Generalized weighted composition operators on Bloch-type spaces

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1 Zhu Joural of Iequalities ad Applicatios 2015) 2015:59 DOI /s R E S E A R C H Ope Access Geeralized weighted compositio operators o Bloch-type spaces Xiaglig Zhu * * Correspodece: jyuzxl@163.com Faculty of Iformatio Techology, Macau Uiversity of Sciece ad Techology, Aveida Wai Log, Taipa, Macau, Chia Abstract I this paper, we give three differet characterizatios for the boudedess ad compactess of geeralized weighted compositio operators o Bloch-type spaces, especially we characterize them i terms of the sequece of Bloch-type orms of the geeralized weighted compositio operator applied to the fuctios I j z)=z j. MSC: 47B38; 30H30 Keywords: geeralized weighted compositio operators; compositio operator; differetiatio operator; Bloch-type space 1 Itroductio Let D be a ope uit dis i the complex plae C ad HD) bethespaceofaalytic fuctios o D. For0<α <, the Bloch-type space or α-bloch space) B α is the space that cosists of all aalytic fuctios f o D such that B α f )= 1 z 2 ) α f z) <. B α becomes a Baach space uder the orm f B α = f 0) + B α f ). Whe α =1,B 1 = B is the well-ow Bloch space. See [1, 2] for more iformatio o Bloch-type spaces. Throughout this paper, ϕ deotes a ocostat aalytic self-map of D. Thecompositio operator C ϕ iduced by ϕ is defied by C ϕ f = f ϕ for f HD). For a fixed u HD), defie a liear operator uc ϕ as follows: uc ϕ f = uf ϕ), f HD). The operator uc ϕ is called the weighted compositio operator. The weighted compositio operator is a geeralizatio of the compositio operator ad the multiplicatio operator defied by M u f = uf. A basic problem cocerig compositio operators o various Baach fuctio spaces is to relate the operator theoretic properties of C ϕ to the fuctio theoretic properties of the symbol ϕ, which attracted a lot of attetio recetly; the reader ca refer to [3]. The differetiatio operator D is defied by Df = f, f HD). For a oegative iteger,wedefie D 0 f ) z)=f z), D f ) z)=f ) z), 1, f HD) Zhu; licesee Spriger. This is a Ope Access article distributed uder the terms of the Creative Commos Attributio Licese which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial wor is properly credited.

2 ZhuJoural of Iequalities ad Applicatios 2015) 2015:59 Page 2 of 9 Let ϕ be a aalytic self-map of D, u HD), ad let be a oegative iteger. Defie the liear operator D ϕ,u, called the geeralized weighted compositio operator, by see [4 6]) D ϕ,u f ) z)=uz) D f ) ϕz) ), f HD), z D. Whe =0aduz) =1,D ϕ,u is the compositio operator C ϕ.if =0,theD ϕ,u is the weighted compositio operator uc ϕ.if =1,uz) =ϕ z), the D ϕ,u = DC ϕ,whichwas studied i [7 10]. For uz)=1,d ϕ,u = C ϕd, which was studied i [7, 11 14]. For the study of the geeralized weighted compositio operator o various fuctio spaces, see, for example, [4 6, 15 19]. It is well ow that the compositio operator is bouded o the Bloch space by the Schwarz-Pic lemma. Compositio operators ad weighted compositio operators o Bloch-type spaces were studied, for example, i [20 28]. The product-type operators o or ito Bloch-type spaces have bee studied i may papers recetly, see [7 11, 13, 14, 18, 29 36] for example. I [27], Wula et al. obtaied a characterizatio for the compactess of the compositio operators actig o the Bloch space as follows. Theorem A Let ϕ be a aalytic self-map of D. The C ϕ : B B is compact if ad oly if lim ϕ j B =0. j I [14], Wu ad Wula obtaied two characterizatios for the compactess of the product of differetiatio ad compositio operators actig o the Bloch space as follows. Theorem B Let ϕ be a aalytic self-map of D, N. The the followig statemets are equivalet. a) C ϕ D : B B is compact. b) lim j C ϕ D I j B =0, where I j z)=z j. c) lim a 1 C ϕ D σ a z) B =0, where σ a z)=a z)/1 az) is the Möbius map o D. Motivated by Theorems A ad B, i this wor we show that D ϕ,u : Bα is bouded respectively, compact) if ad oly if the sequece j α 1 D ϕ,u Ij ) j= is bouded respectively, coverget to 0 as j ), where I j z)=z j. Moreover, we use two families of fuctios to characterize the boudedess ad compactess of the operator D ϕ,u. Throughout the paper, we deote by C a positive costat which may differ from oe occurrece to the ext. I additio, we say that A B if there exists a costat C such that A CB.ThesymbolA B meas that A B A. 2 Mai results ad proofs I this sectio, we give our mai results ad proofs. First we characterize the boudedess of the operator D ϕ,u : Bα. Theorem 1 Let be a positive iteger, 0<α, β <, u HD) ad ϕ be a aalytic selfmap of D. The the followig statemets are equivalet.

