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Composition operators from the space of Cauchy transforms to Bloch and the little Bloch-type spaces on the unit disk. (English) Zbl 1220.30072

Summary: We completely characterize the boundedness and compactness of composition operators from the space of Cauchy transforms on the unit disk 𝔻 into the Bloch-type space ν , i.e., the space of all holomorphic functions f on 𝔻 such that

sup z𝔻 ν(z)|f ' (z)|<,

as well as into the little Bloch-type space ν,0 consisting of all holomorphic functions f on 𝔻 such that

lim |z|1 ν(z)|f ' (z)|=0

for some weight function ν. As a byproduct of our results, the norm of the operator is calculated when ν is replaced by ν /.

MSC:
30H99Spaces and algebras of analytic functions
References:
[1]Allen, R.; Collona, F.: Weighted composition operators on the Bloch space of a bounded homogeneous domain, Oper. theor.: adv. Appl. 202, 11-37 (2010) · Zbl 1213.47026
[2]Avetisyan, K.: Hardy Bloch-type spaces and lacunary series on the polydisk, Glasg. J. Math. 49, 345-356 (2007) · Zbl 1123.32004 · doi:10.1017/S001708950700359X
[3]Bourdon, P.; Cima, J. A.: On integrals of Cauchy – Stieltjes type, Houston J. Math. 14, 465-474 (1988) · Zbl 0684.30034
[4]Choa, J. S.; Kim, H. O.: Composition operators from the space of Cauchy transforms into its Hardy-type subspaces, Rockey mountain J. Math. 31, No. 1, 95-113 (2001) · Zbl 0990.47019 · doi:10.1216/rmjm/1008959670 · doi:http://math.la.asu.edu/~rmmc/rmj/VOL31-1/CONT31-1/CONT31-1.html
[5]Cima, J. A.; Matheson, A. L.: Cauchy transforms and composition operators, Illinois J. Math. 4, 58-69 (1998) · Zbl 0914.30023
[6]Galindo, P.; Lindström, M.: Essential norm of operators on weighted Bergman spaces of infinite order, J. oper. Theor. 64, No. 2, 387-399 (2010) · Zbl 1211.47064
[7]Li, S.; Stević, S.: Generalized composition operators on Zygmund spaces and Bloch type spaces, J. math. Anal. appl. 338, 1282-1295 (2008) · Zbl 1135.47021 · doi:10.1016/j.jmaa.2007.06.013
[8]Madigan, K.; Matheson, A.: Compact composition operators on the Bloch space, Trans. am. Math. soc. 347, No. 7, 2679-2687 (1995) · Zbl 0826.47023 · doi:10.2307/2154848
[9]Montes-Rodríguez, A.: Weighted composition operators on weighted Banach spaces of analytic functions, J. London math. Soc. 61, No. 3, 872-884 (2000) · Zbl 0959.47016 · doi:10.1112/S0024610700008875
[10]Rudin, W.: Function theory in the unit ball of cn, (1980)
[11]H.J. Schwartz, Composition operators on Hp, Thesis, University of Toledo 1969.
[12]Shaabani, M. H.; Robati, B. K.: On the norm of certain weighted composition operators on the Hardy space, Abstr. appl. Anal. 2009, 13 (2009) · Zbl 1176.47023 · doi:10.1155/2009/720217
[13]Sharma, A. K.; Sharma, S. D.: Weighted composition operators between Bergman-type spaces, Comm. korean math. Soc. 21, 465-474 (2006) · Zbl 1160.47308 · doi:10.4134/CKMS.2006.21.3.465
[14]Shields, A. L.; Williams, D. L.: Bounded projections, duality, and multipliers in spaces of analytic functions, Trans. am. Math. soc. 162, 287-302 (1971) · Zbl 0227.46034
[15]Stević, S.: Norm of weighted composition operators from Bloch space to Hμ on the unit ball, Ars combin. 88, 125-127 (2008) · Zbl 1224.30195
