Boundedness of composition operators on some analytic function spaces

A Kamal; S A El-Hafeez; Lalit Mohan Upadhyaya; Ayman Shehata

doi:10.48165/

Authors

A Kamal Department of Mathematics, College of Science and Arts in Muthnib, Qassim University, Qassim, Kingdom of Saudi Arabia.
S A El-Hafeez Department of Mathematics, Faculty of Science, Port Said University, Port Said, Egypt.
Lalit Mohan Upadhyaya Department of Mathematics, Municipal Post Graduate College, Mussoorie, Dehradun, Uttarakhand-248179, India.
Ayman Shehata Department of Mathematics, College of Science and Arts, Unaizah, Qassim University, Qassim, Kingdom of Saudi Arabia.

DOI:

https://doi.org/10.48165/

Keywords:

QK,ω(p, q) spaces, holomorphic functions and weighted α-Bloch space

Abstract

In this paper, we investigate the necessary and sufficient conditions for a composition operator Cφ to be bounded and compact from Bαω to QK,ω(p, q). Moreover, the necessary and sufficient condition for Cφ from the Dirichlet space D to the space QK,ω(p, q) to be compact is also given in terms of the map φ.

References

Ahmed, A. El-Sayed and Bahkit, M.A. Composition operators on some holomorphic Banach func tion spaces, to appear in Mathematica Scandinavica.

Aulaskari, R. and Lappan, P. (1994). Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal, Complex Analysis and its Applications (Eds. Y. Chung Chun et al.), Pitman Research Notes in Mathematics, Longman 305, 136–146.

Aulaskari, R., Lappan, P. and Zhao, R. (2001). On harmonic normal and Q#p functions, Illinois Journal of Mathematics, 45(2), 423–440.

Bourdon, B.S., Cima, J.A. and Matheson, A.L. (1999). Compact composition operators on BMOA, Trans. Amer. Math. Soc., 351, 2183–2169.

Dyakonov, K.M. (2002). Weighted Bloch spaces, Hp, and BMOA, J. Lond. Math. Soc. II. Ser., 65(2), 411–417.

Ess´n, M. and Wulan, H. (2002). On analytic and meromorphic functions and spaces of QK-type space, Illinois J. Math., 46, 1233–1258.

Ess´en, M., Wulan, H. and Xiao, J. (2006). Several function-theoretic characterizations of M¨obius invariant QK spaces, J. Funct. Anal., 230(1), 78–115.

Jiang, L. and He, Y. (2003). Composition operators from Bαto F(p, q, s) , Acta. Math. Sci Ser B, 23, 252–260.

Li, S. and Wulan, H. (2007). Composition operators on QK spaces, J. Math. Anal. Appl. 327, 948–958.

Madigan, P.K. and Matheson, A. (1997). Compact composition operators on Bloch space, Trans. Amer. Math. Soc., 347, 2679–2687.

Makhmutov, S. (1997). Integral characterizations of Bloch functions, New Zealand J. Math., 26, 201–212.

Ohno, S., Stroethoff, K. and Zhao, R. (2003). Weithted composition operators between Bloch type spaces, Rocky Mountain J. Math., 33, 191–215.

Rashwan, R.A., Ahmed, A. El-Sayed and Kamal, Alaa (2009). Some characterizations of weighted holomorphic Bloch space, Eur. J. Pure Appl. Math., 2, 250–267.

Rashwan, R.A., Ahmed, A. El-Sayed and Kamal, Alaa (2009). Integral characterizations of weighted Bloch spaces and QK,ω(p, q) spaces, Mathematica (Cluj), 51(74), No.1, 63–76. [15] Rudin, W. (1987). Real and Complex analysis, McGraw-Hill, New York. [16] Shapiro, J.H. (1993). Composition Operators and Classical Function Theory, Springer-Verlag, New York.

Smith, W. and Zhao, R. (1997). Compact composition operators into Qp spaces, Analysis, 17, 239–263.

Tjani, M. (2003). Compact composition operators on Besov spaces, Trans. Amer. Math. Soc., 355(11), 4683–4698.

Wulan, H. and Zhang, Y. (2008). Hadamard products and QK spaces, J. Math. Anal. Appl., 337, 1142–1150.

Wulan, H. and Zhu, K. (2006). QK type spaces of analytical functions, J. Funct. Spaces Appl., 4, 73–84.

Wulan, H. and Zhu, K. (2007). Derivative-free characterizations of QK spaces, J. Aust. Math. Soc., 82(2), 283–295.

Xiao, J. (2001). Holomorphic Q classes, Lecture Notes in Mathematics 1767, Springer, Berlin. [23] Zhao, R. (1996). On a general family of function spaces, Annales Academiae Scientiarum Fennicae. Series A I. Mathematica. Dissertationes. 105. Helsinki: Suomalainen Tiedeakatemia, 1–56. [24] Stroethoff, K. (1989). Besov-type characterisations for the Bloch space, Bull. Austral. Math. Soc., 39, 405–420.