3 ZhuJoural of Iequalities ad Applicatios 2015) 2015:59 Page 3 of 9 a) The operator D ϕ,u : Bα is bouded. b) j j α 1 D ϕ,u Ij z) <, where I j z)=z j. c) u, 1 z 2 ) β uz) ϕ z) < ad D ϕ,u f a <, D ϕ,u h a <, where f a z)= 1 a 2 1 az) α ad h a z)= 1 a 2 ) 2, 1 az) α+1 z D. d) 1 z 2 ) β uz) ϕ z) 1 z 2 ) β u z) < ad <. 1 ϕz) 2 ) α+ 1 ϕz) 2 ) α+ 1 Proof a) b) This implicatio is obvious, sice for j N, the fuctio j α 1 I j is bouded i B α ad j α 1 I j B α 1. b) c) Assume that b) holds ad let Q = j j α 1 D ϕ,u Ij. For ay a D, itis easy to see that f a ad h a have bouded orms i B α.itisclearthat f a z)= 1 a 2) h a z)= 1 a 2) 2 Ɣj + α) j!ɣα) aj z j, Ɣj +1+α) j!ɣα +1) aj z j. By Stirlig s formula, we have Ɣj+α) j!ɣα) j α 1 as j. Usig liearity we get D ϕ,u f a C 1 a 2) a j j α 1 D ϕ,u Ij Q D ϕ,u h a B β C 1 a 2) 2 Therefore, by the arbitrariess of a D, ad j +1) a j j α 1 D ϕ,u I j Q. D ϕ,u f B a β <, D ϕ,u h B a β <. I additio, applyig the operator D ϕ,u to Ij with j =, +1,weobtai D ϕ,u I ) z)=u z)! ad D ϕ,u I +1) z)=u z) +1)!ϕz)+uz) +1)!ϕ z), while for j <, D ϕ,u Ij ) z) = 0. Thus, usig the boudedess of the fuctio ϕ, wehave u ad 1 z 2 ) β uz) ϕ z) <.

