth saces of the uer half-lane, Al. Math. Comut. 27 (20) [25] S. Stevic, A. K. Sharma, Essential norm of comosition oerators between weighted Hardy saces, Al. Math. Comut. 27 (20) [26] S. Stevic, A. K. Sharma, Weighted comosition oerators between growth saces of the uer-half lane, Util. Math, 84 (20) [27] X. Zhu, Products of differentiation, comosition and multilication from Bergman tye saces to Bers tye sace, Integ. Tran. Sec. Function.,8 (3) (2007),