Surveys in Mathematics and its Applications


ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 9 (2014), 139 -- 147

TOEPLITZ OPERATORS AND MULTIPLICATION OPERATORS IN THE COMMUTANT OF A COMPOSITION OPERATOR ON WEIGHTED BERGMAN SPACES

Mahmood Haji Shaabani and Bahram Khani Robati

Abstract. Let φ be an analytic self-map of D. We investigate which Toeplitz operators and multiplication operators commute with a given composition operator Cφ on Apα(D) for 1 < p <∞ and -1<α<∞. Let S be a bounded linear operator in the commutant of Cφ. We show that under a certain condition on S, S is a polynomial in Cφ.

2010 Mathematics Subject Classification: 47B33; 47B38.
Keywords: Toeplitz operator; Weighted Bergman spaces; Composition operator; Commutant; Multiplication operators.

Full text

References

  1. P. Ahern and Cuckovic, The theorem of Brown-Halmos type for Bergman space Toeplitz operators, Funct. Analysis, 187(2001), 200-210. MR1867348(2002h:47040). Zbl 0996.47037.

  2. B. Cload, Generating the commutant of a composition operator, in: Contemp. Math. 213, Amer. Math. Soc. (1998), 11-15. MR1601052(98i:47030). Zbl 0901.47017.

  3. B. Cload, Composition operators: hyperinvariant subspaces, quasi-normals, and isometries, Proc. Amer. Math. Soc. 127(6)(1999), 1697-1703. MR1476125(99i:47053). Zbl 0917.47027.

  4. B. Cload, Toeplitz operators in the commutant of a composition operator, Studia. Mathematica. 133(2) (1999), 187-196. MR1686697(2001c:47036). Zbl 0924.47017.

  5. C. C. Cowen, The commutant of an analytic Toeplitz operator, Trans. Amer. Math. Soc. 239 (1987), 1-31. MR0482347(58:2420). Zbl 0391.47014.

  6. C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, 1995. MR1397026(97i:47056). Zbl 0873.47017.

  7. C. C. Cowen and B. D. MacCluer, Some problems on composition operators, Contemp. Math. 213, Amer. Math. Soc., (1998), 17-25. MR1601056(99d:47029). Zbl 0908.47025.

  8. H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman Spaces, Springer-Verlag, 2000. MR1758653(2001c:46043). Zbl 0955.32003.

  9. A. E. Richman, Subnormality and composition operator on the Bergman space, Integral Equations Operator Theory, 45(1)(2003), 105-124. MR1952344(2004c:47050). Zbl 1041.47010.

  10. T. S. Worner, Commutant of certain composition operator, Ph.D. thesis, Purdue University, 1998. MR2699464.



M. Haji Shaabani B. Khani Robati
Shiraz University of Technology Shiraz University
Department of Mathematics, Department of Mathematics, College of Sciences,
Shiraz University of Technology, Shiraz University,
Shiraz 71555-313, IRAN. Shiraz 71454, IRAN.
E-mail: shaabani@sutech.ac.ir E-mail: bkhani@shirazu.ac.ir

http://www.utgjiu.ro/math/sma