4 ZhuJoural of Iequalities ad Applicatios 2015) 2015:59 Page 4 of 9 c) d) Assume that c) holds. Let C 1 := D ϕ,u f B a β, C 2 := D ϕ,u h B a β. For w D,set g w z)= 1 w 2 1 wz) α α α + 1 w 2 ) 2, w D. 1 wz) α+1 It is easy to chec that g w B α, g w B α < for every w D.Moreover, D ϕ,u g B w β D ϕ,u f B w β + w D w D I additio, g ) ϕλ) It follows that C 1 + α α + C 2 <. α α + w D D ϕ,u h B w β ) ϕλ) =0, g +1) ) ϕλ) ϕλ) ϕλ) +1 = αα +1) α + 1) 1 ϕλ) 2 ). α+ C 1 + α α + C 2 > D ϕ,u g ϕλ) αα +1) α + 1) 1 λ 2 ) β uλ) ϕ λ) ϕλ) +1 1 ϕλ) 2 ) α+ 2.1) for ay λ D. For ay fixed r 0, 1), from 2.1)wehave 1 λ 2 ) β uλ) ϕ λ) 1 1 λ 2 ) β uλ) ϕ λ) ϕλ) +1 ϕλ) >r 1 ϕλ) 2 ) α+ ϕλ) >r r +1 1 ϕλ) 2 ) α+ C 1 + α α+ C 2 <. 2.2) r +1 αα +1) α + 1) From the assumptio that 1 z 2 ) β uz) ϕ z) <,weget 1 λ 2 ) β uλ) ϕ λ) ϕλ) r 1 λ 2 ) β uλ) ϕ λ) <. 2.3) ϕλ) r 1 ϕλ) 2 ) α+ 1 r 2 ) α+ Therefore, 2.2) ad2.3) yield the first iequality of d). Next, ote that C 1 D ϕ,u f ϕλ) B β αα +1) α + 1) 1 λ 2 ) β u λ) ϕλ) 1 ϕλ) 2 ) α+ 1 αα +1) α + ) 1 λ 2 ) β uλ) ϕ λ) ϕλ) +1 1 ϕλ) 2 ) α+

5 ZhuJoural of Iequalities ad Applicatios 2015) 2015:59 Page 5 of 9 for ay λ D.From2.1)weget 1 λ 2 ) β u λ) ϕλ) 1 ϕλ) 2 ) α+ 1 D ϕ,u f ϕλ) αα +1) α + 1) + α + )1 λ 2 ) β uλ) ϕ λ) ϕλ) +1 1 ϕλ) 2 ) α+ C 1 αα +1) α + 1) + α + )C 1 + αc 2 αα +1) α + 1) α + +1)C 1 + αc 2 αα +1) α + 1). By arbitrary λ D,weget 1 λ 2 ) β u λ) ϕλ) <. 2.4) λ D 1 ϕλ) 2 ) α+ 1 Combiig 2.4)withthefactthatu, similarly to the former proof, we get the secod iequality of d). d) a) For ay f B α,wehave 1 z 2 ) β D ϕ,u f ) z) = 1 z 2) β f ) ϕ)u ) z) 1 z 2) β uz) ϕ z) f +1) ϕz) ) + 1 z 2 ) β u z) f ) ϕz) ) C 1 z 2 ) β uz) ϕ z) 1 ϕz) 2 ) α+ f B α + C 1 z 2 ) β u z) 1 ϕz) 2 ) α+ 1 f B α, 2.5) where i the last iequality we used the fact that for f B α see [2]) Moreover 1 z 2 ) α f z) f 0) + + f ) 0) + 1 z 2 ) α+ f +1) z). D ϕ,u f ) 0) = f ) ϕ0) ) u0) u0) 1 ϕ0) 2 ) f α+ 1 B α. From d) we see that D ϕ,u f = D ϕ,u f ) 0) + 1 z 2 ) β D ϕ,u f ) z) <. Therefore the operator D ϕ,u : Bα is bouded. The proof is complete. ForthestudyofthecompactessofD ϕ,u : Bα, we eed the followig lemma, which ca be proved i a stadard way; see, for example, Propositio 3.11 i [3]. Lemma 2 Let be a positive iteger,0<α, β <, u HD) ad ϕ be a aalytic self-map of D. The D ϕ,u : Bα is compact if ad oly if D ϕ,u : Bα is bouded ad for ay

