Abstract
Countable projective limits of countable inductive limits, so-called PLB-spaces, of weighted Banach spaces of continuous functions have recently been investigated by Agethen, Bierstedt and Bonet. In a previous article, the author extended their investigation to the case of holomorphic functions and characterized when spaces over the unit disc w.r.t. weights of polynomial decay are ultrabornological or barrelled. In this note, we prove a similar characterization for the case of weights which tend to zero logarithmically.
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References
Agethen S., Bierstedt K.D., Bonet J.: Projective limits of weighted (LB)-spaces of continuous functions. Arch. Math. (Basel) 92, 384–398 (2009)
Bierstedt K.D.: A survey of some results and open problems in weighted inductive limits and projective description for spaces of holomorphic functions. Bull. Soc. R. Sci. Liège 70, 167–182 (2001)
Bierstedt K.D., Bonet J.: Weighted (LF)-spaces of continuous functions. Math. Nachr. 165, 25–48 (1994)
Bierstedt K.D., Bonet J.: Weighted (LB)-spaces of holomorphic functions: \({\mathcal{V}H(G) = \mathcal{V}_0H(G)}\) and completeness of \({\mathcal{V}_0H(G)}\) . J. Math. Anal. Appl. 323, 747–767 (2006)
Bierstedt K.D., Bonet J., Galbis A.: Weighted spaces of holomorphic functions on balanced domains. Mich. Math. J. 40, 271–297 (1993)
Bierstedt K.D., Bonet J., Taskinen J.: Associated weights and spaces of holomorphic functions. Stud. Math. 127, 137–168 (1998)
Bierstedt K.D., Meise R., Summers W.H.: A projective description of weighted inductive limits. Trans. Amer. Math. Soc. 272, 107–160 (1982)
Bonet J., Engliš M., Taskinen J.: Weighted L ∞-estimates for Bergman projections. Studia Math. 171, 67–92 (2005)
Braun R., Vogt D.: A sufficient condition for Proj\({\,^1\mathcal{X} = 0}\) . Michigan Math. J. 44, 149–156 (1997)
P. Domański, Classical PLS-spaces: spaces of distributions, real analytic functions and their relatives, Orlicz centenary volume, Banach Center Publ., vol. 64, Polish Acad. Sci., Warsaw, 2004, pp. 51–70.
Frerick L., Wengenroth J.: A sufficient condition for vanishing of the derived projective limit functor. Arch. Math. (Basel) 67, 296–301 (1996)
Meise R., Vogt D.: Introduction to functional analysis, Oxford Graduate Texts in Mathematics, vol. 2. The Clarendon Press Oxford University Press, New York (1997)
V. Palamodov, The projective limit functor in the category of topological linear spaces, Mat. Sb. 75 (1968) 567–603 (in Russian), English transl., Math. USSR Sbornik 17 (1972), 189–315.
Taskinen J.: Compact composition operators on general weighted spaces. Houston J. Math. 27, 203–218 (2001)
Vogt D.: Frécheträume zwischen denen jede stetige lineare Abbildung beschränkt ist. J. Reine Angew. Math. 345, 182–200 (1983)
Vogt D.: Lectures on projective spectra of (DF)-spaces. Seminar lectures, Wuppertal (1987)
D. Vogt, Topics on projective spectra of (LB)-spaces, Adv. in the Theory of Fréchet spaces (Istanbul, 1988), NATO Adv. Sci. Inst. Ser. C 287 (1989), 11–27.
D. Vogt, Regularity properties of (LF)-spaces, Progress in Functional Analysis, Proc. Int. Meet. Occas. 60th Birthd. M. Valdivia, Peñíscola/Spain, North-Holland Math. Stud. 170 (1992), 57–84.
S.-A. Wegner, Notes on a proof of Bonet, Engliš and Taskinen, not intended for submission to a journal, arXiv:1002.3728v1, 2010.
Wegner S.-A.: Projective limits of weighted LB-spaces of holomorphic functions. J. Math. Anal. Appl. 383, 409–422 (2011)
Wengenroth J.: Derived Functors in Functional Analysis, Lecture Notes in Mathematics 1810. Springer, Berlin (2003)
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Wegner, SA. PLB-spaces of holomorphic functions with logarithmic growth conditions. Arch. Math. 98, 163–172 (2012). https://doi.org/10.1007/s00013-011-0339-x
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DOI: https://doi.org/10.1007/s00013-011-0339-x