Hostname: page-component-7479d7b7d-q6k6v Total loading time: 0 Render date: 2024-07-11T10:28:51.957Z Has data issue: false hasContentIssue false
  • English
  • Français

Weighted composition operators with closed range

Published online by Cambridge University Press:  17 April 2009

N. Palmberg
N. Palmberg
Affiliation:
Department of Mathematics, Åbo Akademi University, Fänriksgatan 3 B, FIN-20500 Åbo, Finland, e-mail: niklas.palmberg@abo.fi
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the closed range property of weighted composition operators on weighted Bergman spaces of infinite order (including the Hardy space of infinite order). We give some necessary and sufficient conditions and find a complete characterisation for weighted composition operators associated with conformal mappings. We also give the corresponding results for composition operators on the Bloch-type spaces. Therefore, the results obtained in this paper also improve and generalise the results of Ghatage, Yan, Zheng and Zorboska.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 75 , Issue 3 , June 2007 , pp. 331 - 354
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Bierstedt, K.D., Bonet, J. and Taskinen, J., ‘Associated weights and spaces of holomorphic functions’, Studia Math. 127 (1998), 137–168.Google Scholar
[2]Bonet, J., ‘Weighted spaces of holomorphic functions and operators between them’, in Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), Colecc. Abierta 64 (Univ. Sevilla Secr. Publ., Seville, 2003), pp. 117–138.Google Scholar
[3]Bonet, J., Domański, P. and Lindström, M., ‘Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions’, Canad. Math. Bull. 42 (1999), 139–148.Google Scholar
[4]Bonet, J., Domański, P. and Lindström, M., ‘Pointwise multiplication operators on weighted Banach spaces of analytic functions’, Studia Math. 137 (1999), 177–194.Google Scholar
[5]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), 101–118.Google Scholar
[6]Bjørstad, P. and Grosse, E., ‘Conformal mapping of circular arc polygons’, SIAM J. Sci. Statist. Comput. 8 (1987), 19–32.CrossRefGoogle Scholar
[7]Cichoń, K. and Seip, K., ‘Weighted holomorphic spaces with trivial closed range multiplication operators’, Proc. Amer. Math. Soc. 131 (2002), 201–207.Google Scholar
[8]Contreras, M.D. and Hernandez-Díaz, A.G., ‘Weighted composition operators in weighted Banach spaces of analytic functions’, J. Austral. Math. Soc. Ser. A 69 (2000), 41–60.CrossRefGoogle Scholar
[9]Cowen, C.C. and MacCluer, B.D., Composition operators on spaces of analytic functions (CRC Press, Boca Raton, 1995).Google Scholar
[10]Domański, P. and Lindström, M., ‘Sets of interpolation and sampling for weighted Banach spaces of holomorphic functions’, Ann. Polon. Math. 79 (2002), 233–264.CrossRefGoogle Scholar
[11]Driscoll, T.A. and Trefethen, L.N., Schwarz-Christoffel mapping (Cambridge University Press, Cambridge, 2002).CrossRefGoogle Scholar
[12]Garnett, J.B., Bounded analytic functions (Academic Press, New York, 1981).Google Scholar
[13]Ghatage, P., Yan, J. and Zheng, D., ‘Composition operators with closed range on the Bloch space’, Proc. Amer. Math. Soc. 129 (2001), 2039–2044.Google Scholar
[14]Ghatage, P., Zheng, D. and Zorboska, N., ‘Sampling sets and closed range composition operators on the Bloch space’, Proc. Amer. Math. Soc. 133 (2005), 1371–1377.Google Scholar
[15]Howell, L.H., ‘Numerical conformal mapping of circular arc polygons’, J. Comput. Appl. Math. 46 (1993), 7–28.CrossRefGoogle Scholar
[16]Hedenmalm, H., Korenblum, B. and Zhu, K., Theory of Bergman spaces (Springer-Verlag, New York, 2000).CrossRefGoogle Scholar
[17]Lindström, M. and Palmberg, N., ‘Spectra of composition operators on BMOA’, Integral Equations Operator Theory 53 (2005), 75–86.Google Scholar
[18]Luecking, D., ‘Inequalities on Bergman spaces’, Illinois J. Math. 25 (1981), 1–11.Google Scholar
[19]Lusky, W., ‘On weighted spaces of harmonic and holomorphic functions’, J. London Math. Soc. 51 (1995), 309–320.CrossRefGoogle Scholar
[20]Ohno, S., Stroethoff, K. and Zhao, R., ‘Weighted composition operators between Bloch-type spaces’, Rocky Mountain J. Math. 33 (2003), 191–215.Google Scholar
[21]Ohno, S. and Takagi, H., ‘Some properties of weighted composition operators on algebras of analytic functions’, J. Nonlinear Convex Anal. 2 (2001), 369–380.Google Scholar
[22]Pommerenke, C., Boundary behaviour of conformal maps (Springer-Verlag, Berlin, 1992).Google Scholar
[23]Roan, R.C., ‘Composition operators on Hp with dense range’, Indiana Univ. Math. J. 27 (1978), 159–162.Google Scholar
[24]Shimorin, S., ‘Weighted composition operators associated with conformal mappings’, in Quadrature Domains and Their Applications (The Harold S. Shapiro Anniversary Volume), Oper. Theory Adv. Appl. 156 (Birkhäuser, Basel, 2005), pp. 217–237.CrossRefGoogle Scholar
[25]Smith, W., ‘Composition operators between Bergman spaces and Hardy spaces’, Trans. Amer. Math. Soc. 348 (1996), 2331–2348.CrossRefGoogle Scholar
[26]Shields, A.L. and Williams, D.L., ‘Bounded projections, duality and multipliers in spaces of analytic functions’, Trans. Amer. Math. Soc. 162 (1971), 287–302.Google Scholar
[27]Shields, A.L. and Williams, D.L., ‘Bounded projections and the growth of harmonic conjugates in the disc’, Michigan Math. J. 29 (1982), 3–25.CrossRefGoogle Scholar
[28]Zhu, K., Operator theory in function spaces (Marcel Dekker Inc., New York, 1990).Google Scholar