Summability and duality

• Ghara, Soumitra ^[1] ; Mashreghi, Javad ^[1] ; Ransford, Thomas ^[1]
  1. [1] Université Laval (Quebec, Canadà). Département de mathématiques et de statistique
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 68, Nº. 2, 2024, págs. 407-429
• Idioma: inglés
• DOI: 10.5565/PUBLMAT6822403
• Enlaces
  • Texto completo
• Resumen

  • We formalize the observation that the same summability methods converge in a Banach space X and its dual X∗. At the same time we determine conditions under which these methods converge in weak and weak* topologies on X and X∗ respectively. We also derive a general limitation theorem, which yields a necessary condition for the convergence of a summability method in X. These results are then illustrated by applications to a wide variety of function spaces, including spaces of continuous functions, Lebesgue spaces, the disk algebra, Hardy and Bergman spaces, theBMOA space, the Bloch space, and de Branges–Rovnyak spaces. Our approach shows that all these applications flow from just two abstract theorems.

• Referencias bibliográficas
  • A. Aleman and B. Malman, Hilbert spaces of analytic functions with a contractive backward shift, J. Funct. Anal. 277(1) (2019), 157–199. DOI:...
  • J. M. Anderson, J. Clunie, and Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), 12–37. DOI: 10.1515/crll.1974.270.12
  • O. El-Fallah, E. Fricain, K. Kellay, J. Mashreghi, and T. Ransford, Constructive approximation in de Branges–Rovnyak spaces, Constr. Approx....
  • E. Fricain and J. Mashreghi, The Theory of H(b) Spaces. Vol. 1, New Math. Monogr. 20, Cambridge University Press, Cambridge, 2016.
  • E. Fricain and J. Mashreghi, The Theory of H(b) Spaces. Vol. 2, New Math. Monogr. 21 Cambridge University Press, Cambridge, 2016.
  • J. B. Garnett, Bounded Analytic Functions, Revised first edition, Grad. Texts in Math. 236, Springer, New York, 2007. DOI: 10.1007/0-387-49763-3
  • G. H. Hardy, On the summability of Fourier’s series, Proc. London Math. Soc. (2) 12(1) (1913), 365–372. DOI: 10.1112/plms/s2-12.1.365
  • G. H. Hardy, Divergent Series, With a preface by J. E. Littlewood and a note by L. S. Bosanquet, Reprint of the revised (1963) edition, Editions...
  • Mashreghi, P.-O. Parise, and T. Ransford ´ , Ces`aro summability of Taylor series in weighted Dirichlet spaces, Complex Anal. Oper. Theory...
  • J. Mashreghi, P.-O. Parise, and T. Ransford ´ , Failure of approximation of odd functions by odd polynomials, Constr. Approx. 56(1) (2022),...
  • J. Mashreghi, P.-O. Parise, and T. Ransford ´ , Power-series summability methods in de Branges–Rovnyak spaces, Integral Equations Operator...
  • J. Mashreghi and T. Ransford, Outer functions and divergence in de Branges–Rovnyak spaces, Complex Anal. Oper. Theory 12(4) (2018), 987–995....
  • J. Mashreghi and T. Ransford, Linear polynomial approximation schemes in Banach holomorphic function spaces, Anal. Math. Phys. 9(2) (2019),...
  • M. Riesz, Sur les s´eries de Dirichlet et les s´eries enti`eres, C. R. Acad. Sci. Paris 149 (1910), 909–912.
  • D. Sarason, Doubly shift-invariant spaces in H2 , J. Operator Theory 16(1) (1986), 75–97.
  • D. Sarason, Sub-Hardy Hilbert Spaces in the Unit Disk, Univ. Arkansas Lecture Notes Math. Sci. 10, Wiley-Intersci. Publ., John Wiley &...
  • D. Sarason, Local Dirichlet spaces as de Branges–Rovnyak spaces, Proc. Amer. Math. Soc. 125(7) (1997), 2133–2139. DOI: 10.1090/S0002-9939-97-03896-3
  • D. Vukotic´, The isoperimetric inequality and a theorem of Hardy and Littlewood, Amer. Math. Monthly 110(6) (2003), 532–536. DOI: 10.2307/3647909
  • K. Zhu, Duality of Bloch spaces and norm convergence of Taylor series, Michigan Math. J. 38(1) (1991), 89–101. DOI: 10.1307/mmj/1029004264
  • K. Zhu, Operator Theory in Function Spaces, Second edition, Math. Surveys Monogr. 138, American Mathematical Society, Providence, RI, 2007....