[16]Stević, S.: Norms of some operators from Bergman spaces to weighted and Bloch-type space, Util. math. 76, 59-64 (2008) · Zbl 1160.47027
[17]Stević, S.: Norm and essential norm of composition followed by differentiation from α-Bloch spaces to Hμ, Appl. math. Comput. 207, 225-229 (2009) · Zbl 1157.47026 · doi:10.1016/j.amc.2008.10.032
[18]Stević, S.: Norm of weighted composition operators from α-Bloch spaces to weighted-type spaces, Appl. math. Comput. 215, 818-820 (2009) · Zbl 1181.32011 · doi:10.1016/j.amc.2009.06.005
[19]Stević, S.: On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball, J. math. Anal. appl. 354, 426-434 (2009) · Zbl 1171.47028 · doi:10.1016/j.jmaa.2008.12.059
[20]Stević, S.: On an integral operator from the Zygmund space to the Bloch-type space on the unit ball, Glasg. J. Math. 51, 275-287 (2009) · Zbl 1176.47029 · doi:10.1017/S0017089508004692
[21]Stević, S.: Weighted composition operators between Fock-type spaces in CN, Appl. math. Comput. 215, 2750-2760 (2009) · Zbl 1186.32003 · doi:10.1016/j.amc.2009.09.016
[22]Stević, S.: Weighted composition operators from weighted Bergman spaces to weighted-type spaces on the unit ball, Appl. math. Comput. 212, 499-504 (2009) · Zbl 1186.47020 · doi:10.1016/j.amc.2009.02.057
[23]Stević, S.: Norm and essential norm of an integral-type operator from the Dirichlet space to the Bloch-type space on the unit ball, Abstr. appl. Anal. 2010, 9 (2010)
[24]Stević, S.: Norms of some operators on bounded symmetric domains, Appl. math. Comput. 216, 187-191 (2010) · Zbl 1209.32010 · doi:10.1016/j.amc.2010.01.030
[25]Stević, S.: On an integral operator between Bloch-type spaces on the unit ball, Bull. sci. Math. 134, 329-339 (2010) · Zbl 1189.47032 · doi:10.1016/j.bulsci.2008.10.005
[26]Stević, S.: Weighted differentiation composition operators from the mixed-norm space to the nth weighted-type space on the unit disk, Abstr. appl. Anal. 2010, 15 (2010) · Zbl 1198.30014 · doi:10.1155/2010/246287
[27]Stević, S.: Weighted iterated radial composition operators between some spaces of holomorphic functions on the unit ball, Abstr. appl. Anal. 2010, 14 (2010) · Zbl 1207.47022 · doi:10.1155/2010/801264
[28]Stević, S.; Ueki, S. I.: On an integral-type operator acting between Bloch-type spaces on the unit ball, Abstr. appl. Anal. 2010, 14 (2010)
[29]Ueki, S. I.: Hilbert-Schmidt weighted composition operators on the Fock space, Int. J. Math. anal. 1, No. 16, 769-774 (2007) · Zbl 1160.47306
[30]Ueki, S. I.: Weighted composition operators on the Bargmann – Fock space, Int. J. Mod. math. 3, No. 3, 231-243 (2008) · Zbl 1171.47021
[31]Ueki, S. I.: Weighted composition operators on some function spaces of entire functions, Bull. belg. Math. soc. Simon stevin 17, No. 2, 343-353 (2010) · Zbl 1191.47032 · doi:euclid:bbms/1274896210
[32]Yang, W.; Meng, X.: Generalized composition operators from F(p,q,s) spaces to Bloch-type spaces, Appl. math. Comput. 218, 1-9 (2010)
[33]Zhu, X.: Weighted composition operators from area Nevanlinna spaces into Bloch spaces, Appl. math. Comput. 215, No. 12, 4340-4346 (2010) · Zbl 1185.30058 · doi:10.1016/j.amc.2009.12.064