Harmonic Bloch Function Spaces and their Composition Operators


Download PDF

Authors: S. ESMAEILI, Y. ESTAREMI AND A. EBADIAN

DOI: 10.46793/KgJMat2404.535E

Abstract:

In this paper we characterize some basic properties of composition operators on the spaces of harmonic Bloch functions. First we provide some equivalent conditions for boundedness and compactness of composition operators. In the sequel we investigate closed range composition operators. These results extends the similar results that were proven for composition operators on the Bloch spaces.



Keywords:

Composition operator, Bloch spaces, harmonic function.



References:

[1]   S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, Graduate Texts in Mathematics 137, Springer, New York, 1992.

[2]   H. Chen and P. Gauthier, Boundedness from below of composition operator on α-Bloch spaces, Canad. Math. Bull. 51 (2008), 195–204.

[3]   F. Colonna, The Bloch constant of bounded harmonic mappings, Indiana Univ. Math. J. 38 (1989), 829–840.

[4]   M. Contreras and A. Hernandez-Diaz, Weithed composition operator in Weithed Banach spaces of analytic functions, J. Aust. Math. Soc. 69 (2000), 41–60.

[5]   C. Cowen and B. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995.

[6]   P. Duren, Harmonic Mapping in the Plane, Cambridge Univ. Press, Cambridge, 2004.

[7]   P. Ghatage, D. Zheng and N. Zorboska, Sampling set and closed-range composition operators on the Bloch space, Proc. Amer. Math. Soc. 133 (2005), 1371–1377.

[8]   Z. Lou, Composition operator on Bloch type spaces, Analysis 23(2003), 81–95.

[9]   K. Madigan and A. Matheson, Compact composition operator on the Bloch spaces, Trans. Amer. Math. Soc. 347 (1995), 2679–2687.

[10]   J. H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag, New York, 1993.

[11]   N. Zorboska, Isometric and closed-range composition operators between Bloch-type spaces, Int. J. Math. Math. Sci. (2011), Article ID 132541. https://doi.org/10.1155/2011/132541.