# WEIGHTED COMPOSITION OPERATORS BETWEEN BERGMAN AND BLOCH SPACES

• Sharma, Ajay K. (SCHOOL OF APPLIED PHYSICS AND MATHEMATICS SHRI MATA VAISHNO DEVI UNIVERSITY) ;
• Kumari, Rekha (DEPARTMENT OF MATHEMATICS UNIVERSITY OF JAMMU)
• Published : 2007.07.31
• 99 21

#### Abstract

In this paper, we characterize the boundedness and compactness of weighted composition operators ${\psi}C{\varphi}f=\psi(f^{\circ}\varphi)$ acting between Bergman and Bloch spaces of holomorphic functions on the open unit disk D.

#### Keywords

Bergman spaces;Bloch spaces;little Bloch spaces;weighted composition operator

#### References

1. K. R. M. Attele, Multipliers of composition operators, Tokyo J. Math. 15 (1992), 185-198 https://doi.org/10.3836/tjm/1270130260
2. S. J. Axler, Zero multipliers of Bergman spaces, Canad. Math. Bull. 28 (1985), 237-242 https://doi.org/10.4153/CMB-1985-029-1
3. Z. Cuckovic and R. Zhao, Weighted composition operators on the Bergman space, J. London Math. Soc. 70 (2004), 499-511 https://doi.org/10.1112/S0024610704005605
4. M. D. Contreras and A. G. Hernandez-Diaz, Weighted composition operators on Hardy spaces, J. Math. Anal. Appl. 263 (2001), 224-233 https://doi.org/10.1006/jmaa.2001.7610
5. C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic functions, CRC Press Boca Raton, New York, 1995
6. F. Forelli, The isometries of HP spaces, Canad. J Math. 16 (1964), 721-728 https://doi.org/10.4153/CJM-1964-068-3
7. K. Hoffman, Banach spaces of analytic functions, Dover Publications, Inc., 1988
8. H. Kamowitz, Compact operators of the form ${\upsilon}C{\varphi}$, Pacific J. Math. 80 (1979),205-211 https://doi.org/10.2140/pjm.1979.80.205
9. V. Matache, Compact composition operators on Hardy spaces of a half-plane, Proc. Amer. Math. Soc. 127 (1999), 1483-1491 https://doi.org/10.1090/S0002-9939-99-05060-1
10. G. Mirzakarimi and K. Seddighi, Weighted composition operators on Bergman and Dirichlet spaces, Georgian. Math. J. 4 (1997), 373-383 https://doi.org/10.1023/A:1022946629849
11. R. K. Singh and S. D. Sharma, Composition operators on a functional Hilbert space, Bull. Austral. Math. Soc. 20 (1979), 277-284 https://doi.org/10.1017/S0004972700011084
12. S. Ohno, K. Stroethoff, and R. Zhao, Weighted composition operators between Bloch-type spaces, Rocky Mountain J. Math. 33 (2003), 191-215 https://doi.org/10.1216/rmjm/1181069993
13. S. Ohno and H. Takagi, Some properties of weighted composition operators on algebras of analytic functions, J. Nonlinear Convex Anal. 2 (2001), 369-380
14. S. Ohno and R. Zhao, Weighted composition operators on the Bloch spaces, Bull. Austral Math. Soc. 63 (2001), 177-185 https://doi.org/10.1017/S0004972700019250
15. K. Zhu, Operator theory in function spaces, Marcel Dekker, New York, 1990
16. H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman spaces, Springer, New York-Berlin, 2000
17. M. D. Contreras and A. G. Hernandez-Diaz, Weighted composition operators on spaces of functions with derivatiae in a Hardy space, J. Operator Theory 52 (2004), 173-184

#### Cited by

1. New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space vol.11, pp.1, 2013, https://doi.org/10.2478/s11533-012-0097-4