Linear topological properties of the Lumer-Smirnov class of the polydisc

Marek Nawrocki

Studia Mathematica (1992)

  • Volume: 102, Issue: 1, page 87-102
  • ISSN: 0039-3223

Abstract

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Linear topological properties of the Lumer-Smirnov class L N ( n ) of the unit polydisc n are studied. The topological dual and the Fréchet envelope are described. It is proved that L N ( n ) has a weak basis but it is nonseparable in its original topology. Moreover, it is shown that the Orlicz-Pettis theorem fails for L N ( n ) .

How to cite

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Nawrocki, Marek. "Linear topological properties of the Lumer-Smirnov class of the polydisc." Studia Mathematica 102.1 (1992): 87-102. <http://eudml.org/doc/215916>.

@article{Nawrocki1992,
abstract = {Linear topological properties of the Lumer-Smirnov class $LN_∗(^n)$ of the unit polydisc $^n$ are studied. The topological dual and the Fréchet envelope are described. It is proved that $LN_∗(^n)$ has a weak basis but it is nonseparable in its original topology. Moreover, it is shown that the Orlicz-Pettis theorem fails for $LN_∗(^n)$.},
author = {Nawrocki, Marek},
journal = {Studia Mathematica},
keywords = {linear topological properties of the Lumer-Smirnov class; topological dual; Fréchet envelope; weak basis; Orlicz-Pettis theorem},
language = {eng},
number = {1},
pages = {87-102},
title = {Linear topological properties of the Lumer-Smirnov class of the polydisc},
url = {http://eudml.org/doc/215916},
volume = {102},
year = {1992},
}

TY - JOUR
AU - Nawrocki, Marek
TI - Linear topological properties of the Lumer-Smirnov class of the polydisc
JO - Studia Mathematica
PY - 1992
VL - 102
IS - 1
SP - 87
EP - 102
AB - Linear topological properties of the Lumer-Smirnov class $LN_∗(^n)$ of the unit polydisc $^n$ are studied. The topological dual and the Fréchet envelope are described. It is proved that $LN_∗(^n)$ has a weak basis but it is nonseparable in its original topology. Moreover, it is shown that the Orlicz-Pettis theorem fails for $LN_∗(^n)$.
LA - eng
KW - linear topological properties of the Lumer-Smirnov class; topological dual; Fréchet envelope; weak basis; Orlicz-Pettis theorem
UR - http://eudml.org/doc/215916
ER -

References

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  1. [1] L. Drewnowski, Un théorème sur les opérateurs de ( Γ ) , C. R. Acad. Sci. Paris Sér. A 281 (1975), 967-969. 
  2. [2] L. Drewnowski, Topological vector groups and the Nevanlinna class, preprint. Zbl0835.46018
  3. [3] E. Dubinsky, Basic sequences in stable finite type power series spaces, Studia Math. 68 (1980), 117-130. Zbl0476.46009
  4. [4] P. L. Duren, Theory of H p Spaces, Academic Press, New York 1970. Zbl0215.20203
  5. [5] N. J. Kalton, Subseries convergence in topological groups and vector measures, Israel J. Math. 10 (1971), 402-412. Zbl0226.22005
  6. [6] N. J. Kalton, Basic sequences in F-spaces and their applications, Proc. Edinburgh Math. Soc. 19 (1974), 151-167. Zbl0296.46010
  7. [7] N. J. Kalton, Quotients of F-spaces, Glasgow Math. J. 19 (1978), 103-108. Zbl0398.46002
  8. [8] N. J. Kalton, The Orlicz-Pettis theorem, in: Contemp. Math. 2, Amer. Math. Soc., 1980, 91-100. Zbl0566.46008
  9. [9] N. J. Kalton, N. T. Peck and J. W. Roberts, An F-Space Sampler, London Math. Soc. Lecture Note Ser. 89, Cambridge Univ. Press, 1984. 
  10. [10] G. Lumer, Espaces de Hardy en plusieurs variables complexes, C. R. Acad. Sci. Paris 273 (1971), 151-154. Zbl0216.16101
  11. [11] M. Nawrocki, On the Orlicz-Pettis property in nonlocally convex F-spaces, Proc. Amer. Math. Soc. 101 (1987), 492-496. Zbl0645.46002
  12. [12] M. Nawrocki, Multipliers, linear functionals and the Fréchet envelope of the Smirnov class N ( n ) , Trans. Amer. Math. Soc. 322 (1990), 493-506. Zbl0713.46015
  13. [13] M. Nawrocki, The Fréchet envelopes of vector-valued Smirnov classes, Studia Math. 94 (1989), 61-75. Zbl0702.46021
  14. [14] M. Nawrocki, Linear functionals on the Smirnov class of the unit ball in n , Ann. Acad. Sci. Fenn. 14 (1989), 369-379. 
  15. [15] M. Nawrocki, The Orlicz-Pettis theorem fails for Lumer’s Hardy spaces ( L H ) p ( ) , Proc. Amer. Math. Soc. 109 (1990), 957-963. Zbl0718.46009
  16. [16] S. Rolewicz, Metric Linear Spaces, PWN, Warszawa, and Reidel, Dordrecht 1984. 
  17. [17] L. A. Rubel, Internal-external factorization in Lumer's Hardy spaces, Adv. in Math. 50 (1983), 1-26. Zbl0539.32004
  18. [18] W. Rudin, Function Theory in Polydiscs, Benjamin, New York 1969. Zbl0177.34101
  19. [19] W. Rudin, Lumer's Hardy spaces, Michigan Math. J. 24 (1977), 1-5. 
  20. [20] W. Rudin, Function Theory in the Unit Ball of n , Grundlehren Math. Wiss. 241, Springer, 1980. 
  21. [21] J. H. Shapiro, Mackey topologies, reproducing kernels, and diagonal maps on Hardy and Bergman spaces, Duke Math. J. 43 (1976), 187-202. Zbl0354.46036
  22. [22] J. H. Shapiro, Some F-spaces of harmonic functions for which the Orlicz-Pettis theorem fails, Proc. London Math. Soc. 50 (1985), 299-313. Zbl0603.46001
  23. [23] J. H. Shapiro, Linear topological properties of the harmonic Hardy spaces h p for 0 < p < 1, Illinois J. Math. 29 (1985), 311-339. Zbl0624.46011
  24. [24] J. H. Shapiro and A. L. Shields, Unusual topological properties of the Nevanlinna Class, Amer. J. Math. 97 (1975), 915-936. Zbl0323.30033
  25. [25] N. Yanagihara, The containing Fréchet space for the class N + , Duke Math. J. 40 (1973), 93-103. 
  26. [26] N. Yanagihara, Multipliers and linear functionals for the class N + , Trans. Amer. Math. Soc. 180 (1973), 449-461. Zbl0243.46036

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