Norm of a composition operator from the space of Cauchy transforms into Zygmund-type spaces

Ajay K.  Sharma

Ajay K. Sharma Central Univ. Jammu, (Bagla), Raya-Suchani, Samba, India)

Keywords: composition operator, space of Cauchy transforms, Zygmund-type space, little Zygmund-type space

Abstract

The problem of evaluation of the norm of a composition operator acting between the spaces of holomorphic functions is quite difficult.
In the Hardy and weighted Bergman spaces, the norm of the composition operator is unknown even for the choice of a fairly simple symbol.
We compute the operator norm of composition operator acting between the space of Cauchy transforms and Zygmund-type spaces.
We also characterize bounded and compact composition operators acting between these spaces.

References

Avramidou, P.; Jafari, F. On norms of composition operators on Hardy spaces. Function spaces (Edwardsville, IL, 1998), 47--54, Contemp. Math., 232, Amer. Math. Soc., Providence, RI, 1999. MR1678319, DOI 10.1090/conm/232/03383

Bourdon, Paul; Cima, Joseph A. On integrals of Cauchy-Stieltjes type. Houston J. Math. 14 (1988), no. 4, 465--474. MR0998448

Choa, Jun Soo; Kim, Hong Oh. Composition operators from the space of Cauchy transforms into its Hardy-type subspaces. Rocky Mountain J. Math. 31 (2001), no. 1, 95--113. MR1821370, DOI 10.1216/rmjm/1008959670

Cima, Joseph A.; Matheson, Alec. Cauchy transforms and composition operators. Illinois J. Math. 42 (1998), no. 1, 58--69. MR1492039

Cima, J. A.; MacGregor, T. H. Cauchy transforms of measures and univalent functions. Complex analysis, I (College Park, Md., 1985–86), 78--88, Lecture Notes in Math., 1275, Springer, Berlin, 1987. MR0922293, DOI 10.1007/BFb0078345

Cowen, Carl C.; MacCluer, Barbara D. Composition operators on spaces of analytic functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995. {rm xii}+388 pp. ISBN: 0-8493-8492-3 MR1397026

Hammond, Christopher. The norm of a composition operator with linear symbol acting on the Dirichlet space. J. Math. Anal. Appl. 303 (2005), no. 2, 499--508. MR2122233, DOI 10.1016/j.jmaa.2004.08.049

Hibschweiler, R. A. Composition operators on spaces of Cauchy transforms. Studies on composition operators (Laramie, WY, 1996), 57--63, Contemp. Math., 213, Amer. Math. Soc., Providence, RI, 1998. MR1601068, DOI 10.1090/conm/213/02849

Hibschweiler, R. A. Composition operators on spaces of fractional Cauchy transforms. Complex Anal. Oper. Theory 6 (2012), no. 4, 897--911. MR2947965, DOI 10.1007/s11785-010-0104-3

Hibschweiler, R. A.; MacGregor, T. H. Closure properties of families of Cauchy-Stieltjes transforms. Proc. Amer. Math. Soc. 105 (1989), no. 3, 615--621. MR0938912, DOI 10.2307/2046908

Hibschweiler, R. A.; MacGregor, T. H. Multipliers of families of Cauchy-Stieltjes transforms. Trans. Amer. Math. Soc. 331 (1992), no. 1, 377--394. MR1120775, DOI 10.2307/2154014

Hibschweiler, R. A.; MacGregor, T. H. Bounded analytic families of Cauchy-Stieltjes integrals. Rocky Mountain J. Math. 23 (1993), no. 1, 187--202. MR1212736, DOI 10.1216/rmjm/1181072616

Madigan, Kevin; Matheson, Alec. Compact composition operators on the Bloch space. Trans. Amer. Math. Soc. 347 (1995), no. 7, 2679--2687. MR1273508, DOI 10.2307/2154848

Stević, Stevo. Generalized composition operators between mixed-norm and some weighted spaces. Numer. Funct. Anal. Optim. 29 (2008), no. 7-8, 959--978. MR2450200, DOI 10.1080/01630560802282276

Stević, Stevo; Sharma, Ajay K. Composition operators from the space of Cauchy transforms to Bloch and the little Bloch-type spaces on the unit disk. Appl. Math. Comput. 217 (2011), no. 24, 10187--10194. MR2806404, DOI 10.1016/j.amc.2011.05.014