6 ZhuJoural of Iequalities ad Applicatios 2015) 2015:59 Page 6 of 9 bouded sequece f j ) j N i B α which coverges to zero uiformly o compact subsets of D, D ϕ,u f j 0 as j. Theorem 3 Let be a positive iteger, 0<α, β <, u HD) ad ϕ be a aalytic selfmap of D such that D ϕ,u : Bα is bouded. The the followig statemets are equivalet. a) D ϕ,u : Bα is compact. b) lim j j α 1 D ϕ,u Ij =0, where I j z)=z j. c) lim ϕa) 1 D ϕ,u f ϕa) =0ad lim ϕa) 1 D ϕ,u h ϕa) =0. d) 1 z 2 ) β uz) ϕ z) 1 z 2 ) β u z) lim =0 ad lim =0. ϕz) 1 1 ϕz) 2 ) +α ϕz) 1 1 ϕz) 2 ) +α 1 Proof a) b) Assume that D ϕ,u : Bα is compact. Sice the sequece {j α 1 I j } is bouded i B α ad coverges to 0 uiformly o compact subsets, by Lemma 2 it follows that j α 1 D ϕ,u Ij 0asj. b) c)supposethatb)holds.fixε >0adchooseN N such that j α 1 D ϕ,u Ij < ε for all j N. Letz D such that ϕz ) 1as. Arguig as i the proof of Theorem 1,wehave D ϕ,u f ϕz ) C 1 ϕz ) 2 ) ϕz ) j j α 1 D ϕ,u I j = C 1 ϕz ) 2 ) N 1 ϕz ) j j α 1 D ϕ,u I j + ϕz ) ) j j α 1 D ϕ,u I j CQ 1 ϕz ) N ) + Cε, j=n where Q = j j α 1 D ϕ,u Ij.Sice ϕz ) 1as,fromthelastiequalityad the arbitrariess of ε,wegetlim D ϕ,u f ϕz ) =0,i.e., lim ϕa) 1 D ϕ,u f ϕa) =0. Notice that N 1 j +1)r j = 1 rn Nr N 1 r), 0 r <1, 1 r) 2 arguigasitheproofoftheorem1,weget D ϕ,u h ϕz ) C 1 ϕz ) 2 ) 2 ϕz ) j j α D ϕ,u I j C 1 ϕz ) 2 N 1 ) 2 j +1) ϕz ) j j α 1 D ϕ,u Ij + C 1 ϕz ) 2 ) 2 j +1) ϕz ) j j α 1 D ϕ,u I j j=n C1 ϕz ) N N ϕz ) N 1 ϕz ) ) + Cε.

7 ZhuJoural of Iequalities ad Applicatios 2015) 2015:59 Page 7 of 9 Therefore, lim D ϕ,u h ϕz ) Cε. By the arbitrariess of ε, weobtaithedesired result. c) d) To prove d) we oly eed to show that if z ) N is a sequece i D such that ϕz ) 1as,the 1 z 2 ) β uz ) ϕ z ) 1 z 2 ) β u z ) lim =0, lim =0. 1 ϕz ) 2 ) α+ 1 ϕz ) 2 ) α+ 1 Let z ) N be such a sequece that ϕz ) 1as. Arguig as i the proof of Theorem 1,weobtai lim D ϕ,u g ϕz ) lim D ϕ,u f ϕz ) + α + α lim D ϕ,u h ϕz ) =0. Hece lim D ϕ,u g ϕz ) = 0. Similarly to the proof of Theorem 1,wehave!1 z 2 ) β uz ) ϕ z ) ϕz ) +1 D 1 ϕz ) 2 ) α+ ϕ,u g ϕz ) 0 as, which implies 1 z 2 ) β uz ) ϕ z ) 1 z 2 ) β uz ) ϕ z ) ϕz ) +1 lim = lim =0. 2.6) 1 ϕz ) 2 ) α+ 1 ϕz ) 2 ) α+ I additio, D ϕ,u f ϕz ) + + 1)!1 z 2 ) β uz ) ϕ z ) ϕz ) +1 1 ϕz ) 2 ) α+!1 z 2 ) β u z ) ϕz ) 1 ϕz ) 2 ) α+ 1. From 2.6)adtheassumptiothat D ϕ,u f ϕz ) 0as,wehave 1 z 2 ) β u z ) 1 z 2 ) β u z ) ϕz ) lim = lim =0, 1 ϕz ) 2 ) 1 ϕz ) 2 ) α+ 1 as desired. d) a) Assume that f ) N is a bouded sequece i B α covergig to 0 uiformly o compact subsets of D. By the assumptio, for ay ε >0,thereexistsδ 0, 1) such that 1 z 2 ) β ϕ z) uz) 1 ϕz) 2 ) α+ < ε ad 1 z 2 ) β u z) < ε 2.7) 1 ϕz) 2 ) α+ 1 whe δ < ϕz) <1.SupposethatD ϕ,u : Bα is bouded, by Theorem 1,wehave ad C 3 = 1 z 2 ) β u z) < 2.8) C 4 = 1 z 2 ) β uz) ϕ z) <. 2.9)

