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

Nawrocki, Marek

Studia Mathematica, Tome 103 (1992), p. 87-102 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $LN_∗(^n)$.

Détail

Publié le : 1992-01-01

Zbl 0814.46018

EUDML-ID : urn:eudml:doc:215916

Bibliographie

[00000] [1] L. Drewnowski, Un théorème sur les opérateurs de $ℓ_∞(Γ)$, C. R. Acad. Sci. Paris Sér. A 281 (1975), 967-969.

[00001] [2] L. Drewnowski, Topological vector groups and the Nevanlinna class, preprint. | Zbl 0835.46018

[00002] [3] E. Dubinsky, Basic sequences in stable finite type power series spaces, Studia Math. 68 (1980), 117-130. | Zbl 0476.46009

[00003] [4] P. L. Duren, Theory of $H^p$ Spaces, Academic Press, New York 1970. | Zbl 0215.20203

[00004] [5] N. J. Kalton, Subseries convergence in topological groups and vector measures, Israel J. Math. 10 (1971), 402-412. | Zbl 0226.22005

[00005] [6] N. J. Kalton, Basic sequences in F-spaces and their applications, Proc. Edinburgh Math. Soc. 19 (1974), 151-167. | Zbl 0296.46010

[00006] [7] N. J. Kalton, Quotients of F-spaces, Glasgow Math. J. 19 (1978), 103-108. | Zbl 0398.46002

[00007] [8] N. J. Kalton, The Orlicz-Pettis theorem, in: Contemp. Math. 2, Amer. Math. Soc., 1980, 91-100. | Zbl 0566.46008

[00008] [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.

[00009] [10] G. Lumer, Espaces de Hardy en plusieurs variables complexes, C. R. Acad. Sci. Paris 273 (1971), 151-154. | Zbl 0216.16101

[00010] [11] M. Nawrocki, On the Orlicz-Pettis property in nonlocally convex F-spaces, Proc. Amer. Math. Soc. 101 (1987), 492-496. | Zbl 0645.46002

[00011] [12] M. Nawrocki, Multipliers, linear functionals and the Fréchet envelope of the Smirnov class $N_∗(^n)$, Trans. Amer. Math. Soc. 322 (1990), 493-506. | Zbl 0713.46015

[00012] [13] M. Nawrocki, The Fréchet envelopes of vector-valued Smirnov classes, Studia Math. 94 (1989), 61-75. | Zbl 0702.46021

[00013] [14] M. Nawrocki, Linear functionals on the Smirnov class of the unit ball in $ℂ^n$, Ann. Acad. Sci. Fenn. 14 (1989), 369-379.

[00014] [15] M. Nawrocki, The Orlicz-Pettis theorem fails for Lumer’s Hardy spaces $(LH)^p()$, Proc. Amer. Math. Soc. 109 (1990), 957-963. | Zbl 0718.46009

[00015] [16] S. Rolewicz, Metric Linear Spaces, PWN, Warszawa, and Reidel, Dordrecht 1984.

[00016] [17] L. A. Rubel, Internal-external factorization in Lumer's Hardy spaces, Adv. in Math. 50 (1983), 1-26. | Zbl 0539.32004

[00017] [18] W. Rudin, Function Theory in Polydiscs, Benjamin, New York 1969. | Zbl 0177.34101

[00018] [19] W. Rudin, Lumer's Hardy spaces, Michigan Math. J. 24 (1977), 1-5.

[00019] [20] W. Rudin, Function Theory in the Unit Ball of $ℂ^n$, Grundlehren Math. Wiss. 241, Springer, 1980.

[00020] [21] J. H. Shapiro, Mackey topologies, reproducing kernels, and diagonal maps on Hardy and Bergman spaces, Duke Math. J. 43 (1976), 187-202. | Zbl 0354.46036

[00021] [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. | Zbl 0603.46001

[00022] [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. | Zbl 0624.46011

[00023] [24] J. H. Shapiro and A. L. Shields, Unusual topological properties of the Nevanlinna Class, Amer. J. Math. 97 (1975), 915-936. | Zbl 0323.30033

[00024] [25] N. Yanagihara, The containing Fréchet space for the class $N^+$, Duke Math. J. 40 (1973), 93-103.

[00025] [26] N. Yanagihara, Multipliers and linear functionals for the class $N^+$, Trans. Amer. Math. Soc. 180 (1973), 449-461. | Zbl 0243.46036