Abstract
We investigate the continuity, compactness and invertibility of weighted composition operators \(W_{\psi ,\varphi }{:}\, f \rightarrow \psi (f \circ \varphi )\) when they act on the classical Korenblum space \(A^{-\infty }\) and other related Fréchet or (LB)-spaces of analytic functions on the open unit disc which are defined as intersections or unions of weighted Banach spaces with sup-norms. Some results about the spectrum of these operators are presented in case the self-map \(\varphi \) has a fixed point in the unit disc. A precise description of the spectrum is obtained in this case when the operator acts on the Korenblum space.
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Abramovich, Y.A., Aliprantis, C.D.: An invitation to operator theory. Graduate Studies in Mathematics. Amer. Math. Soc., 50 (2002)
Albanese, A.A., Bonet, J., Ricker, W.J.: The Cesàro operator in the Fréchet spaces \(\ell ^{p+}\) and \(L^{p-}\). Glasgow Math. J. 59, 273–287 (2017)
Albanese, A.A., Bonet, J., Ricker, W.J.: The Cesàro operator on Korenblum type spaces of analytic functions. Collect. Math. 69(2), 263–281 (2018)
Albanese, A.A., Bonet, J., Ricker, W.J.: Operators on the Fréchet sequence spaces \(ces(p+), 1\le p\le \infty \). Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113(2), 1533–1556 (2019)
Albanese, A.A., Bonet, J., Ricker, W.J.: Linear operators on the (LB)-sequence spaces \(ces(p-), 1\le p\le \infty \). Descriptive topology and functional analysis. II, 43–67, Springer Proc. Math. Stat., 286, Springer, Cham (2019)
Arendt, W., Chalendar, I., Kumar, M., Srivastava, S.: Powers of composition operators: asymptotic behaviour on Bergman, Dirichlet and Bloch spaces. J. Austral. Math. Soc. 1–32. https://doi.org/10.1017/S1446788719000235
Aron, R., Lindström, M.: Spectra of weighted composition operators on weighted Banach spaces of analytic funcions. Israel J. Math. 141, 263–276 (2004)
Bierstedt, K.D., Summers, W.H.: Biduals of weighted Banach spaces of analytic functions. J. Austral. Math. Soc., Ser. A, 54(1), 70–79 (1993)
Bonet, J.: A note about the spectrum of composition operators induced by a rotation. RACSAM 114, 63 (2020). https://doi.org/10.1007/s13398-020-00788-5
Bonet, J., Domański, P., Lindström, M., Taskinen, J.: Composition operators between weighted Banach spaces of analytic functions. J. Austral. Math. Soc., Ser. A, 64(1), 101–118 (1998)
Bourdon, P.S.: Essential angular derivatives and maximum growth of Königs eigenfunctions. J. Func. Anal. 160, 561–580 (1998)
Bourdon, P.S.: Invertible weighted composition operators. Proc. Am. Math. Soc. 142(1), 289–299 (2014)
Carleson, L., Gamelin, T.: Complex Dynamics. Springer, Berlin (1991)
Cowen, C., MacCluer, B.: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton, FL (1995)
Contreras, M., Hernández-Díaz, A.G.: Weighted composition operators in weighted Banach spacs of analytic functions. J. Austral. Math. Soc., Ser. A 69, 41–60 (2000)
Eklund, T., Galindo, P., Lindström, M.: Königs eigenfunction for composition operators on Bloch and \(H^\infty \) spaces. J. Math. Anal. Appl. 445, 1300–1309 (2017)
Hedenmalm, H., Korenblum, B., Zhu, K.: Theory of Bergman Spaces. Grad. Texts in Math. 199. Springer, New York (2000)
Jarchow, H.: Locally Convex Spaces. Teubner, Stuttgart (1981)
Kamowitz, H.: Compact operators of the form \(uC_{\varphi }\). Pac. J. Math. 80(1) (1979)
Korenblum, B.: An extension of the Nevanlinna theory. Acta Math. 135, 187–219 (1975)
Köthe, G.: Topological Vector Spaces II. Springer, New York Inc (1979)
Lusky, W.: On the isomorphism classes of weighted spaces of harmonic and holomophic functions. Stud. Math. 75, 19–45 (2006)
Meise, R., Vogt, D.: Introduction to functional analysis. Oxford Grad. Texts in Math. 2, New York, (1997)
Montes-Rodríguez, A.: Weighted composition operators on weighted Banach spaces of analytic functions. J. Lond. Math. Soc. 61(3), 872–884 (2000)
Queffélec, H., Queffélec, M.: Diophantine Approximation and Dirichlet series. Hindustain Book Agency, New Delhi (2013)
Shapiro, J.H.: Composition Operators and Classical Function Theory. Springer, New York (1993)
Shields, A.L., Williams, D.L.: Bounded projections, duality and multipliers in spaces of analytic functions. Trans. Amer. Math. Soc. 162, 287–302 (1971)
Zhu, K.: Operator Theory on Function Spaces, Math. Surveys and Monographs, Amer. Math. Soc. 138 (2007)
Acknowledgements
This paper is part of the PhD thesis of the author, which is supervised by J. Bonet and P. Galindo. The author is thankful to them for their guidance and helpful suggestions. She also thanks the referees for the very careful reading of the manuscript.
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This research was partially supported by the research project MTM2016-76647-P and the grant BES-2017-081200.
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Gómez-Orts, E. Weighted composition operators on Korenblum type spaces of analytic functions. RACSAM 114, 199 (2020). https://doi.org/10.1007/s13398-020-00924-1
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DOI: https://doi.org/10.1007/s13398-020-00924-1
Keywords
- Weighted composition operator
- Compact operator
- Spectrum
- analytic functions
- Growth Banach spaces
- Korenblum space
- Fréchet spaces
- (LB)-spaces