8 ZhuJoural of Iequalities ad Applicatios 2015) 2015:59 Page 8 of 9 Let K = {z D : ϕz) δ}.theby2.8)ad2.9)wehavethat 1 z 2 ) β D ϕ,u f ) z) i.e.,weget z K 1 z 2 ) β uz) ϕ z) f +1) ) ϕz) + 1 z 2 ) β u z) f ) ) ϕz) z K 1 z 2 ) β uz) ϕ z) + C \K C 4 z K f +1) D ϕ,u f = C 4 w δ f 1 ϕz) 2 ) α+ B α + C \K ) ϕz) + Cε f B α, ϕz) ) + C 3 z K f +1) f ) w) + C 3 w δ + Cε f B α + u0) f ) f ) w) 1 z 2 ) β u z) 1 ϕz) 2 ) α+ 1 f B α ϕ0) ). 2.10) Sice f coverges to 0 uiformly o compact subsets of D as, Cauchy s estimate gives that f ) 0as o compact subsets of D.Hece,lettig i 2.10)ad usig the fact that ε is a arbitrary positive umber, we obtai D ϕ,u f 0as. Applyig Lemma 2 the result follows. Competig iterests The author declares that they have o competig iterests. Acowledgemets The author was partially ported by the Macao Sciece ad Techology Developmet Fud No. 098/2013/A3), NSF of Guagdog Provice No. S ) ad NNSF of Chia No ). Received: 5 December 2014 Accepted: 28 Jauary 2015 Refereces 1. Zhu, K: Operator Theory i Fuctio Spaces. Deer, New Yor 1990) 2. Zhu, K: Bloch type spaces of aalytic fuctios. Rocy Mt. J. Math. 23, ) 3. Cowe, CC, MacCluer, BD: Compositio Operators o Spaces of Aalytic Fuctios. CRC Press, Boca Rato 1995) 4. Zhu, X: Products of differetiatio, compositio ad multiplicatio from Bergma type spaces to Bers type space. Itegral Trasforms Spec. Fuct. 18, ) 5. Zhu, X: Geeralized weighted compositio operators o weighted Bergma spaces. Numer. Fuct. Aal. Optim. 30, ) 6. Zhu, X: Geeralized weighted compositio operators from Bloch spaces ito Bers-type spaces. Filomat 26, ) 7. Hibschweiler, R, Portoy, N: Compositio followed by differetiatio betwee Bergma ad Hardy spaces. Rocy Mt. J. Math. 35, ) 8. Li, S, Stević, S: Compositio followed by differetiatio betwee Bloch type spaces. J. Comput. Aal. Appl. 9, ) 9. Li,S,Stević, S: Compositio followed by differetiatio betwee H ad α-bloch spaces. Houst. J. Math. 35, ) 10. Yag, W: Products of compositio differetiatio operators from Q K p, q) spaces to Bloch-type spaces. Abstr. Appl. Aal. 2009, Article ID ) 11. Liag, Y, Zhou, Z: Essetial orm of the product of differetiatio ad compositio operators betwee Bloch-type space. Arch. Math. 100, ) 12. Stević, S: Products of compositio ad differetiatio operators o the weighted Bergma space. Bull. Belg. Math. Soc. Simo Stevi 16, ) 13. Stević, S: Norm ad essetial orm of compositio followed by differetiatio from α-bloch spaces to H μ. Appl. Math. Comput. 207, ) 14. Wu, Y, Wula, H: Products of differetiatio ad compositio operators o the Bloch space. Collect. Math. 63, ) 15. Li, H, Fu, X: A ew characterizatio of geeralized weighted compositio operators from the Bloch space ito the Zygmud space. J. Fuct. Spaces Appl. 2013, Article ID )

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