Abstract

Given a weight vv on an open subset UU of Cn{\bf C}^n, Hv(U){\cal H}_v(U) (resp. Hvo(U){\cal H}_{v_o}(U)) denotes the Banach space of holomorphic functions ff on UU such that vfvf is bounded on UU (resp. converges to 00 on the boundary of UU). We show that Hv(U){\cal H}_v(U) is canonically isometrically isomorphic to the bidual of Hvo(U){\cal H}_{v_o}(U) if and only if Hvo(U){\cal H}_{v_o}(U) is an M-ideal in Hv(U){\cal H}_v(U) and the associated weights v~o\tilde v_o and v~\tilde v coincide

Author information

Contact details are reproduced from the original publication and may be historical.

Christopher Boyd

School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland

Pilar Rueda

Dep. de Análisis Matemático, Facultad de Matemáticas, Universidad de Valencia, 46100 Burjasot, Valencia, Spain

References

  1. E. M. Alfsen: M -structure and intersection properties of balls in Banach spaces, Isr. J. Math. 13 (1973) 235–245.https://doi.org/10.1007/BF02762797Google Scholar
  2. E. M. Alfsen, E. G. Effros: Structure in real Banach spaces, Parts I and II, Ann. Math. 96 (1972) 98–128 and 129–173.https://doi.org/10.2307/1970896Google Scholar
  3. J. M. Anderson, J. Duncan: Duals of Banach spaces of entire functions, Glasg. Math. J. 32 (1990) 215–220.https://doi.org/10.1017/S0017089500009241Google Scholar
  4. T. Ando: Closed range theorems for convex sets and linear liftings, Pac. J. Math. 44 (1973) 393–410.https://doi.org/10.2140/pjm.1973.44.393Google Scholar
  5. E. Behrends: M-Structure and the Banach-Stone Theorem, Lecture Motes in Mathematics 736, Springer, Berlin (1979).https://doi.org/10.1007/BFb0063153Google Scholar
  6. K. D. Bierstedt, J. Bonet, A. Galbis: Weighted spaces of holomorphic functions on balanced domains, Mich. Math. J. 40 (1993) 271–297.https://doi.org/10.1307/mmj/1029004753Google Scholar
  7. K. D. Bierstedt, J. Bonet, J. Taskinen: Associated weights and spaces of holomorphic functions, Stud. Math. 127(2) (1998) 137–168.https://doi.org/10.4064/sm-127-2-137-168Google Scholar
  8. K. D. Bierstedt, W. H. Summers: Biduals of weighted Banach spaces of analytic functions, J. Aust. Math. Soc., Ser. A 54 (1993) 70–79.https://doi.org/10.1017/S1446788700036983Google Scholar
  9. J. Bonet, E. Wolf: A note on weighted Banach spaces of holomorphic functions, Arch. Math. 81 (2003) 650–654.https://doi.org/10.1007/s00013-003-0568-8Google Scholar
  10. C. Boyd, P. Rueda: The v-boundary of weighted spaces of holomorphic functions, Ann. Acad. Sci. Fenn., Math. 30 (2005) 337–352.https://doi.org/10.1215/ijm/1258138315Google Scholar
  11. C. Boyd, P. Rueda: Complete weights and v-peak points of spaces of weighted holomorphic functions, Isr. J. Math. 155 (2006) 57–80.https://doi.org/10.1007/BF02773948Google Scholar
  12. C. Boyd, P. Rueda: Isometries between spaces of weighted holomorphic functions, Stud. Math. 190(3) (2009) 203–231.https://doi.org/10.4064/sm190-3-1Google Scholar
  13. C. Boyd, P. Rueda: Isometries of weighted spaces of holomorphic functions on unbounded domains, Proc. R. Soc. Edinb., Sect. A, Math. 139 (2009) 253–271.https://doi.org/10.1017/S0308210507001230Google Scholar
  14. J. Clunie, T. Kövari: On integral functions having prescribed asymptotic growth. II, Can. J. Math. 20 (1968) 7–20.https://doi.org/10.4153/CJM-1968-002-1Google Scholar
  15. J. Diestel, J. J. Uhl: Vector Measures, Mathematical Surveys 15, AMS, Providence (1977).https://doi.org/10.1090/surv/015Google Scholar
  16. G. Godefroy: Espaces de Banach: existence et unicité de certains préduax, Ann. Inst. Fourier 28 (1978) 87–105.https://doi.org/10.5802/aif.702Google Scholar
  17. P. Harmand, A. Lima: Banach spaces which are M-deals in their biduals, Trans. Amer. Math. Soc. 283 (1984) 253–264.https://doi.org/10.1090/S0002-9947-1984-0735420-4Google Scholar
  18. P. Harmand, D. Werner, W. Werner: M-Ideals in Banach Spaces and Banach Algebras, Lecture Notes in Mathematics 1547, Springer, Berlin (1993).https://doi.org/10.1007/BFb0084355Google Scholar
  19. W. Lusky: On the isomorphism classes of spaces of harmonic and holomorphic functions, Stud. Math. 175(1) (2006) 19–45.https://doi.org/10.4064/sm175-1-2Google Scholar
  20. K.-F. Ng: On a theorem of Dixmier, Math. Scand. 29 (1971) 279–280.https://doi.org/10.7146/math.scand.a-11054Google Scholar
  21. T. Ransford: Potential Theory in the Complex Plane, London Mathematical Society Student Texts 28, Cambridge University Press, Cambridge (1995).https://doi.org/10.1017/CBO9780511623776Google Scholar
  22. L. A. Rubel, A. L. Shields: The second duals of certain spaces of analytic functions, J. Aust. Math. Soc. 11 (1970) 276–280.https://doi.org/10.1017/S1446788700006649Google Scholar
  23. D. Werner: New classes of Banach spaces which are M-ideals in their biduals, Math. Proc. Camb. Philos. Soc. 111 (1992) 337–354.https://doi.org/10.1017/S0305004100075447Google Scholar

Suggested citation

C. Boyd, P. Rueda. “The Biduality Problem and M-Ideals in Weighted Spaces of Holomorphic Functions.” Journal of Convex Analysis 18 (2011), No. 4, 1065–1074. https://doi.org/10.68381/jca18065

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2011. Rights now held by Banach Press.