Composition operators on weighted spaces of holomorphic functions on the upper half plane

  • Wolfgang Lusky

Abstract

We consider moderately growing weight functions $v$ on the upper half plane $\mathbb G$ called normal weights which include the examples $(\mathrm{Im} w)^a$, $w \in \mathbb G$, for fixed $a > 0$. In contrast to the comparable, well-studied situation of normal weights on the unit disc here there are always unbounded composition operators $C_{\varphi }$ on the weighted spaces $Hv(\mathbb G)$. We characterize those holomorphic functions $\varphi \colon \mathbb G \rightarrow \mathbb G$ where the composition operator $C_{\varphi } $ is a bounded operator $Hv(\mathbb G) \rightarrow Hv(\mathbb G)$ by a simple property which depends only on $\varphi $ but not on $v$. Moreover we show that there are no compact composition operators $C_{\varphi }$ on $Hv(\mathbb G)$.

References

Ardalani, M. A. and Lusky, W., Bounded operators on weighted spaces of holomorphic functions on the upper half-plane, Studia Math. 209 (2012), no. 3, 225–234. https://doi.org/10.4064/sm209-3-2

Ardalani, M. A. and Lusky, W., Weighted spaces of holomorphic functions on the upper halfplane, Math. Scand. 111 (2012), no. 2, 244–260. https://doi.org/10.7146/math.scand.a-15226

Bierstedt, K. D., Bonet, J., and Taskinen, J., Associated weights and spaces of holomorphic functions, Studia Math. 127 (1998), no. 2, 137–168.

Bonet, J., Domański, P., Lindström, M., and Taskinen, J., Composition operators between weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Ser. A 64 (1998), no. 1, 101–118.

Duren, P. L., Theory of $H^p$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970.

Harutyunyan, A. and Lusky, W., A remark on the isomorphic classification of weighted spaces of holomorphic functions on the upper half plane, Ann. Univ. Sci. Budapest. Sect. Comput. 39 (2013), 125–135.

Lusky, W., On weighted spaces of harmonic and holomorphic functions, J. London Math. Soc. (2) 51 (1995), no. 2, 309–320. https://doi.org/10.4064/sm175-1-2

Lusky, W., On the isomorphic classification of weighted spaces of holomorphic functions, Acta Univ. Carolin. Math. Phys. 41 (2000), no. 2, 51–60.

Shields, A. L. and Williams, D. L., Bonded projections, duality, and multipliers in spaces of analytic functions, Trans. Amer. Math. Soc. 162 (1971), 287–302. https://doi.org/10.2307/1995754

Shields, A. L. and Williams, D. L., Bounded projections, duality, and multipliers in spaces of harmonic functions, J. Reine Angew. Math. 299/300 (1978), 256–279.

Shields, A. L. and Williams, D. L., Bounded projections and the growth of harmonic conjugates in the unit disc, Michigan Math. J. 29 (1982), no. 1, 3–25.

Stanev, M. A., Weighted Banach spaces of holomorphic functions in the upper half plane, preprint arxiv:math/9911082 [math.FA], 1999.

Published
2018-02-20
How to Cite
Lusky, W. (2018). Composition operators on weighted spaces of holomorphic functions on the upper half plane. MATHEMATICA SCANDINAVICA, 122(1), 141-150. https://doi.org/10.7146/math.scand.a-97126
Section
Articles