Abstract
We determine the spectra of weighted composition operators acting on the weighted Banach spaces of analytic functionsH ν p ∞ when the symbolφ has a fixed point in the open unit disk. Further, we apply this result to give the spectra of composition operators on Bloch type spaces. In particular, we answer in the affirmative a conjecture by MacCluer and Saxe.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
K. D. Bierstedt, J. Bonet and J. Taskinen,Associated weights and spaces of holomorphic functions, Studia Mathematica127 (1998), 137–168.
J. Bonet, P. Domański, M. Lindström and J. Taskinen,Composition operators between weighted Banach spaces of analytic functions, Journal of the Australian Mathematical Society (Series A)64 (1998), 101–118.
J. Bonet, P. Domański and M. Lindström,Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions, Canadian Mathematical Bulletin42 (1999), 139–148.
P. Bourdon and J. Shapiro,Mean growth of Koenigs eigenfunctions, Journal of the American Mathematical Society10 (1997), 299–325.
M. D. Contreras and A. G. Hernandez-Díaz,Weighted composition operators in weighted Banach spaces of analytic functions, Journal of the Australian Mathematical Society (Series A)69 (2000), 41–60.
C. Cowen and B. MacCluer,Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995.
C. Cowen and B. MacCluer,Spectra of some composition operators, Journal of Functional Analysis125 (1994), 223–251.
H. Kamowitz,Compact operators of the form uC ϕ , Pacific Journal of Mathematics80 (1979), 205–211.
H. Kamowitz,The spectra of some composition operators, Journal of Functional Analysis18 (1975), 132–150.
B. MacCluer and K. Saxe,Spectra of composition operators on the Bloch and Bergman spaces, Israel Journal of Mathematics128 (2002), 325–354.
A. Montes-Rodríguez,Weighted composition operators on weighted Banach spaces of analytic functions, Journal of the London Mathematical Society61 (2000), 872–884.
A. Montes-Rodríguez,The essential norm of a composition operator on Bloch spaces, Pacific Journal of Mathematics188 (1999), 339–351.
J. H. Shapiro,Composition Operators and Classical Function Theory, Springer, Berlin, 1993.
A. L. Shields and D. L. Williams,Bounded projections, duality, and multipliers in spaces of analytic functions, Transactions of the American Mathematical Society162 (1971), 287–302.
P. Wojtaszczyk,Banach Spaces for Analysts, Cambridge University Press, Cambridge, 1991.
L. Zheng,The essential norms and spectra of composition operators on H ∞, Pacific Journal of Mathematics203 (2002), 503–510.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of the second author was partially supported by the Academy of Finland Project No. 51906; the research of this paper was carried out while this author was visiting Kent State University, whose hospitality is acknowledged with thanks.
Rights and permissions
About this article
Cite this article
Aron, R., Lindström, M. Spectra of weighted composition operators on weighted banach spaces of analytic functions. Isr. J. Math. 141, 263–276 (2004). https://doi.org/10.1007/BF02772223
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02772223